Wikipedia talk:Requests for mediation/Monty Hall problem/Archive 5

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Next steps (September)

A word of explanation: I am traveling right now and haven't had reliable access to the Internet. Will has also been otherwise occupied. We hope to be able to move forward with the mediation next week. However, we are open to suggestions as to the best way to go. We find that, given the intricacy of the disagreement and the volubility with which participants express themselves, it is hard to make gains. One suggestion is that we deploy the wiki equivalent of a talking stick. I think this might work. To use it we could have moderated questions and then give each person one response. The mediators could ask follow-up questions and then return to the list of agreed upon questions. Would participants like to try this? If so, would each of you be able to respond? We will pick it up on Monday. Sunray (talk) 13:53, 10 September 2010 (UTC)

1. Whatever. Glkanter (talk) 14:26, 10 September 2010 (UTC)

Seems reasonable to me. This is more or less wp:arbcom style, which seems to work there. I think it might be a good idea to continue the discussion from above as the first topic to be handled this way. -- Rick Block (talk) 15:24, 10 September 2010 (UTC)

+1 glopk (talk) 18:48, 10 September 2010 (UTC)

Fine. Martin Hogbin (talk) 19:09, 10 September 2010 (UTC)

Nice idea. BTW I also have a suggestion for a topic/issue. It's the question whether short (unconditional, intuitive) solutions to the MHP may be considered equally correct as long (conditional) solutions. Of course all the editors here have their own personal opinions on this matter, we are divided in two camps, and a majority believe that the only really good solutions are the more complex conditional solutions, which can be mainly found in the statistics literature. A minority believe that the short unconditional solutions close the matter. Another tiny minority believe that both classes of solutions are equally legitimate and that there yet other legitimate solutions (e.g., coming from a decision theoretic perspective). Somehow we have to disentangle our personal convictions from what can be concluded from studying the reliable sources. This leads to a new level of conflict and in fact a vicious circle since if for instance you adhere to the full conditional solutions you will tend to find the statistics literature which endorses them more reliable and authoritative than the less specialist literature. You will also see different parts of the history of MHP as more significant (e.g. the pre-Parade statistics literature). Gill110951 (talk) 09:53, 11 September 2010 (UTC)

Agree. – And +1, that's what it is about: "What is the MHP" or "What is the MHP in teaching conditional probability". Gerhardvalentin (talk) 18:10, 11 September 2010 (UTC)

Sounds like we have consensus on the talking stick approach. Thanks for suggestions for questions. It is a good sign that you are agreeing on this. Tomorrow, I will get more time at a computer and will try to get this rolling. In the meantime, I hope Will and I will have a chance to confer. Sunray (talk) 15:50, 13 September 2010 (UTC)

I think that seems like a great approach. And I just want to say that although I don't take part in a lot of the mathematics discussion I still consider myself a participant and am trying, to the best of my limited ability, to follow along. Cheers, Colincbn (talk) 01:17, 14 September 2010 (UTC)

Could we get an update from the mediators? I'm all for doing this slowly and with deliberation, but the current pace seems absolutely glacial. The parties involved are simply using other avenues, see for example talk:Monty Hall problem, to continue the argument. -- Rick Block (talk) 13:35, 16 September 2010 (UTC)

The commments there originally just concerned Boris his comment of the discussion length and the possibility to get him involved as an independent expert. It was not meant to start a new discussion. However it seems to me that Gill110951 and I are largely agreeing there. If others here essentially agree to the description there (many of the involved actually have done in past stements anyway) maybe that can be taken as a base for a compromise and avoid then we get bogged down and sidetracked by ultimately marginally things (like the order of various section). In the big picture it matters only that the various sections or more precisely described aspects are transparant and correct, whereas the exact location of a particular aspect in the article doesn't really matter all that much.--Kmhkmh (talk) 14:29, 16 September 2010 (UTC)

Sorry Rick, I need my daily fix of MHP argument every day ... so if nothing much has happened on this page I look for action somewhere else. But I agree with Kmhkmh that things seem to be getting a whole lot clearer. Gill110951 (talk) 21:08, 16 September 2010 (UTC)

Yep, things are a lot easier when two reasonable people talk to each other. When it's 4 or 6, it may be a little harder. We are in mediation because more and/or less reasonable people want a piece of the action, and there has been no way around it on the article talk page. glopk (talk) 21:56, 16 September 2010 (UTC)

I am not so sure things are getting clearer. Firstly, the distinction between a conditional and an unconditional problem is arbitrary. Despite many failed attempts, no one has yet defined exactly what a condition is in probability. Martin Hogbin (talk) 22:45, 16 September 2010 (UTC)

I am not aware of MHP-relevant and reputable sources that (a) mention or use a conditional probability AND (b) use a definition of conditional probability other than the usual one of Probability Theory (or an equivalent one). Are there any other "attempts" that are relevant herein? glopk (talk) 23:03, 16 September 2010 (UTC)

Just to be clear (mostly for others who may be reading this), Martin is referring to previous discussions (like this one), not anything that appears in sources. -- Rick Block (talk) 02:29, 17 September 2010 (UTC)

And to be even clearer, I am not interested in discussing on this mediation page any Probability Theory, except where strictly needed to clarify a source. Further, I believe that any and all discussions of philosophical interpretations of probability are out of scope. Please, let's focus on the issues at hand. glopk (talk) 04:13, 17 September 2010 (UTC)

I disagree with you there, Glopk; but of course what we talk about right here will depend on the instructions of the mediators! Many sources on MHP do discuss what they mean by probability. Jason Rosenhouse's splendid book "The Monty Hall Problem" (Oxford University Press, 2009) has a whole chapter on this. What one means by conditional probability depends on what one means by probability, and whether or not one "must" use conditional probability depends on this too. It is not a philosophical question at all. If you want to use probability in the real world, e.g. when choosing a strategy on a game show, you had better have some idea what you are talking about. Thanks Rick for the link and thanks Martin for raising the question again. I think that resolving that issue is germane to the disagreements or misunderstandings which have led to this mediation. If you want to edit wikipedia pages on MHP you will need to do research on what the sources mean by probability and conditional probability. To do justice to any source you'll have to make some attempt to see their writings in their own context. Usually there will be silent assumptions about what probability is all about, and hence what conditional probability is all about. BTW my day job right now is about the use of statistics in the forensic and the legal context. A context where there is a dispute about information, about its meaning, and about its relevance. All the time, these questions are the questions which need to be brought out of the dark and put on the table. Gill110951 (talk) 05:04, 17 September 2010 (UTC)

I see nothing wrong with having a (separate) section that deals with question, how different interpretations of probability may influence the MHP or be meaningful for it. Rosenhouse is doing that indeed, so we would have him as a source at least. However there is a danger, that we lose focus here and get lost in yet another neverending discussion thread. We should treat this subject as Rosenhouse does in separate section and not somewhat needlessly mix it with the other issues, where most of us were already reaching common ground (at least my impression) and now are threatened to lose it again or at least to get sidetracked. For the mediation to be successful, we need a goal oriented focused discussion.--Kmhkmh (talk) 08:42, 17 September 2010 (UTC)

@Kmhkmh: I did not say we ought to write a huge amount on the MHP page on this particular issue. I'm saying that awareness of the issue might help us editors to understand our mututal differences better, and understand our sources better. In the meantime we'll see if and when it raises its head in the mediation. Gill110951 (talk) 13:42, 20 September 2010 (UTC)

I agree that it is important to stay focused. If it would help, we could try to organize this page better. Sunray (talk) 13:33, 17 September 2010 (UTC)

Re: comments about the glacial pace: I apologize for my part in this. I'm not finding travel conducive to mediation. Sometimes it is very difficult to predict when I'm going to be able to get online. With that caveat, I am willing to continue. However, if my absences become intolerable, we may have to suspend the mediation.

I am encouraged to read comments from several participants that progress is being made. Would someone be willing to summarize any consensus that seems to be emerging? Sunray (talk) 13:33, 17 September 2010 (UTC)

Re pace - would it be possible for you and Will to jointly run this? It seems between the two of you it should be possible for at least one of you to check in on how we're doing here at least once a day. Questions addressed to mediators going without any response for a week at time doesn't seem very conducive to any sort of progress. If the volume here is simply overwhelming, maybe we should set up a separate page for questions to mediators or take the detailed discussion per topic to a separate page leaving this page for "meta" issues like questions to mediators or send questions to you via email or do something to strictly control the volume. There are several questions in the discussions above that have (still) not been addressed.

I'm not sure what common ground Kmhkmh is referring to (just above), but I think the discussion about the suggested change to the sentence referenced to Carlton has pretty much run its course. I think it would be very helpful if you or Will could read through this discussion at some point and summarize whether you think we've agreed to anything and, if so, what. This is the topic I was suggesting we start with in our new, more controlled, mode of discussion. I definitely agree with Kmhkmh that we need a goal oriented, focused discussion. I think we more or less did this with this one focused topic - to let it now just sit as yet another unresolved issue and move on to other questions is very discouraging. -- Rick Block (talk) 14:57, 17 September 2010 (UTC)

The common ground was not about carlton issue, but the original conflict and more specifically the discussion between Gill110951 and me on the discussion page of the MHP article.--Kmhkmh (talk) 15:32, 17 September 2010 (UTC)

For reference, the unaddressed questions I'm referring to are

• [1]
• [2]
• [3]
• [4]
• [5]
• [6]

-- Rick Block (talk) 05:21, 18 September 2010 (UTC)

Questions

Please add suggested questions for discussion below. Responses to questions should be brief and limited to one per participant, until all participants have had a chance to respond.

1. May short (unconditional, intuitive) solutions to the MHP be considered equally correct as long (conditional) solutions?
2. What is the MHP in teaching conditional probability?
3. Do we have consensus on the suggested change to the "simple solution" referenced to Carlton (per the discussion above)?
4. Do we have consensus that the unconditional answer to the question can be presented as long as sources indicate the issues surrounding it?
5. Is there agreement that the article should reflect the treatment in textbooks as an introductory conditional probability problem (without the article itself taking a textbook approach)?
6. What criticism is there of the conditional approach?
7. Is the MHP a real world problem?
8. What more "givens" are needed for the (conditional) solution?
9. Would participants be willing to begin working collaboratively on article sections? (I assume we would move segments of the article here for discussion).
10. What is the preferred order of sections in the article?

Question 1

Yes. Martin Hogbin (talk) 15:27, 17 September 2010 (UTC)

No, insofar most reputable sources for the simple (unconditional) solutions do not address which question are they answering (average chance of winning over a population Vs. chance of winning as judged by a particular player), and/or what assumptions they make on the host behavior, AND there are reputable sources pointing out these shortfalls of the simple solutions. However, they are useful as intuitive explanations and aids to understand the correct solutions. glopk (talk) 16:03, 17 September 2010 (UTC)

As per the reliable sources, including Selvin, who originated the puzzle, Yes. Glkanter (talk) 16:54, 17 September 2010 (UTC)

To be NPOV about it, the article must not take a stance on this question but rather must present the short (intuitive) solutions, long (conditional) solutions, and the published criticisms of the short solutions. In a sense, this means the answer to this question (as stated) is no - but only because there are sources that present short solutions and sources that criticize these solutions, but no corresponding sources that criticize long solutions. -- Rick Block (talk) 18:47, 17 September 2010 (UTC)

In what manner is this statement meaningful to this discussion and this article?:

"...but no corresponding sources that criticize long solutions." - Rick Block

In nearly two years of this conversation, I'm not aware of any editor suggesting the existence of any such reliable sources. Glkanter (talk) 07:03, 18 September 2010 (UTC)

That there are no such sources is the exact point. We have sources presenting solution A, sources criticizing solution A, sources presenting solution B, and no sources criticizing solution B. A and B are each presented by reliable sources so in one sense they are equally correct, but since A is criticized (by reliable sources) and B is not there is another sense in which they are not equally correct. -- Rick Block (talk) 22:52, 18 September 2010 (UTC)

As Wikipedia editors, we are interested in what reliable sources actually say, not adjudicating 'right' from 'wrong'. Therefore, the conclusion above: "...there is another sense in which they are not equally correct." is without merit or purpose to this discussion and article. Glkanter (talk) 01:14, 19 September 2010 (UTC)

Yes, based on sources that do not consider the question to be a very tricky question (accepting that the player already knew about the location of the car when selecting her door, and/or that the host just may have offered to switch because the guest's first choice per chance was the car, and/or accepting any "famous host's whistle-blowing tattletale bias" in opening his unfavored door, and by that showing the car actually was behind his preferred door, i.e. giving a closer hint on the actual "hidden" (but no more secret!) location of the car etc. etc. etc.) – Falk e.g. shows the simple solution to be correct "if no host's bias exists and is known to exist". And based on sources that say that, even any extreme host's bias to be given, staying never can be better. And on sources that say that the standard version does not ask for a probability, but for a decision. Gerhardvalentin (talk) 19:39, 17 September 2010 (UTC)

Yes. Comments: indeed, we have sources A who propose solution A, and sources B who propose solution B and criticise A. This all needs to be reported on wikipedia in an intelligent and unbiased and informative way. The fact that B criticises A whereas A does not criticise B does not, in my mind, mean automatically that B is therefore better. One can argue that B has ulterior motives and sees the problem in a different context from A. I would also argue that B typically understands the meaning of probability differently from A. I would also humbly point out that B not only gives a more complicated solution but also needs to make more, and more complicated, assumptions. Those people who prefer A can rightly prefer A because one gets a clear and convincing answer while not having to make so many assumptions. The reason most A's don't criticise B is that they find it an uninteresting technical complication. Proponent's of B tend to be dogmatic "you *have* to compute a conditional probability". Since it is not clear that the extra assumptions are well warranted, this gives A an edge: their solution is more widely applicable than B's. I think this is one of the reason whe A's rarely bother to criticise B. BTW I belong to neither party. I think that both A and B present valid solutions to different valid mathematizations of a verbal problem. I like to mention that there are both amateurs and professionals who say "the player should choose their door at random" so that switching gives a 2/3 probability of winning; nothing more needs to be assumed! Gill110951 (talk) 11:41, 19 September 2010 (UTC)

No, and the criticism of some sources should clearly be mentioned. Nijdam (talk) 14:30, 19 September 2010 (UTC)

Mediator's comment

There seem to be some common threads in the responses. Whether or not participants agreed with this question, there seems to be consensus to let the sources be the guide. It seems that one doesn't have to answer the question and that an NPOV article wouldn't even try.

Further question

Do we have consensus that the unconditional answer to the question can be presented as long as sources indicate the issues surrounding it? Sunray (talk) 12:44, 18 September 2010 (UTC)

The majority of sources do not indicate any issues surrounding the simple solutions, but I agree that the article should indicate those issues. The question is, 'Where in the article should they be indicated?'. Martin Hogbin (talk) 16:57, 18 September 2010 (UTC)

As long as this is done in an appropriate balanced way. The first question many people want to answer going to the MHP page is, "is it smart to switch? If so, why ?" They shouldn't get the impression that there is an easy argument for dumbo's for switching, which smart people actually know is flawed; and that smart people have to learn algebra or go through tiring computations, only to get "the same answer"! I think that the solution is for us editors to realise that there are different mathematizations of the MHP, they each have their pro's and cons. The simple solution is the correct and complete solution to a legitimate mathematization of the problem (according to many reliable sources). The mere fact that we spend years here not being able to convince our co-editors that our way of seeing things is the only good way, and the mere fact of the diversity of opinions in reliable sources, makes this absolutely clear (to me, at least). Of course it's our duty to report that a lot of sources are very decided on "what is the MHP". If those sources give interesting reasons for their opinion, or if they are working in a context which makes it understandable, we'll report that too. We obviously can't take sides. Gill110951 (talk) 13:03, 20 September 2010 (UTC)

Richard, it can't be all that obvious that we can't take sides. Look at the MHP article today, and two years ago. Read some of the arguments on this page regarding which solutions are 'more right'. Somehow, that message is not getting through to all the editors. Glkanter (talk) 13:34, 20 September 2010 (UTC)

Question 2

The MHP is simple probability puzzle of which a slight variant (the host does not chose uniformly between goat-hiding doors) may be used to illustrate the pitfalls of making unwarranted assumptions in conditional probability. Martin Hogbin (talk) 15:27, 17 September 2010 (UTC)

The question as posed is irrelevant to the MHP article and this mediation page, as the MHP article's purpose is not to teach about conditional probability. Martin Hogbin's answer above, however, is a rephrasing of one argument used by reputable sources (e.g. Falk) to show why some of the simple solutions are incomplete, namely that the reasoning they use fails when applied to a slight variant of the standard MHP interpretation, a variant which is however compatible with the problem statement. glopk (talk) 16:03, 17 September 2010 (UTC)

Insofar as the MHP is a standard example in many introductory probability textbooks, the article should reflect the treatment the problem is given in these textbooks (as it should also reflect the treatment the problem is given in other sources as well). In these textbooks the problem is nearly universally interpreted as an elementary problem in conditional probability. -- Rick Block (talk) 18:47, 17 September 2010 (UTC)

Besides showing the existence of very tricky variants, in a separate later section, and showing there the remarkable iridescent history of the MHP, you can show there that the unconditional solution is not to be stated as a complete solution to the various "conditional problems" of such tricky variants, and because of that offer there one example of conditional probability, too. And it should be mentioned there that, because the "MHP example" is widely used as an easy and very suitable example exercise in teaching and training conditional probability maths, it's naturally one relevant object in many introductory probability textbooks therefore. But finally stop to confuse and finally put a stop to sell the full MHP article as a textbook in introductory conditional probability theory. Because it isn't. Gerhardvalentin (talk) 19:39, 17 September 2010 (UTC)

The MHP was born (Selvin) as a probability puzzle with a natural place in teaching conditioning, in an introductory statistics or probability course, before it came into the public domain with vos Savant. The wikipedia article should give one of those solutions. Fortunately the mathematical situation has got a lot simpler in the last few years especially with the publication of Morgan et al's correction to their earlier paper. We know (from reliable sources, and from simple calculations which can be repeated on the wikipedia pages) that *provided* all three doors are initially equally likely to hide the car, the posterior odds on the "other door" hiding the car are 1:q, where q is the chance the quizmaster would have opened the door which he did open, given the car being behind the specific door which was chosen by the player. Thus it is never unfavourable to switch doors, we don't have to worry our heads about what q is, or try to come up with reasons why it might be 1/2. Depending on how you interpret probability, q might be "obviously" 1/2 (the subjectivist interpretation); but with other interpretations (the frequentist), it is simply unknown. If q=1/2 then the posterior odds are 2:1. The q=1/2 situation corresponds to the subjectivist's position of totally symmetry with respect to the door numbers. Since their prior information makes the door numbers exchangeable, the specific door numbers in any particular instance are irrelevant. The subjectivist computes the unconditional probability because the door numbers are a priori irrelevant, just as much as the day of the week. Note that the solution where you say: "you choose your door initially completely at random and thereafter switch" does not require that you assume all doors are equally likely to hide the car. You *make* your door hide the car with probability 1/3 by your own actions, and you don't assume anything else whatsoever. You do not assume that given your choice, all three doors are equally likely to hide the car. People who assume all doors are initially equally likely to hide the car are usually (implicitly or explicitly) using probability in its subjective sense, and the chance is 1/3 for each door since their information does not give any door numbers a special status. So their beliefs about any door hiding the car are the same, hence their subjective probabilities are the same. Their answer "2/3" is a measure of their rational subjective belief. It means no more, and no less. Frequentists who want to use equal probabilities 1/3, 1/3, 1/3 have to be told in advance "the car has been hidden completely at random". This is indeed information which is added in the elementary statistics text-books, because it has to be added in order for the approach which they want the student to take to work. Personally I don't think this is an assumption which is likely to be true about a real game show. Iit is added by many of the text book accounts since they want the student to solve the problem using certain tools and hence the student has to be given the information which is necessary to solve the problem with these tools. Other solutions using different tools need different information. Gill110951 (talk) 13:08, 19 September 2010 (UTC)

Mediator's comment

I think that there may be agreement that the article, as a whole, should not be constructed as a "textbook in introductory conditional probability theory." The article should, however, reflect the treatment in textbooks. Let's test that.

Further question

Is there agreement that the article should reflect the treatment in textbooks as an introductory conditional probability problem (without the article itself taking a textbook approach)? Sunray (talk) 13:00, 18 September 2010 (UTC)

• Yes, textbook coverage should be reflected as well. However I don't quite see how that really matters that much as far as the content of the article is concerned. Unconditional and conditional treatment can both be found in journal articles and books on MHP and they both can be found in probability textbooks. I.e. there isn't really such a thing as a unique textbook approach, that would yield a different coverage of the subject. It might be worthwhile though to simply mention that the topic (MHP) itself has become a popular (real life) example in probability textbooks and introductionsry probability courses/lectures at universities or in highschool since the mid 90s.--Kmhkmh (talk) 13:13, 18 September 2010 (UTC)

• Yes. The use of a slight variant of the problem as a text book introduction to conditional probability problem should be covered in the article. Martin Hogbin (talk) 17:01, 18 September 2010 (UTC)

• Yes Gill110951 (talk) 13:09, 19 September 2010 (UTC)

• Yes, exactly, but: reflecting the treatment of various versions in textbooks (in adopting additional assumptions for that purpose), as apparently very suitable exercises in teaching conditional probability, should also emphasize that all of that obviously never does address the original "standard question", that was and still is clearly asking for a decision and not for any conditional probability. Gerhardvalentin (talk) 14:38, 19 September 2010 (UTC)

Question 3

We have (as of my last count) a majority-based consensus that the decision tree picture suggested for inclusion above (As per Carlton 2005) should not be included in the article. We do not have a clear consensus on whether the direct quote from Section 5 of Carlton's paper should be included as is without the surrounding context, or whether it should be paraphrased as proposed by Rick Block (and/or as modified by myself). glopk (talk) 16:03, 17 September 2010 (UTC)

The question of whether the decision tree is OR has not been resolved. Glkanter (talk) 16:54, 17 September 2010 (UTC)

As the solution will be criticized in the article, my interpretation of NPOV leads me to conclude the 'proper' or 'fair' approach is to let the reader decide for himself whether the criticism has merit or not (provided this is not too unwieldy, and in this instance, Carlton's simple solution is very brief). Accordingly, the reader should be exposed to the actual solutions provided by the reliable sources, not an editor's paraphrasing. In the same spirit, the decision tree lets the reader judge the mathematical merits of the criticisms of Carlton's simple solution for himself. Are there any reliable sources published subsequent to Carlton that directly criticize his simple solution? Glkanter (talk) 22:41, 17 September 2010 (UTC)

I believe we have a consensus (not unanimity) that the decision tree picture should not be included (making the question of whether it is OR effectively moot, although we haven't really talked about this question directly), and a consensus that the direct quote should not be included (with or without the surrounding context). There is not yet a consensus (either way) about the alternate suggestion to revise the paraphrase to more closely match what Carlton actually says. -- Rick Block (talk) 18:47, 17 September 2010 (UTC)

As to Change to the "simple solution" referenced to Carlton: Not yet, see also question 1.
And in my opinion it was better to show (later) the clear full complete tree ( [[Image:Monty tree door1.svg|right|thumb|350px]] ), pointing out that it pictures only 1/3 (4 of 12) possible outcomes: Only the one case where the player initially has selected door 1. Gerhardvalentin (talk) 23:13, 17 September 2010 (UTC)

To me the whole carlton image issue seems like completely unnecessary distraction. Issues regarding formal OS and exactly citing Carlton aside, I see absolutely no need to include this particular image. It is not essential to the simple solution, which was fine without it so far, nor does it imho provide new insight or more clarity. All it had done so far was triggering another controversy between editors.--Kmhkmh (talk) 13:20, 18 September 2010 (UTC)

Using a decision tree to show that the short unconditional solutions to MHP are at least as mathematically formalizable as the conditional solutions appears to me to have played a useful role in clearning the air and clearing minds. The question now however, is not whether the solution as such is incorrect or unmathematical; the important question (among many reliable sources as well as among many editors) is whether the formal mathematical problems which the unconditional and conditional solutions address, are equally valid mathematizations of vos Savant/Whitaker's verbal question. Well, we have already established that that is a matter of opinion. So it seems to me that using a decision tree in a formal way also in the unconditional approach is pretty superfluous, provided we take care that the reader doesn't get the impression that just because the conditional solution uses complicated decision trees or long mathematical formauls means that the solution is somehow more scientific or correct. No. It is simply the case that if you pose a more subtle question you'll have to do more work (and add more prior information) to get an answer. Gill110951 (talk) 13:18, 19 September 2010 (UTC)

Mediator's comment

There is no consensus to include the Carleton decision tree image. Sunray (talk) 13:03, 18 September 2010 (UTC)

I don't believe this item has been discussed in the proper context yet. Some editors argue it adds no value. Other editors argue it is OR. As the OR question is unresolved, I have not been attempting to address the merits of its inclusion in the article. Glkanter (talk) 13:43, 18 September 2010 (UTC)

I believe there is a clear consensus against inserting Glkanter's (proper attribution) image, as unnecessary/redundant. The author has already stated his case for inclusion above, and it has obviuosly proved unconvinging. As others have noted, this makes the issue of whether it is OR effectively moot. glopk (talk) 03:39, 19 September 2010 (UTC)

I think the question of whether or not this tree is OR is a red herring. If we were all agreed that it would be useful for readers of the article then no-one would object to including it. It's a cute graphical representation of the important elementary fact that conditioning on a probability one event does not change probabilities. Is that needed in the article? I think that if the article as a whole is written in an unbiased and balanced way, then we'll see easily enough that the picture is superfluous. Gill110951 (talk) 13:24, 19 September 2010 (UTC)

Question 4

• Yes, but I must say both questions (4,5) are imho almost beyond discussion. Both approaches are common and well sourced treatments of the problem. So as long as their description in the article is accurate I see absolutely no grounds to exclude them.--Kmhkmh (talk) 13:24, 18 September 2010 (UTC)

My concern about the criticisms of the unconditionals solutions has always been been 'where' and 'how frequently' will they appear in the article. I believe the current article violates NPOV on this (without question, UNDUE is being disregarded in the article). The criticisms come from reliable sources, so they are included. Which brings up a related topic: Morgan, et al's paper and letter. Does one negate the other? Are they still considered a 'criticizing' source? Glkanter (talk) 13:43, 18 September 2010 (UTC)

• "can be presented as long as sources indicate the issues surrounding it" is curious wording. Perhaps what is meant is "can be presented as long as sources the article indicates the issues surrounding it that are raised in reliable sources". I suspect we're going to be talking about exactly how and exactly where in great detail, but it seems nearly axiomatic that the unconditional approach should be presented. -- Rick Block (talk) 16:57, 18 September 2010 (UTC)
• The majority of sources do not indicate any issues surrounding the simple solutions, but I agree that the article should indicate those issues. The question is, 'Where in the article should they be indicated?'. Martin Hogbin (talk) 17:06, 18 September 2010 (UTC)
• Pardon, I know my English not being that great, for this reason my question to the moderator:
  You asked "Do we have consensus that the unconditional answer to the question can be presented as long as sources indicate the issues surrounding it."
  Can I read your words "as long as sources indicate the issues surrounding it" to be addressing e.g. Ruma Falk's detection, that the unconditional solution can only be stated to be an incomplete solution "if the host is biased (and you know about this bias)" – in: Ruma Falk "A closer look at the probabilities of the notorious three prisoners", 1991, - addressing Marilyn vos Savant in Parade Dec.2, 1990, p. 25 - concerning her "shell-example" for the MHP? Is Falks indication "if the host is biased (and you know about this bias" what you mean with "indicating the issues surrounding the source"?
  Or do you mean that any cited source referring to the unconditional solution imperatively must also reflect other issues that say that the unconditional solution cannot be stated to be a complete solution, e.g.?
  Or do you mean that citing a source saying that the standard version does not ask for a probability, but for a decision, in effect do show conditional solutions as well? Thank you. Gerhardvalentin (talk) 18:23, 18 September 2010 (UTC)
• Yes. But please note that also the conditional approach is open to criticism. In a textbook, the author can always write "the student may assume that...". But in the real world, you can't assume anything. Those who push the conditional approach are obliged to add more "givens" into the problem, in order to get a solution. Many of those assumptions are kind of standard in elementary probability textbooks. That doesn't mean to say that they will be typically justified in the real world. Quite the contrary, in fact. Making the same assumptions in the real world can be dangerous if not disastrous. "Standard assumptions" of independence, equal probabilities, normal distributions, and so on, are the cause of many miscarriages of justice occuring when laypersons (medical people, lawyers, judges and jurors) pretend they are experts in applying probability to the real world! Gill110951 (talk) 13:41, 19 September 2010 (UTC)

1. What criticism is there on the conditional approach?

2. Is the MHP a real world problem?

3. What more "givens" are needed for the (conditional) solution?

Nijdam (talk) 09:36, 20 September 2010 (UTC)

Good questions! I'll write some answers on my talk page at [7] . Gill110951 (talk) 09:38, 20 September 2010 (UTC)

Mediator's comment

Sorry about the byzantine wording of the question. It was a follow-up question to the discussion of Question 1 and a simple attempt to check consensus (which is confirmed). The additional point that has been raised is about weighting of sources. A concern has been raised about undue weight being given to certain sources. Sunray (talk) 20:56, 19 September 2010 (UTC)

Question 5

• Yes and see my comment above--Kmhkmh (talk) 13:24, 18 September 2010 (UTC)
• Yes. Also nearly axiomatic. -- Rick Block (talk) 16:57, 18 September 2010 (UTC)
• Yes. The article should say that the MHP, because of being an appropriate "example exercise" in teaching and training conditional probability mathematics, of course is widely treated in textbooks as an introductory conditional probability question. But more important for the lemma is to accent the fact that, even though it is an adequate example for training purpose in conditional probability, the standard version does not even ask for a probability at all, but the standard version is asking for a decision. And that it never will be of disadvantage to switch, even if the host should be extremely biased. And for this reason the clear answer can be given without consulting conditional probability theory that only presents exactly the same answer. Gerhardvalentin (talk) 23:54, 18 September 2010 (UTC)
• Yes, the article should, at the appropriate place, show how a slight variant of the problem may be used as an instructive conditional probability problem, as is done in several text books. Martin Hogbin (talk) 10:50, 19 September 2010 (UTC)
• Yes, and (but) with the focus placed by Gerhardvalentin. vos Savant/Whitaker's words ask for a decision or action, not for a probability. If you had taken fate into your own hands and chosen your initial door completely at random (many sources, both amateur and professional, take it as "obvious" that you did do that) and then switch, you'll win the car 2/3 of the time. If you didn't do that, then you'll have to trust to your intuition or to supplementary information that all doors are initially equally likely to hide the car. Then it doesn't matter which door you initially chose nor how you chose it, and it's never to your disadvantage to switch, and often very much to your advantage. On average you can hope to go home with the car 2/3 of the time, which for the randomizer is guaranteed. The difference between them is whether "2/3" is based on intuition, or on supplementary "givens", or engineered and guaranteed by your own actions. Gill110951 (talk) 13:33, 19 September 2010 (UTC)

Mediator's comment

Clear consensus here. I'm impressed with the ability of participants to find areas they can agree on. This bodes well for the outcome of this mediation. I've added three new questions, above (suggested by Nijdam), and one by me. BTW I am currently traveling in Newfoundland and seem to be getting more reliable access here on the rock (go figure). Sunray (talk) 15:14, 20 September 2010 (UTC)

Question 6

What criticism is there of the conditional approach?

• Keeping the focus on sources, I believe this question should be answered with references to sources that criticize the conditional approach. I am not aware of any such sources. -- Rick Block (talk) 15:24, 20 September 2010 (UTC)

• Every economics and game theory source which doesn't bother with conditional probabilities but which exhibits the minimax solution (player chooses a door uniformly at random and then switches; no further assumptions made at all!) gives implicit criticism of the conditional approach. Every reliable source which seems to present the unconditional solution as "the answer" gives implicit criticism. Last time I looked through my library of books about or with large sections on MHP, I found several, including several books of mathematical puzzles written or compiled by mathematicians. My personal criticism is written out at length on my talk page here: [8]. In short I would say that the conditional approach makes artificial assumptions which are pulled out of the air in order to enable the solution which the conditionalists are pushing. It is an example of solution driven science, not problem driven science. They are pushing that solution because they are usually writing in the context of an elementary probability class, and want their students to learn about conditional probability and about Bayes' theorem. In that context, there is nothing much wrong with that. However personally, as a teacher of mathematics, I like to use MHP as an exercise in mathematical model building and mathematization. That accent automatically opens up the field to alternative approaches. Mathematical modelling is an art, not a science; or more precisely, it is a craft, not something which can be done by following a rule book. I wrote a couple of reliable sources propounding this point of view myself (one of them is in Springer's online encyclopeadia of statistics, or will be soon) but I'm not pushing OR here. Just trying to help us see things at a higher level so we can be more objective about the task ahead. I have no objection whatever to placing the "standard problem" prominently in the article. I would just like us to be careful not to suggest it thereby has some kind of legitimacy which other versions don't have. Sorry for my verbosity and lack of references in [9]. I would suggest that my respected co-editors try simply to understand what I am trying to say there. If they understand, then maybe they'll suddenly realise that they already know the reliable sources themselves, or that we are talking such common-sense common-knowledge that it doesn't need sourcing. This exercise might make a number of sources which so far they found incomprehensible, suddenly make sense. (Or might suddenly make some of their co-editors make sense instead of nonsense). I'll try to add sources myself, certainly if and when we start composing parts of the article itself. Gill110951 (talk) 16:52, 20 September 2010 (UTC)

• Famous mathematician Ruma Falk did serious work on the 'conditional solution' of the MHP, and she criticized vos Savant's shell example for a very good reason. As I read Falk, she clearly is saying that the opening of one door by the host cannot change the odds of the door selected by the guest, unless the host is biased to open his favored door, "and you know about this bias", and she says that – without such a given and known host's bias, the odds of the door selected by the guest remain unchanged, because we’ve learned nothing to allow us to revise the odds. There is no other way to gather what Falk clearly is saying: she clearly indicates just this unique reason, and no other reason, because there is no other reason. Without a given and known bias of the host, by opening of one door, you have learned nothing to revise the odds of the door selected by the guest. As long as no other reliable sources are contradicting and disagreeing with Falk in this respect, the 'conditional solution' is wrongly presented in this lemma. We should pay attention to the fact that – without expressively mentioning of such a given and known host's bias – the opening of a door by the host has no influence on the odds of the door selected by the guest and thus never can be any relevant condition to change these odds. – "Using" of the door opened by the host as a so called "new condition" without mentioning the host's given and known bias, evidently emerges just as an utilitarian and practical method to provide an alleged "new condition" for the purpose of teaching conditional probability maths. That is good and welcome. But all of that isn't addressing the famous 50:50 paradoxon "MHP" itself. We should pay regard to the fact that any alleged "influence" can only result from the given and known bias of the host to open his preferred door. Otherwise not. We should not confusingly be selling 'conditional solutions' without clearly mentioning of their reason: A given and known bias of the host. Gerhardvalentin (talk) 14:02, 21 September 2010 (UTC)

  Per talk:Monty Hall problem#Some suggestions for a compromise this is not what Falk actually says (and, BTW, she's a psychologist, not a mathemtician). -- Rick Block (talk) 14:43, 21 September 2010 (UTC)

Rick is right, no "famous mathematician" - (Ruma Falk, Departement of Psychology and School of Education), so I was wrong, sorry. But nevertheless, inevitably she is mathematician, too, and we should pay respect to her conclusions. Gerhardvalentin (talk) 14:58, 21 September 2010 (UTC)

• Principally there's nothing wrong with including criticism of the conditional approach from reputable sources. However from papers/literature I've seen and recall on top of my head there's probably not that much explicit criticism. There are some that criticise Morgan's tone and assertiveness, but they do not explicitly reject his conditional approach and are somewhat indifferent towards favouring a particular approach.--Kmhkmh (talk) 15:21, 21 September 2010 (UTC)
• The archetypal conditional solution by Morgan et al is significantly criticised in the same journal in which it is published by Seymann, who points out that it should be the intent of the questioner that should be addressed rather than his precise words. This is particularly important as Whitaker's question is a letter from a member of the public, published in a general interest magazine. It is quite likely that he did not want the answer to the conditional problem at all. The conditional problem is essentially an academic extension of the MHP.

In addition, the conditional solution has been criticised by many editors and readers of this article for being overly complicated and obfuscating the central MHP paradox.

The conditional solution generally presented is not much more complete that the unconditional solution in that it does not show all the doors that the player might have chosen.

It can also be argued that, for the standard problem in which the host chooses a goat-door uniformly, even if the problem is to be considered conditional, the full conditional solution is not required due to the obvious and intuitive symmetry of the situation. Martin Hogbin (talk) 23:10, 22 September 2010 (UTC)

Question 7

Is the MHP a real world problem?

• Focusing on sources, most sources (by far) treat the problem as a puzzle that has the (surprising) answer that switching wins with probability 2/3 rather than 1/2 even though you're deciding whether to switch standing in front of two closed doors and one open door. There are some that explore real world (game theoretic) aspects of the problem relaxing various constraints imposed by the "fully explicit" version - to my knowledge the constraint most often relaxed is the requirement that the host always makes the offer to switch (leading to a minimax solution that switching doesn't matter, not that you should switch). -- Rick Block (talk) 15:45, 20 September 2010 (UTC)

• The 'real world' aspect of the puzzle is that it begins, "Suppose you're on a game show". With that, the common expectations of the contestant (any contestant, really) being unaware of the placement of the car and that the host would never 'tip off' the location of the car to the contestant exist. Glkanter (talk) 15:57, 20 September 2010 (UTC)

• I would say that the MHP is a puzzle about a imaginable real world, which has the surprising answer "switch". The real world aspect of "suppose you're on a game show" is part of what appeals to people and motivates people (including all editors here, I submit). If it would *only* be a mathematical puzzle with a surprising answer 2/3 no one would need a wikipedia article on it. Reasonable expectations about the real world certainly do play a role in formulating appealing solutions, and selling books about MHP. I think this holds equally for the usual passive unconditional solution, for the usual passive conditional solution, and for the usual (game theorist's) active unconditional solution (this is not the game theoretic version of a much more exotic nature mentioned by Rick). I can see good "real world" reasons for being interested in all these solutions, and good "mathematical recreation" reasons for being interesting in all three, and very good pedagogical reasons for being interested in all three. There is not such a world of difference between Bayes' theorem and von Neumann's minimax theorem. Both can be explained to lay persons in a simple context like this. Isn't it wonderful that such a simple paradox opens up a view to such a rich world. Gill110951 (talk) 16:15, 20 September 2010 (UTC)

• Yes and no. Obviously you can consider the game show scenario as a real world scenario, nevertheless many of the publication on the subject focus on theoretical aspect nit necessarily bear a strong real world connection.--Kmhkmh (talk) 15:25, 21 September 2010 (UTC)

• Primarily no. It is notable for being a simple probability puzzle. As Rick says above, 'most sources (by far) treat the problem as a puzzle that has the (surprising) answer that switching wins with probability 2/3 rather than 1/2 even though you're deciding whether to switch standing in front of two closed doors and one open door'.

On the other hand the article should address some of the real world issues and complications in the appropriate place. Martin Hogbin (talk) 23:14, 22 September 2010 (UTC)

Question 8

What more "givens" are needed for the (conditional) solution?

• The fully explicit (Krauss and Wang) version, already in the "Problem" section, describes everything necessary to make the conditional solution 2/3 and is what many sources say is the standard version. There are sources that make fewer assumptions, but I think these should be considered variants - for example the Morgan et al. and Gillman solution where the host preference is treated as an unknown variable is already treated as a variant. -- Rick Block (talk) 16:01, 20 September 2010 (UTC)

  Martin says below that the solution "should" assume uniform car placement AND uniform player pick AND uniform host choice between two goats - "should" according to what source? In the literature, uniform car placement and uniform host choice between goats are described as "standard" assumptions (no doubt because they make the conditional probability of winning by switching 2/3). I'm quite happy for the article to primarily focus on solutions using these "standard" assumptions, however asserting sources "should" make these assumptions is absurd. We don't really know what assumptions any particular source is making unless the source explicitly says, but if it does say we do know and we "should" accept what it says. -- Rick Block (talk) 15:06, 23 September 2010 (UTC)

I do not exactly say that. A say that there are two logically consistent ways of dealing with the vague problem statement. One is to say that all the undefined distributions are unknown and may not be uniform. In that case, the problem is insoluble. The other logically consistent option is to assume that they are all uniform, in which case the answer is exactly 2/3. As Rick agrees, taking the unknown distributions as uniform at random, has become the standard. The only logical alternative is to say that the problem cannot be solved (by probability theory)Martin Hogbin (talk) 17:46, 24 September 2010 (UTC)

• That's right, you need uniform distribution of location of car ,and uniform conditional distribution of host's choice, then the conditional probabilities are all 2/3 and the answer is "switch". You need much less to get an unconditional probability 2/3 and the answer "switch", namely, you only need that your initial choice has 1/3 chance of hitting the car. There is avenue opened up by this realisation, namely that one can actively engineer that that assumption is correct, rather than passively have to "know" it. You don't need anything given, you just need the opportunity to choose your initial door at random, in order to justify "switch" and (unconditional) 2/3. In between, you need a uniform distribution of location of car to ensure that all conditional probabilities are at least 1/2 and the answer "switch". A lot of sources uses words like "standard version" to describe the version which they want to focus on, for whatever reasons, indeed usually the one with everything completely specified and everything uniform. Usually this is merely a question of useful terminology, or a pedagogical choice, and often intended rather to exclude truly exotic variants like four doors, Monty sometimes revealing a car, and so on, as can be seen by consulting the sources themselves. Occasionally the phrase is used in order to self-justify the author, as can also easily be seen by consulting the source! The author wants to say that he is the only guy giving the right answer to the right question. Well, that's what some reliable sources like to say, and we can report their opinion of themselves, but we don't have to fall for it. (It is wonderful to read how Rosenhouse slams Morgan et al (first paper) for their arrogance in this direction.)Gill110951 (talk) 16:29, 20 September 2010 (UTC)
• I do not disagree with Rick and Gill but also mention the need for consistency when making assumptions. As the distribution of none of: the initial car placement, the player's initial door choice, and the host's goat-door choice is given in the Whitaker statement, they should either all be taken as unknown, in which case the problem is insoluble, or all taken as uniform, in which case the answer is 2/3 and the conditional problem becomes essentially indistinguishable from the unconditional one. Martin Hogbin (talk) 23:22, 22 September 2010 (UTC)

Martin, the question is not what you think but what the sources think on this topic. And the sources all have opinions (different ones). Gill110951 (talk) 15:19, 24 September 2010 (UTC)

I though you were suggesting that we allowing reality ('The Truth') to cloud our judgment. I might add that there are no sources at all which explicitly suggest that the unknown distributions should be treated differently. Martin Hogbin (talk) 17:46, 24 September 2010 (UTC)

In what ways do the sources disagree on the premises, say, as put together by K&W? Are you referring to Host Bias, or Random/Forgetful Monty? I don't think anyone, (maybe Morgan, but who can tell anymore) claims those things are the MHP. Glkanter (talk) 15:23, 24 September 2010 (UTC)

Question 9

Would participants be willing to begin working collaboratively on article sections? (I assume we would move segments of the article here for discussion).

• Yes. -- Rick Block (talk) 16:04, 20 September 2010 (UTC)

• For what purpose? The major issue of frequency & placement of the reliably sourced criticisms of the simple solutions remains unaddressed, let alone having reached any sort of consensus. What about the specific issues relating to Carlton's simple solution? In short, what's changed that will lead to successful collaboration? Glkanter (talk) 16:18, 20 September 2010 (UTC)
  • I was thinking that we could address the issues as we come to them (perhaps even starting with a less contentious segment of the article). When one has specific text to deal with it tends to give a focus to discussions. It will only work, though, if participants agree to give it a good try. Sunray (talk) 12:49, 21 September 2010 (UTC)

With all due respect, having given this article countless hours of thought, the frequency and placement of the criticisms *is* the only issue that the mediation really has to resolve. The rest can be inferred from that conclusion. And without this, nothing else can be accomplished. Other than a lot of postings that lead nowhere. IMHO, anyways. Glkanter (talk) 12:57, 21 September 2010 (UTC)

And yet participants remain deadlocked and have come to mediation. During the course of this mediation, (according to some participants) there has been, as Richard puts it, "an increase of mutual understanding... and trust." I am suggesting this approach because it has worked before in difficult mediations. Would you be willing to give it a try? Sunray (talk) 12:00, 22 September 2010 (UTC)

I certainly agree that effective mediation is required to end the stalemate. I disagree with that particular suggestion, though, and shared my reasons. By all means, move forward in any manner you like. I will cautiously participate to the extent I feel I have something to contribute to the discussions, as long as those discussions bring us closer to a resolution(s). Glkanter (talk) 14:18, 22 September 2010 (UTC)

• Yes. (What's changed? Increase of mutual understanding, hence growth of mutual respect and trust). Gill110951 (talk) 16:36, 20 September 2010 (UTC)

• yes, but imho for a collaboration going anywhere Rick and Martin in particular need to come to some understanding first. I'm not interested in investing time in article work/edits that have a good chance of going nowhere. At least as long as that is a likely outcome from my perspective my editing time is better spend on other articles.--Kmhkmh (talk) 15:31, 21 September 2010 (UTC)
  • I agree with you that all (most?) participants would have to be on board. We have heard from several in the affirmative (including Rick). Would others be willing to comment now? Sunray (talk) 12:00, 22 September 2010 (UTC)
    • The order of the sections is almost the only thing that Rick and I disagree about. I am reasonable happy with the article as it is now but Rick would prefer to revert to the earlier order. As you can see, other editors also disagree on the best order. For most editors, where the section occurs in the article affects what they would be wish to say in that section. I therefore think it would be best to discuss the order first. Martin Hogbin (talk) 17:32, 24 September 2010 (UTC)

I see a question on this subject has been added below. I will comment on it there. Martin Hogbin (talk) 17:48, 24 September 2010 (UTC)

• Yes, but before work on anything else can start, one item has to be solved, and that is the conflict between
  * correct solutions based on "implicitly given correct conditions" and
  * correct solutions based on mathematics by means of conditional probability theorems, reputedly including the use of the "positive number of the door opened".
  Alas! Gerhardvalentin (talk) 13:25, 22 September 2010 (UTC)

• The first issue to resolve, in my opinion, is the order of the sections. I prefer the article as it is now, with all the simple stuff first, with no complications based on conditions, followed by a full discussion of the more complex issues. If we can agree to that format then I am happy to work on the article a section at a time. Martin Hogbin (talk) 23:27, 22 September 2010 (UTC)

• I agree with Martin. If we can agree on the format then we can indeed start working section by section. Or even in parallel, by giving some different "task groups" different sections to work on. Gill110951 (talk) 10:34, 23 September 2010 (UTC)

• Disagree. The current order of sections enforces a structural POV by relegating the conditional/probability-theoretical solution to the bottom of the page. I much prefer an ordering similar to the one the article had as of the last FA review, i.e. with the two classes of solutions immediately following each other. glopk (talk) 19:46, 23 September 2010 (UTC)

** I don't see why the current order necessarily enforces a POV. The simple solutions are all favoured by many reliable sources. The complex solutions are all favoured by many reliable sources. We can point out right in the beginning that not all sources are agreed what constitutes a solution and say that we start with the simple ones for pedagogical reasons. We can also point out that the simpler solutions actually make less assumptions and therefore are more widely applicable, while the complex solutions make more assumptions, are therefore less widely applicable, but on the other hand do give stronger conclusions. It does seem to be a matter of taste / culture / context which assumptions are felt to be reasonable and hence how detailed an analysis can be given. (I do hope everybody agrees with truth of my last statements). Gill110951 (talk) 13:05, 30 September 2010 (UTC)

Question 10

What is the preferred order of sections in the article?

• The surface level disagreement is about the placement of the "Aids to understanding" section, although I think we're actually talking about NPOV. To keep this brief, I believe the section order favored by some is for the initial sections of the article to present the problem (with the standard assumptions of uniform car placement and uniform host choice between two goats) and then an extended section describing only the simple solutions (without mentioning anything about the published criticisms of these solutions or that many, many sources present conditional solutions) and only then present a conditional solution including the criticisms of the simple solutions in this section (or deferring the criticisms to an even later section characterized as "academic concerns"). IMO, this effectively creates a structural POV endorsing the simple solutions and a much more NPOV approach is to present both simple and conditional solutions in a single Solution section, more like the approach as of the last FAR (i.e. this version). The counter argument is that this favors the "conditionalist" POV. I have drafted numerous Solution sections (e.g. this one in the show/hide section, or this one) attempting to address this "pro-conditionalist" POV concern by presenting both simple and conditional solutions in a scrupulously NPOV manner (leaving the criticism of the simple solutions for a later section). This issue is the meat of this mediation. -- Rick Block (talk) 15:53, 24 September 2010 (UTC)
• I support keeping the "Aids to understanding" section where it is now. Even if I took Rick's POV that the simple solutions are deficient/incomplete/answer the wrong question, and the conditional solutions are complete/rigorous/answer the question as asked, I would still support the current section order. This is because, in common with most text books and many good WP articles on technical subjects, the current order starts with the simple and moves towards the complex. This has nothing to do with 'structural POV' it is simply the best way to present complex information.

It is quite common for an article or book to start with a simplified explanation which glosses over some of the subtleties of the subject and follow this with a more complete explanation which explains in full exactly when and how the first explanation might be deficient. That is exactly my proposal for this article.

I proposed the current section order as a solution to the disagreement here. To my mind it should be good for both simplists and conditionalists. Martin Hogbin (talk)

Please do not characterize this as "Rick's POV". I'm talking about the POV of sources - in particular the many, many sources that present conditional solutions without directly criticizing "simple" solutions and the POV of the relatively few sources (but still not nearly an insignificant number) that explicitly criticize the simple solutions. I'm fine with the order being simple to more complex, but NOT fine with inserting multiple sections of "aids to understanding" attempting to convince the reader the simple solutions are correct between these types of solutions. This structure conveys the impression that the simple solutions are considered to be correct and undisputed. IMO, this does not represent the significant published views of reliable sources fairly, proportionately, and without bias as mandated by fundamental Wikipedia policy (specifically, WP:NPOV). -- Rick Block (talk) 19:35, 24 September 2010 (UTC)

The point I was making was that, regardless of your POV or anyone else's POV concerning the correctness of various solutions, the current order is preferable. Martin Hogbin (talk) 21:39, 24 September 2010 (UTC)

And, the point I'm making is that regardless of your POV or anyone else's POV the current order violates NPOV. Do you disagree with this, or are you suggesting it's OK to violate NPOV? -- Rick Block (talk) 23:10, 24 September 2010 (UTC)

Of course the current order does not violate NPOV. It is the order that most good technical WP articles and most text books use. No one suggest that a physics book is pushing a Newtonian POV because it starts with a chapter on Newtonian physics and then proceeds with a chapter on relativity that explains that Newtonian physics only is a low-speed weak-field approximation. This is the normal way to present a complex topic. Martin Hogbin (talk) 10:26, 25 September 2010 (UTC)

Two comments: (1) There is at least one editor that would want to ban all criticisms of the simple solutions from the article (in the above analogy, that'd be banning the mention of the fact that Newtonian physics is a low-speed weak-field approximation), regardless of the section ordering. In my view this is wrong, and I think we need to reach a consensus that the sources criticizing the simple solutions deserve space in the article. (2) The current order is unsatisfactory because it does much more than proceed from simple to complex: it pushes, through the article's presentation, the idea that all the reputable sources that matter think the simple solutions are A-OK, except for a few published nutcases that may as well be pushed to the bottom, as if they don't really matter. glopk (talk) 02:07, 27 September 2010 (UTC)

It is merely your interpretation of Wikipedia's NPOV policy that simple solutions followed by conditional solutions is a violation. As I have for 2 years, I find your interpretation of this Wikipedia policy to be unsupported by the policy itself, or by existing practices throughout the civilized world. Glkanter (talk) 00:43, 25 September 2010 (UTC)

"::::I strongly suspect the absolute probability of a layperson understanding a conditional probability analysis is near 0. -- Rick Block (talk) 04:05, 15 February 2008 (UTC)"

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=next&oldid=191572133

Old diff copied by Glkanter (talk) 06:57, 25 September 2010 (UTC)

":What's wrong about it? The probability of winning by staying is 1/3. The probability of winning by switching is 2/3, i.e. double 1/3. -- Rick Block (talk) 03:01, 10 October 2007 (UTC)"

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=163485140&oldid=163427476

"Old diff copied by Glkanter (talk) 07:12, 25 September 2010 (UTC)

This one is too long to copy. Bullet points 5 & 6 and the paragraphs that follow are the most relavent.

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=194958681&oldid=194824904

Old diff copied by Glkanter (talk) 07:17, 25 September 2010 (UTC)

• I have no real preference regarding a particular order. I would however insist that various solution section contain a precise, accurate and sourced description, that establishes the exact context for a particular approach instead of just being fuzzily presenting it as a solution of "the MHP".--Kmhkmh (talk) 19:41, 24 September 2010 (UTC)

Well, before Rick knew about Morgan, the simple solutions were all that was in the article. Then another editor brought up Morgan, and the previously FA reviewed article took a dramatic 180 degree turn. At one point, there were 0 simple solutions in the solution section of the article. The reliable sources hadn't changed. What caused the article to change? Rick's POV, of course. Glkanter (talk) 19:42, 24 September 2010 (UTC)

"::::I strongly suspect the absolute probability of a layperson understanding a conditional probability analysis is near 0. -- Rick Block (talk) 04:05, 15 February 2008 (UTC)"

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=next&oldid=191572133

Old diff copied by Glkanter (talk) 06:57, 25 September 2010 (UTC)

":What's wrong about it? The probability of winning by staying is 1/3. The probability of winning by switching is 2/3, i.e. double 1/3. -- Rick Block (talk) 03:01, 10 October 2007 (UTC)"

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=163485140&oldid=163427476

Old diff copied by Glkanter (talk) 07:12, 25 September 2010 (UTC)

This one is too long to copy. Bullet points 5 & 6 and the paragraphs that follow are the most relevantBullet points 5 & 6 and the paragraphs that follow are the most relavent..

http://en.wikipedia.org/w/index.php?title=Talk:Monty_Hall_problem&diff=194958681&oldid=194824904

Old diff copied by Glkanter (talk) 07:17, 25 September 2010 (UTC)

Since you copied these twice (???), I assume you think these old diffs have something to do with what we're talking about here. Frankly, combined with this and this I'm struggling to find a way to interpret this other than as a personal attack. Once again, I'd really like Sunray to respond to this. -- Rick Block (talk) 00:39, 26 September 2010 (UTC)

Me, too. Glkanter (talk) 01:08, 26 September 2010 (UTC)

I'm not attacking you, Rick Block. I have no interest in that. I want your interference in the editing of the MHP to stop. Yesterday, right here on this mediation talk page, you provided what was needed to prove what you've been doing all along. You used the very same arguments 2 years ago that you called 'NPOV violation!' yesterday. Crying 'wolf' because you're not getting your way. Same exact claim of 'NPOV violation!' you used in December, 2009 when the consensus didn't go your way. Of course, it wasn't an NPOV violation 2 years ago when you championed that approach. That's offensive (worse actually, but I'll stay within the ground rules) to the other editors and Wikipedia's readers. You need to be stopped. Escorted off the premises, if need be. Glkanter (talk) 03:37, 26 September 2010 (UTC)

Are the ground rules from above still in effect? I would very much like Sunray to comment on this comment. -- Rick Block (talk) 22:55, 24 September 2010 (UTC)

Which ground rule has been violated, Rick? Glkanter (talk) 00:43, 25 September 2010 (UTC)

I repeat, I'd like Sunray to respond to this. -- Rick Block (talk) 01:50, 25 September 2010 (UTC)

I'm not seeing these comments as being personal attacks. Statements are in bounds as long as they focus on observations and content. I would suggest that participants keep trying real hard to avoid personalizing comments. Sunray (talk) 16:33, 26 September 2010 (UTC)

Seriously? I thought groundrule #1 was "Focus on content rather than the contributor. Note: This is to be interpreted literally, as worded."

To clarify, I was only reading the comments in this section. On re-reading, I see that you are referring to comments in a section below. Another editor also raised an objection to those comments and I agreed that they were inappropriate, and removed them. Sunray (talk) 03:16, 27 September 2010 (UTC)

OK, so fine. I'll respond to Glkanter's accusations.

The article after the first FAR looked like this. After the second FAR it looked like this. There are two significant differences between these two versions of the article. One is the addition of the "Morgan POV" that some editors claim still dominates the article. The other is the addition of numerous references - essentially every independent thought in the article was referenced to a source.

The addition of the Morgan POV was the result of a long discussion on the talk page, initiated by an anonymous user, that I initially resisted. The first quote above "I strongly suspect the absolute probability of a layperson understanding a conditional probability analysis is near 0" is from this discussion. The user eventually offered up the Morgan et al. source as support for his position. After reading and understanding this source, I indeed changed my position on this.

The second quote "What's wrong about it? The probability of winning by staying is 1/3. The probability of winning by switching is 2/3, i.e. double 1/3." was a response to a user on the talk page questioning the sentence in the lead that says switching doubles the probability of winning the car. I have no idea what Glkanter thinks this quote shows. I've never said the probability of winning by switching for the usual variant is anything other than 2/3. He seems to be implying I'm being inconsistent about something.

The last link is to a suggestion that Glkanter is apparently misreading. What it suggests is we change the solution to be a conditional solution (but without making a big deal about it).

I understand perfectly. You wanted *only* the conditional solution. No mention of the simple solutions any more in the Solution section. They would exist only later in the article as you suggested, and clearly accompanied by Morgan's criticisms, as per bullet point #6:

"Add a discussion of the conditional and unconditional consideration, referencing Morgan et al, to a later section of the article.".

But again, you didn't consider eliminating the simple solutions entirely from the Solution section to be an NPOV violation at the time. Going as far back as the consensus reached in November/December 2009, no editors have suggested that the conditional solutions are not worthy of inclusion in the Solution section. Just looking for consistency and honesty, that's all. Glkanter (talk) 22:03, 26 September 2010 (UTC)

Most of this discussion was BEFORE the second FAR, i.e. before the extensive inline referencing was added. As part of that effort I personally read at least 50 reliable sources in an effort to source the article's content (most of which I did not originally write). My convictions about what is and is not NPOV are based on this effort. I believe I have consistently attempted to focus discussions on sources, rather than personal beliefs. -- Rick Block (talk) 18:08, 26 September 2010 (UTC)

Simple To Complex - 'NPOV Violation' Eliminated As An Argument

I'd like to see the merits discussed without 'NPOV violations' being improperly brought up. Martin, care to resume the discussion? Glkanter (talk) 20:12, 26 September 2010 (UTC)

Presenting solutions from simpler to more complex is fine with me. Separating them by a lengthy "Aids to understanding" section, or in any way implying the "simple" solutions are universally considered to be correct and undisputed is not. -- Rick Block (talk) 21:23, 26 September 2010 (UTC)

Certainly, and as you suggested on February 29, 2008 in bullet point #6:

"Add a discussion of the conditional and unconditional consideration, referencing Morgan et al, to a later section of the article." Rick Block (talk) 19:49, 29 February 2008 (UTC)

That's been my point for a long time, the criticisms (especially Morgan's) don't *have* to accompany the simple solutions. They can come later in the article. Now, if you want to discuss where the Aids to understanding belong, I'm very open to that discussion. Glkanter (talk) 21:42, 26 September 2010 (UTC)

Aids to understanding after solutions (all solutions). This keeps the different solutions at the same level of emphasis, in a solution section approximately like this one (in the show/hide section). Simpler solutions are first, but they're presented along with a conditional solution. Within the solution section, the intent is to say different sources say different things, some sources say <the first thing>, some say <the second thing>, some say <the third thing> - without the article editorially picking which one is "right". -- Rick Block (talk) 22:48, 26 September 2010 (UTC)

My wish is that the simple solution section is accompanied by the criticism on it, in order that no one reads the simple solution and comes to think it is complete. Nijdam (talk) 22:56, 26 September 2010 (UTC)

The complete picture of MHP consists of a range of solutions, which step by step make stronger assumptions and therewith obtain stronger conclusions, but at the same time are of more and more limited applicability. The most simple solution makes a minimal assumption (your initial choice has 1/3 chance to hit the car - an assumption which you could have forced to be true by your own action!) and gives you an excellent reason to switch. Selvin quotes with approval Monty Hall's own rendering of that solution saying "I could not have said it better myself". His two notes already present both conditional and unconditional solutions and he does not appear to have a preference for either, rather they seem to sit together in his mind in peaceful coexistence. What Nijdam calls *the* complete solution makes a whole load more assumptions whose validity is hotly debated by the sources, and moreover whose meaning depends strongly on what the source understands under the word "probability". The article should be written in a constructive and positive way starting simple and working up to complex. It is not the editors' job to criticise any solution at all. Our job is to give an impartial overview of solutions which many sources have found useful and natural in different contexts. Gill110951 (talk) 10:39, 1 October 2010 (UTC)

Yes, that approach certainly promotes your POV, Nijdam. My opinion has long been that that would be an NPOV violation. Neither side is more 'right' or 'wrong' or 'complete' or 'incomplete'. I prefer a single Solution section. With no sub-headings. Without eliminating any of the simple solutions currently in the article. Without any criticisms or helpful explanations. And eliminating anything in the Aids to understanding that isn't properly sourced, which is mostly everything, actually. Glkanter (talk) 23:01, 26 September 2010 (UTC)

Is it your opinion that including any published criticism with a simple solution would be an NPOV violation? Please explain. -- Rick Block (talk) 02:34, 27 September 2010 (UTC)

Yes, that has been my opinion for a long time. As I just stated, only solutions should be in a Solution section. Each reliable source has offered his contribution to the MHP problem based on his idea of the MHP without qualification. Which I think is the most important point. Every source's POV is that his solution is correct and complete. So, offering each solution without comment is obviously NPOV. The criticisms come from reliable sources, and belong in the article. But presenting them alongside the solution diminishes the perceived value of the solution. That's where the NPOV violation occurs. I have previously suggested that an early, neutral comment about the lack of unanimity in the sources with a link to a later section would give the reader a heads up (and a chance to look ahead) that there's something more to the story. Glkanter (talk) 03:14, 27 September 2010 (UTC)

Actually, WP:NPOV says: articles "must be written from a neutral point of view, representing fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources". Making judgments about this seems to me to require knowing not just what significant views have been published, but the proportion of sources which hold those views. If the simple solutions were criticized by, say, every published academic article on the MHP then it would clearly be POV to NOT include such criticisms. My point is the POV of sources who criticize other sources is just as valid as the POV of the sources they're criticizing and this boils down to a weight decision. I'm curious - how much of the literature on the MHP have you read, is it more like 5 articles/books, or 50, or 500? -- Rick Block (talk) 03:59, 27 September 2010 (UTC)

I think there are 6 significant issues with your approach.

1. One method of not taking a POV is to give each source his own voice. At least once. You disagree.

2. We do not agree which sources are critical of the simple solutions. In fact, we don't even agree which sources *offer* simple solutions.

3. This approach could and has turned into off-tangent discussions right within the article. If it's valid to offer the criticisms of the simple solutions alongside them, then the criticisms of those criticisms would appropriately follow those criticisms. I just don't think the reader who came for 'is it 50/50 or 1/3 & 2/3' is well served by encountering the various camps so early: the simple solutions; the critics; uninvolved sources that have provided conditional solutions; and the critics of the critics. And there's at least 4 critics of the critics: Seymann; vos Savant; Rosenhouse; and Morgan. This is why I want to stay away from discussions of each source's intents.

4. Any survey will be incomplete, and flawed. Prior to February, 2008 such a survey performed by Wikipedia editors would not have included Morgan, despite the paper being published in 1991. I can go so far as to acknowledge the existence of the various POVs, but I don't think it's plausible to determine the proportions. In fact, unless you can find a reliable source that has done such a survey, any descriptions in the article of 'POV prevalence' among the sources derived from a survey conducted by Wikipedia editors would be OR, and inappropriate.

5. Rick, you could answer the question about deferring the criticisms, too. Back in February, 2009, you suggested having only a conditional solution, and defer even mentioning the simple solutions until later in the article. And then only as part of a discussion of both POVs. What were *your* justifications at that time?

6. The standard being held up is not consistent with past practices of the MHP article.

As of May 11, 2008 there were no reliably sourced simple solutions in the solution section.

There was a 2 or 3 sentence unsourced simple explanation that was prominently accompanied by the table that is now the 2nd image in the Condition Solution section.

In April of 2009, the unsourced simple explanation was re-written and attributed to some sources.

For a period of time, the simple solution was followed by a section called 'Criticisms of the simple solutions'.

The reliably sourced simple solutions were relegated to the Aids to understanding section which came after the Solutions, Criticisms, and Sources of confusion section.

Not until April 10, 2009 was a reliably sourced simple solution, the Combining Doors solution (attributed to Devlin, Williams, and Stibel, et al) put in the solution section. It was the last item in the Simple solution section, after the generic explanation (now attributed to Wheeler, Mack, Schwager, and vos Savant) and a large, offensive, complicated, unsourced image. This was immediately followed by the Conditional solution section, which began with a criticism of the simple solutions.

How much have I read? Not as much as you. More than you think, probably. Enough that I understand there are conflicting sources that deserve proper representation in the article. My main disagreement with you, at least on this issue, is how to interpret Wikipedia's policy on NPOV and reflect that policy properly in the article. As even your own interpretations of what constitutes NPOV have changed over time, the cautious route is to let the sources speak without interruption of warnings. Glkanter (talk) 04:27, 27 September 2010 (UTC)

My first wish is to have a single "Solutions" section, with subsections as follows:

• Simple solutions. Sources: vos Savant, Adams, Devlin, Williams, Stibel et al.
• Probabilistic solutions, including analysis of the question's ambiguity, difference between conditional and unconditional answers and criticisms of the simple solutions. Sources: Selvin, Morgan et al., Gillman , Grinstead and Snell, Gill, Chun, plus Henze or Gill for the Mathematical Formulation subsection.

Following, Aids to understanding, cut to what is properly sourced.

My second wish is for Glkanter to stop disrupting this conversation with personal attacks and past quotes from other editors lacking context. We can all play this game, and mining the talk page's history for Glkanter's dicta would yield rich pickings indeed. But I really really really (did I say really?) do not wish to waste valuable time doing it, and further sink this mediation in the process. glopk (talk) 01:42, 27 September 2010 (UTC)

I don't agree with your Solution section suggestion, as I prefer solutions sections that are unadorned with criticisms and other commentary. Phrased another way, Only solutions should be in a Solution section.

Otherwise, I post things here that imho are appropriate and that add value to the debate in some way. You, of course, will decide for yourself what to post here. Glkanter (talk) 01:56, 27 September 2010 (UTC)

Yes, we all understand what you are saying. Would you now please give a chance to other editors to chime in? (Personal comment removed per agreement on this talk page) glopk (talk) 02:16, 27 September 2010 (UTC)

I fail to see how the current article order does not represent a NPOV. We have some simple solutions followed by a section 'Aids to understanding' explaining the simple solutions, we then have the conditional solutions with some discussion about them. If you read the A to U section you will see that it clearly refers to the simple solutions and belongs with them. The A to U section explains why the answer is 2/3 rather than 1/2, it says nothing about host goat-door choice and why this might affect the answer.

I would have no objection to having a second A to U section after the conditional solutions explaining what the conditional business is all about but the current A to U section is clearly about the simple solutions and should be kept with them.

To those who think that this is a dastardly plot to impose a particular POV by stealth I can only repeat two things. The article quite clearly states that some sources find deficiencies with the simple solutions, in other words, both POVs are clearly and openly given in the article. Secondly, please have a look at some mathematics (or other technical subject) text books (or WP articles) and you will see that it is quite common to get a simplified exposition of a subject first, maybe with some points glossed over for the sake of clarity, followed by a fuller explanation. Although weasel words (for example 'we start with a simplified discussion') are sometimes included in the simple explanation, quite often they are not. In the case of the MHP such comments are particularly unhelpful as the basic paradox is in understanding why the answer is 2/3. It is too easy for editors here to loose sight of just how hard that question is for most people to understand. Anything that detracts from that understanding is undesirable in this article.Martin Hogbin (talk) 09:15, 27 September 2010 (UTC)

IMO, a more integrated presentation will be both more understandable to the reader and more NPOV.

The very first Aid to understanding (why the probability is 2/3 and not 1/2) is clearly attempting to describe conditional probability. The probability people think about is a specific case where the player has picked a door (say door 1) and the host has opened a specific door (say door 3) leaving the player standing in front of two closed doors and one open door (97% of Krauss and Wang's sample approach the problem this way). The "probability" of the open door (e.g. door 3) is obviously 0 at this point, leading many people to think the "probabilities" of the other two doors are 1/2 (since there are two doors and the probability must add up to 1). These probabilities are conditional probabilities (this is both factually true and undisputed by any source). The case the reader is thinking about (again, 97% of K&W's sample!) is this conditional case. Explaining why the probability is 2/3 rather than 1/2 seems to me to require discussing conditional probability (at some level). It is possible (with great difficult according to K&W) to change the reader's mental model of the problem from one where they're thinking about the conditional case to one where they're thinking about the unconditional case (which is what makes the "simple" solutions understandable), but IMO what this is actually doing is forcing the reader to buy into the POV that the simple solution is addressing the "right" problem. The difference between these problems is confusing to many (per Morgan et al. - "the distinction between the conditional and unconditional situations here seem to confound many") and after "seeing" the simple solution as correct switching back to focusing on the conditional problem is also extremely difficult (per Morgan et al. - "F1 [which says the strategy of never switching wins 1/3 of the time, so switching must win 2/3 of the time] is immediately appealing, and we found its advocates quite reluctant to capitulate").

The conditional approach is presented (more or less) as often as any of the simple approaches. Treating this as an NPOV issue means the article must not present one approach as preferable to the other. Deferring any discussion of conditional probability until after the reader has been bludgeoned into believing in the "simple" solutions IMO creates a structural POV, see WP:STRUCTURE. The point of the conditional approach is NOT that the host's choice between two goats might matter, but that the probability AFTER the host opens a door is (by definition) a conditional probability. It is obvious to everyone that the probability BEFORE the host opens a door is 1/3. The entire point of the problem is the difference caused by the host opening a door, which is what the conditional approach analyzes. With the usual assumptions the host opening, say door 3, does not change the numeric probability of the player's initially chosen door but it does change the probabilities of the other two doors (one doubles and one goes to 0). These are all conditional probabilities, which the simple solutions (simply) do not address this. If you read them carefully, they ALL change the decision point from AFTER the host opens a door to BEFORE the host opens a door (at which point the probability of the car being behind any of the doors is 1/3, including the one the host opens!!!) - which is a perfectly fine way to approach the problem but it does not match how 97% of K&W's sample were thinking about it. -- Rick Block (talk) 16:42, 27 September 2010 (UTC)

Rick- I agree with your description but imho this is leading us nowhere at least as far as the mediation is concerned. You are exchanging those exact arguments with Martin for over a year now. Maybe the mediator is needed here with some suggestions, but reiterating already known viewpoints for the gazillions's time is not providing us with any new insight nor is it leading to a conflict resolution. Even in the (unlikely) case that some of the involved might "still haven't understood it yet", they are unlikely to understand it at the nth repetition right now.--Kmhkmh (talk) 22:41, 27 September 2010 (UTC)

I too would welcome assistance on this subject from the mediators. Martin Hogbin (talk) 08:59, 28 September 2010 (UTC)

• I prefer simple to complex. I don't see any conflict whatsoever between unconditional and conditional solutions. Conditional solutions need to put in more assumptions in order to get out their stronger conclusions. I identify three different sets of assumptions which are just enough to give you three different levels of conclusions. They all have reliable sources behind them and they are all self-consistent (based on correct logic/mathematics). Most wikipedia readers will be more than satisfied with the simple solution. I also notice that the different understandings of what probability actually means, is a major factor influencing which assumption set appeals most to you. We editors of the MHP article have to stand "above" this issue, just as we have to stand above all issues on which the sources disagree. We have to do be able to do this independently of our own personal taste in the matter. Gill110951 (talk) 09:22, 28 September 2010 (UTC)

Hey, Richard, most wikipedia readers will also be more than satisfied with the explanation that it doesn't matter whether to switch or not, as the odds are 50-50, don't you think?Nijdam (talk) 15:59, 28 September 2010 (UTC)

I agree Nijdam. Also the point of view that you should not switch because it doesn't make a difference, should be reported in the article. Many reliable sources have said so (eg Paul Erdos). It was my own initial reaction when I was first told the problem. Our job is to make a useful overview of what reliable sources have said about MHP. We are here to discuss how to organise all that material, not to promote our own opinion. We can't write the article from the point of view that there is only one true correct proper solution to MHP, even if it happens to be any of the editors' personal point of view. It's not my point of view, but apparently it is yours. Gill110951 (talk) 07:43, 30 September 2010 (UTC)

Simple To Complex, Continued

Kmhhmh and Martin have asked for a mediator's assistance on this. I don't know whether it will be helpful, but I can give my reaction to the article. More importantly, I think, I could give my thoughts on writing a good article. Many of you (certainly editors who have edited a variety of articles) will be familiar with what I am about to say.

My personal reaction was that the explanation of the simple solution was what first drew me in. I began to think: "What is the problem." It whetted my appetite for more. I've probably taken more statistics than the average reader, but I can tell you that I think that the article could be more clearly written (and better organized).

As an experienced editor, I would say that readability is tremendously important in creating a good article. There are readability level tools, but the real secret is to use simple, understandable language and to tell a story. If the majority of editors value readability, perhaps we could us that as a guide. Sunray (talk) 12:49, 29 September 2010 (UTC)

That seems good advice to me, especially as the MHP is notoriously difficult for most people to understand when they first see it. My opinion is that mention of 'difficult things' like 'conditional probability' too early in the article is detrimental to readability. Martin Hogbin (talk) 15:34, 29 September 2010 (UTC)

But why can't we have our cake, and eat it too? As far as I can see there aren't any objections to placing the simple solutions at the beginning of the Solutions section. Rick Block, Nijdam, Kmhkmh and I are simply requiring that the reader be not "bludgeoned" by a long chain of "Simple Solutions" followed by "Aids to Understanding" into believing that this is all there is to it. What's wrong with having the simple solutions followed by a subsection with the probabilistic solutions, roughly of the same lenght, with the "Aids to Understanding" following? I am personally willing to stipulate that the Conditional Solution section could be made more readable, and would be OK with moving the Mathematical Formulation at the end appendix-like, as it was at the last FA. Wording about the limits of the simple solutions (that sources say that they not address the problem as stated in view or reputable sources) could be place in the subsections of the Aids to Understanding section as well, as a motivation and introduction for the conditional solution(s). Would you be willing to work on a compromise along these lines? glopk (talk) 03:59, 30 September 2010 (UTC)

I too want to have a cake and eat it too. Personally I think that both simple and complicated solutions are completely valid. No-one should be bludgeoned into believing that *either* is the whole true story. I think that those who "know" that there is one and only one truly correct solution to MHP are not going to be helpful editors of the MHP page. But understanding how people come to such positions is a key to understanding the literature. Of course we must report that some "reliable souces" take rather dogmatic views. They get bashed on the head by other reliable sources for their arrogance and narrow-mindedness. It seems to me to be a prerequisite to editing the MHP page that one does not hold a dogmatic view oneself. It seems to me a second prerequist that one understands the internal logic of different solutions and the logical relationships between different solutions. We are talking about a famous puzzle, right, which needs to be solved by clear logical thinking and a dash of probability theory! Consider only the mainstream solutions, each with many reliable sources behind them from varying fields, not exotic academic or specialist solutions. I perceive two main solutions: the simple, and the "standard", full, symmetric, conditional. Between them is the slightly more complicated solution allowing host bias. Uno: simple, due: intermediate, tre: standard/conditional. Note that the simple solution makes less assumptions than the intermediate which in its turn makes less assumptions than the "standard" full statement with standard full conditional solution. We have to report that there are different points of view which is the best solution. Well, those different points of view must come down to the question whether the more complex assumptions are justified or not, since the internal logic of uno, due, tre is indisputible. Evidently, opinions are strongly divided on the key question "what are the good assumptions?"; both among reliable sources and among editors. That's good, the community of editors is a good reflection of the community of reliable sources, so the MHP page has a good chance to be complete and to be balanced (as long as we don't bludgeon any of our colleagues into leaving our collective enterprise). Our job is merely to organise and report, and of course we want to do this in an attractive way since MHP is such a great problem! Therefore we are going to have to be clean and efficient in our organising and reporting. In my humble opinion, it could help us a lot in that enterprise to separate the opinions about what probability means and about what it is reasonable to assume about a quizshow or about various authors' intentions or context from the indisputible simple logical facts. The latter can offer a key to the organization of the story we have to tell. Gill110951 (talk) 06:13, 30 September 2010 (UTC)

Yes, stop selling underhand hidden assumptions in order to improve readability. Gerhardvalentin (talk) 12:26, 1 October 2010 (UTC)

Readability

Readability was the suggested basis on which we should review the current article. My opinion is that the lead sets a good example of the way to go. It has rarely been disputed. Who agrees? Martin Hogbin (talk)

I agree.Gill110951 (talk) 10:45, 1 October 2010 (UTC)

Point of order

There are discussions going on at talk:Monty Hall problem that I think the mediators have already suggested should take place mainly here, see /Archive 3#Continue discussions on the talk page, or not?. If the mediators could comment about these off-mediation threads and possibly put a reminder on the article talk page it might be helpful. -- Rick Block (talk) 15:59, 22 September 2010 (UTC)

I have presumed that it is better for arguments on the MHP to take place elsewhere, or are we requested not to talk about the subject at all except here? Martin Hogbin (talk) 23:28, 22 September 2010 (UTC)

I too understand that the mediation page is not the place to "talk MHP". Here we talk about editing a wikipedia article, and we try to be brief and remain within the structure of the mediation. Gill110951 (talk) 10:32, 23 September 2010 (UTC)

The usual practice, during a mediation, is to suspend discussions on the article talk page. Article talk pages are really intended for discussion of article editing and that is suspended during a mediation. So it would be best to confine the discussion to the mediation talk page, IMO. Sunray (talk) 14:08, 23 September 2010 (UTC)

Next points of order

Let me try to get some points straight in order to help the mediation process. I want to clarify some issues and ask the participants whether they agree or not. If not, just very shortly, argue why. I don't ask for your personal opinions, but your opinions about the sources the article will be based on.

As several people have difficulty with reading, I extend the statement.

1. The main treatment of the MHP in the relevant sources focus on the MHP as a simple probability problem.

AGREE
Nijdam (talk) 09:37, 23 September 2010 (UTC)

NOT AGREE
I have suggested on the talk pages that the first sentence in the article, describing the MHP as a 'probability puzzle' should be corrected. Many reliably published solutions take advantage of other sciences to solve the MHP. Glkanter (talk) 09:54, 23 September 2010 (UTC)

Certainly a lot of reliable sources focus on the MHP as a simple probability problem, in the sense of a probability problem for a probability class learning about conditional probability. But also a lot of reliable sources, which I think are relevant, don't have this focus. So I don't agree that the "exercise in conditional probability" approach is dominant in the relevant sources. Gill110951 (talk) 16:36, 29 September 2010 (UTC)

• Somewhat agree. Personally I view it primarily as a probability problem given as a puzzle or brain teaser. However in doubt the article should cover any aspect or viewpoint that is treated in reputable sources (in particular psychology and game theory).--Kmhkmh (talk) 12:28, 28 September 2010 (UTC)

Of course, no problem, I only made a statement about how most sources treat the MHP, and, IMHO, how most readers will view it. Nijdam (talk) 13:21, 29 September 2010 (UTC)

Why not? The MHP is known as one prime example of a "psychological problem", fallacy more than 90 %. Mentally caused by misdirection. Isn't it?

Richard: You say "... is right with probability 1/3, then simple logic tells you that switching gives you the car with probability 2/3." Please take heed of the tattletale host, in opening of his "unfavored door" showing that the car actually is behind his "favored door". Gerhardvalentin (talk) 15:17, 28 September 2010 (UTC)

Gerhard: that's why you should choose your door initially completely at random, and switch. Ignore door numbers. Don't bother your head with wondering whether the host is trying to trick or treat you. Don't worry about probabilities which you don't know. Gill110951 (talk) 16:28, 29 September 2010 (UTC)

Yes, of course. Because it doesn't matter. But some editors (see above) do insist on disclosure of secrets resp. at least do believe in telepathy and clairvoyance. That's the "real problem". Gerhardvalentin (talk) 16:38, 29 September 2010 (UTC)

There are editors here who believe in telepathy and clairvoyance? Wow! Please disclose their names... Of course all sensible solutions to MHP take the sensible point of view that we will rule out telepathy and clairvoyance, collusion and secrets... Gill110951 (talk) 06:20, 30 September 2010 (UTC)

The guest has selected her door (say #1). As she had no knowledge on the location of the car, the odds of her door will be 1/3. Isn't it so?
Under this given condition, the host has opened another door, showing a goat. That's a fact.
If the host (in 1/3) should have had two goats to choose from, he will not have been biased to give additional tattletale info on the actual location of the car.
Thus, independent of the host's showing of one goat behind door #2 or behind door #3, this will have no influence whatsoever on the odds of the door selected by the guest, because by his opening of a door we’ve "learned nothing" to allow us to revise the odds of the door selected by the guest (Falk).
Repeat: No influence whatsoever on the odds of the door selected by the guest, the odds of her door did remain 1/3.

Yes, the "MHP" is a widely used example in teaching probability theory, and yes, the MHP has great and prominent influence in textbooks as a useful example.
Nevertheless some editors are claiming that, vice versa, Bayes' theorem has important influence on the odds of the door selected by the guest. And they are claiming that (by telepathy or by clairvoyance?) there must be an appropriate influence, and that without Bayes the "problem cannot really be solved".
The door #3, opened by the host, still must have a huge impact on the odds of her selected door, otherwise it was not wise to take the door #3 as an important "condition" to revise the odds of her door, to show the door #3 as the decisive influence, the elementary key on the odds of her door. Divination or fallacy?

Again, the MHP example has some great influence in teaching conditional probability, this fact remains undisputed. But also vice versa?
Will it really be possible to rule out telepathy and clairvoyance, collusion and secrets and divinity? Or will conditional probability textbook sources continue to serve as a justification for the vice-versa reverse? Gerhardvalentin (talk) 14:31, 30 September 2010 (UTC)

Gerhard, you are using probability in the subjectivist sense. Within your interpretation of probability, I can agree with you. But it is not obligatory to solve MHP with subjective probability. The conflict between unconditionalists and conditionalists in this mediation coincides with a hidden difference of opinion concerning the meaning of probability. Is it in your mind, or is it out there in the physical world? Gill110951 (talk) 05:52, 1 October 2010 (UTC)

Can't we use The Truth to guide the organization of the article?

Why don't we use The Truth here to guide our discussions? It's a fact that different reliable sources think that Whitaker asks different questions, different sources find it more or less reasonable to make different supplementary assumptions. (Sometimes because they have give different real world meanings to the word "probability"; sometimes because they think of the guest as an active player, not as a passive player; sometimes because of semantic ambiguity). Our job is to survey this in a coherent way. At least the underlying logic (call it math or probability if you like) is very very clear and I think indisputable.

We all agree that we are told (or may assume) that the host always opens a door revealing a goat and can do this because he knows where the car is hidden.

Do we all agree that the following (uno due tre) are Logically Correct Implications?

Do we all agree that each starts by formulating a stronger assumption than the previous one, and (not suprisingly) is able to draw a stronger conclusion?

• Uno: IF you *only* assume player's initial choice is correct with probability 1/3 THEN you know that switching gives him the car with probability 2/3.

• Due: IF you actually assume that *all* doors are initially equally likely to hide the car THEN not only is the preceding assumption (and hence also conclusion) true, but we also are guaranteed that all conditional probabilities are at least 1/2. And on average 2/3 because of "uno". This tells us that not only is "always switching" a good idea, but also there is no strategy (sometimes switching, sometimes not, depending on the door numbers in question) which is better. We needs Bayes for this. The odds form as used by Rosenthal, in his article and book, does the job here with least pain and makes intuitively clear why the answer is tricky: given you chose door 1, the chance the host opens door 3 given the car is behind 2 is typically different from the chance the host opens door 3 given the car is behind 1. In the symmetric case twice as big, but in any case, it's always at least as big. That's why his opening of door 3 actually does change the odds the car is behind door 2 versus behind door 1, in the direction favouring door 2.

• Tre: IF you not only assume that all doors are initially equally likely to hide the car but ALSO assume that either choice of Monty is equally likely when he has a choice, THEN the door numbers are irrelevant by symmetry. The car is behind the other door with probability 2/3. Both unconditionally, and conditionally given the specific doors. You know in advance, or at least, you can figure out in advance, that you can ignore the door numbers. (You can also ignore the day of the week - it doesn't matter if it's Tuesday or Wednesday).

That's basically all there is to say regarding the maths content of MHP.

The sequence here goes: Uno: very easy; Due: a bit subtle (and needs conditional probability); Tre: quite easy (and refers to conditional probability). I think it would be most pedagogical to first present the list of assumptions in order of strength, and then work through the solutions in the order 1, 3, 2. The reader can stop when they get tired (or bored) without missing anything important (for that kind of reader).

All that remains is to add the references to the sources, say that they tend to quarrel about the right assumptions to make, and to organise the whole thing into a nice story for wikipedia readers. Next to the maths story there is the historical story of the origins and development and fights, which we have been simply repeating in the MHP talk pages, round and round in the same circles, for years. Gill110951 (talk) 09:32, 24 September 2010 (UTC)

Exactly. Uno: True (because there is only one car, but the player had no knowledge of where it is. That is known only by the host and his staff, and nobody else knows.)
Due: True (probability from at least 1/2 (in 2/3) to max. 1 (in 1/3), so on average 2/3, as long as nothing else is evident.)
Tre: True (for as long as you have no knowledge whatsoever about the host's behaviour, you impossibly can declare this behaviour to be "evident". So in opening of one door, no matter which "number", you are absolutely out of position to "assume better information".) – But you can speculate about it (if the host should be extremely biased either in 2/3 the odds of both doors are min. 1/2 each, or in 1/3 the chance for the second closed door will be max "1". But, for this one single game, in the situation given, you have no knowledge whatsoever on that, and so all you can say is: Pws will double your chance to 2/3. Anything more would be an arrogant presumption. Gerhardvalentin (talk) 14:42, 23 September 2010 (UTC)

Gerhardvalentin, when you say "True" you are saying why *you* would consider each assumption in turn to be true: that's a matter of opinion, and also depends on what you mean by probability. All opinions (and sorts of probability) are supported by different reliable sources. Clearly you are a subjectivist (probability is in your information), not a frequentist (probability is out there objectively in the world). This makes it easier for you to go further. I am not presenting my choice, I am merely showing the menu including price list. In each case the conclusion (successively stronger) follows from the assumption (successively stronger). It is up to the consumer to choose which product to buy. And some people always buy PC's, others always buy Macs, and the difference is a matter of religion. A few go for Linux. If you have more money you can afford a more powerful car. But you have to get the money from somewhere, first. There are no free lunches. And a three course lunch costs more than two and that costs more than one. Gill110951 (talk) 09:43, 24 September 2010 (UTC)

The truth? As I am sure you all know, Wikipedia is interested in verifiability, not truth. Several participants have expressed the need to agree on the order of sections. I've added that as Question 10, above. Would someone be willing to suggest an order? From the discussion thus far, I've got the idea that the progression should be from simpler to more complex, but there are various ways of achieving this. I look forward to discussion in the section above. Sunray (talk) 14:02, 24 September 2010 (UTC)

Well Sunray, all my statements are easily verifiable either by 1+1=2 logic or by extremely reliable sources so that is not the problem. They are universal logical truths (hence essentially tautologous), they are not opinions. They are all "if..then.." statements. The disagreement among the editors and among the sources, concerns the matter of opinion or taste or context... as to which "if's" it is reasonable to assume. I offer an agreed collection of logically necessary true steps arranged in order to difficulty as suggestion of how to order the article. Gill110951 (talk) 15:17, 24 September 2010 (UTC)

This not really germane. If such statements are tautologically true, then there will be sources expressing them, and picking which ones to report upon is an editorial decision. If there are no such reputable sources, then those statements are irrelevant to the present discussion. Regardless, the order of the sections need not follow an order of increasing complexity - we are not writing a list of claims for a patent. glopk (talk) 15:46, 24 September 2010 (UTC)

It doesn't need to follow them but it can. I'd regard that as a rather marginal point that should not block us from reaching a compromise.--Kmhkmh (talk) 19:45, 24 September 2010 (UTC)

It is germane, because each of the three tautologies is the complete mathematical basis of the solutions to MHP offered by a whole heap of reliable sources. And I think that this list of three covers all the important ones. Please note that I do not ask anyone to buy any of the "if"'s. I just say that if you buy any one of the "if"'s, then you're entitled to take the corresponding "then" home with you. Understanding this is a big step towards understanding the sources. A second step is to understand that not everyone thinks about probability in the same way. Different understandings of probability make it more or less natural to buy different assumptions. If you want to understand the sources, you also have to get your mind around this one. Gill110951 (talk) 08:57, 28 September 2010 (UTC)

Richard, I'd be grateful if you stopped trying to teach us on this mediation page of a "right" way to mathematize the MHP. I appreciate the elegance of your formulation, but we have gone down this route for a long time now, and it hasn't helped. The archive of the talk page is paved with dialogs in which one side literally refused to read equations screaming Truth to their face, just because they happened to contradict their long-held Belief. This is why I entirely agree with Rick Block that the only way out is to stick to the sources and treat this whole mess purely as a POV issue, and work as hard as possible to make the article report their content espousing a NPOV. glopk (talk) 04:25, 30 September 2010 (UTC)

Glopk, I am not trying to teach you a "right way" to mathematize the MHP. I am suggesting a way to organise all the important mainstream existing points of view, all the important mainstream existing mathematizations, all those espoused by many reliable sources in a variety of contexts, in a way which makes the relationships between them transparent and exposes their strengths and weaknesses (as observed in the reliable sources). I too am bored by the endess repetitions of arguments on these pages where someone pushes a particular POV and appears totally unable to comprehend the existence of any other. I have worked very trying to understand the roots of such apparent inability to comprehend other people's points of views. I think I made a lot of progress. I try to share it with all both individually and collectively. Please just tell me if you can agree with my three statements - three claims of the kind: IF you assume "A" THEN you may conclude "B". Is that so difficult?

Richard - are you suggesting a combined solution section, perhaps somewhat like the ones I've drafted (i.e. this one, in the show/hide section)? Would you say presenting "simple" solutions in the article first, in their own lengthy section (without mentioning anything about the interpretation of the problem statement as asking about the conditional probability until later in the article) represents the views of published reliable sources neutrally, fairly, proportionately, and without bias (per WP:NPOV)? This is the Truth we're after here. -- Rick Block (talk) 16:26, 24 September 2010 (UTC)

I would prefer a lengthy and careful presentation of simple solutions first, since that is what most readers will find most useful. I don't think we should make a big thing about the fact that some sources are dogmatic about what is the correct way to solve the problem. A sensible presentation will show that there is complete harmony between the different approaches. If you make stronger assumptions you can draw stronger conclusions. Different sources find it more or less natural to make different assumptions. We must report that in a neutral way. Gill110951 (talk) 17:19, 29 September 2010 (UTC)

I'm suggesting that we focus on what we ought all to be able to agree on. We all agree that 1+1=2, right? Is there seriously anyone around here who doesn't agree with the following statement?

* IF your initial choice is correct with probability 1/3 THEN switching gives the car with probability 2/3

I think everyone agrees that that, because I suppose everyone agrees:

* the switcher wins if and only if the stayer loses.

and

* 1-1/3=2/3.

If we would agree on the hard mathematical facts, the eternal truths, true independently of what is the MHP, what is probability, then we might be able to have a civilized discussion about how to organise the article in order to serve the interests of our readers, to reflect the reliable sources, and to get the article back to being a great Featured article again.

Now my Personal Opinion is that there are different mathematizations of vos Savant/Whitaker's words which all have reliable sources behind them, all are self-consistent and meaningful, and that they roughly correspond to Uno, Due and Tre. If you are prepared to assume such and such, then you are entitled to conclude bla bla bla. From this point of view there is no conflict at all. The only conflict comes when people want to push their personal opinions rather than reflect the world out there of MHP. The different sources belong in different contexts, different traditions, within each of which the stated assumptions are reasonable and conventional.

I think the article should be organized first presenting the problem, mentioning the different assumptions which different sources find natural (those are the IF parts of Uno, Due, Tre). Then present the simple solutions which are essentially the solutions based only on assuming the IF of Uno. Then present the conditional probability approach first based on the IF of Tre (the so-called standard problem). Then present the conditional probability approach starting from the IF of Due. Gill110951 (talk) 08:17, 28 September 2010 (UTC)

Rick, I just looked at the draft by yourself which you referred to. It is nicely written and organized but I don't agree with some of the statements which are made. Much more importantly, I think there are reliable sources "pro" and reliable sources "contra" a number of the remarks which you take for granted. It seems to me that you are deeply at heart a frequentist while Garry and Martin are deeply at heart subjectivists. In the texts of either side I see hidden assumptions which make sense to the writer but just can't be accepted by the other. I'm afraid that different conceptions of probability, together with difficulties in comprehending some basic logical truths, are what eternally bedevil our discussions here. Gill110951 (talk) 09:11, 28 September 2010 (UTC)

Richard - I would appreciate it if you would stop trying to characterize me as a "frequentist". I completely understand both the frequentist and subjectivist views. And, rather than unspecific comments about "some of the statements" you disagree with or "hidden assumptions" in the texts, it would be helpful if you were much more specific. What statements in particular? What assumptions? I think it would also be helpful if you could be more brief in your posts (less lecturing). -- Rick Block (talk) 14:48, 28 September 2010 (UTC)

Rick, I didn't say that you don't understand both views. I only said that your draft seemed to me to be built around a frequentist understanding of MHP (corresponding to the understanding of many but not all reliable sources), so subjectivists might feel uncomfortable with it. I have annotated your draft on your talk page in order to support my claims [10] The whole explanation is focussed on a frequentist picture and it is written from the point of view that the conditional solution is the only complete solution. That's your opinion and the opinion of other editors, it's the opinion of many reliable sources. But not all editors and not all sources agree.Gill110951 (talk) 13:09, 30 September 2010 (UTC)

The 'Truth'? Not in this article. Today's reverts were unconscionable. Selvin as a reference for 'it must be solved conditionally'? He gives a simple solution. "Simple solutions are 'before a goat is revealed' false solutions" - then Carlton's non-paraphrased actual quoted solution stating 'the host has opened a door to reveal a goat' is removed. The 'Truth'? Don't make me laugh. Glkanter (talk) 06:25, 30 September 2010 (UTC)

If it's not 50/50, then what is it?

Selvin said the host chooses randomly when faced with 2 goats. I will infer he meant the contestant is aware that the host chooses randomly.

PpWhitaker/vos Savant, of course, didn't mention it in her column.

No other values (a stated host bias) have ever been given as a premise by any other sources as being a part of the MHP, although some sources say we can't just assume it's 50/50.

So, using Chun's conditional tree, if despite Selvin's statement (and a lot of other logical arguments), we can't assume 50/50, what values *are* used? Per which source? Will the contestant be aware of these figures? Glkanter (talk) 13:32, 25 September 2010 (UTC)

If it's not 50:50 it's something else. Call it a:b for host opens door 2 versus host opens door 3, where a and b are whole numbers between 0 and 100 adding to 100, for the case that the car was behind door 1 and the guest chose door 1. Call it c:d in the similar door 2 case, call it e:f in the similar door 3 case (door 1 : door 3, and door 1 : door 2). Assume that all doors were initially equally likely to hide the car. It follows by simple book-keeping / decision trees / odds form of Bayes law (and confirmed by many reliable sources) ... that the conditional odds that the car is behind door 2 versus behind 1, given that guest chose door 1 and host opened door 3, are 100:b. Since b is a whole number between 0 and 100, it follows that even if you don't know it, you'ld be wise to switch. You know for sure that b is not larger than 100. I think that Glkanter, like many reliable sources, understands probability in a rational subjective Bayesian sense. So he does "know" all probabilities, by introspection. They reflect his information. Given vos Savant's words, we don't know anything to prefer any of the doors, the door numbers are explicitly stated to be totally arbitrary labels ("say Door 1 and say Door 3",) so it is, for him, and all such sources, 50:50. However there also exist a lot of reliable sources for which probability is out there in the real world, in the biophysics, say, underlying Monty's brain processes. For such sources probabilities exist out there but are necessarily often unknown to the person who is considering them. I also don't know Monty's brain mass and the temperature on the stage and whether his dinner agreed with him or not, or if he carries that gene for migraine. Fortunately in this case there is a rational and unequivocal solution for which it is not necessary to know these probabilities! The answer is "switch"; always switching beats any other strategy, mixed or pure. To conclude this, it is necessary and it is sufficient to know that the quiz team hides the car uniformly at random. This is the "IF" of what I called "Due". Reliable sources a-plenty who promote this point of view and who do the computation.

Fortunately for us, Islamic scientists gave algebra to the world at a time when "the West" was a load of barbarians! Gill110951 (talk) 08:44, 28 September 2010 (UTC)

Richard, approach this as a logical question only, and for our discussion purposes only. What numeric values, other than 50/50, could be used in the conditional decision tree, based on the sources? Not algebraic equations. Numeric values. The 50/50 comes from Selvin, and from symmetry, indifference, being a game show, etc. Once the choice is made to use that decision tree method to solve the problem, what other values can be used to choose between 2 goats, based on the sources?

There are no such values. No source has, as a premise, any other values. The only possible values for the MHP are the 50/50 for the reasons I just gave. Otherwise, not even 2/3 is a valid outcome. This is what's called a 'logical argument'. In many cases, it's how disputes are settled. Glkanter (talk) 09:28, 28 September 2010 (UTC)

Based on the Parade version of the problem and vos Savant's subsequent clarifications in Parade, Morgan et al. say the values are p/q where they are both non-negative and p+q=1. They also explore extreme variants where p=0 and q=1. Gillman says the same thing based on vos Savant's problem description. Rosenthal explores the p=0 and q=1 case as well, and also generalizes this to p and q are unknowns in the range [0,1]. Lucas, Rosenhouse, and Schepler explore the p=0 and q=1 case. Eisenhauer examines the p=0 and q=1 case. Many of these sources also examine the variant where the host opened a door completely randomly but happened to not reveal the car. -- Rick Block (talk) 15:12, 28 September 2010 (UTC)

Are you saying 0 and 1 are valid values for p & q in the MHP? Did Selvin or vos Savant provide these? Is each source allowed to add his own unique premises to the problem? Is there an 'end date' on that, or is the problem still open to new premises? Why are these later editors simply allowed to ignore and overrule Selvin's unambiguous statement in his 2nd letter to the American Statistician that the host chooses equally between the 2 doors when faced with 2 goats? If 0 & 1 are valid values for the MHP, why doesn't the Solution section have a second decision tree with 0 & 1, rather than 50/50? By the way, do those sources say the contestant is aware of the 0 & 1 preference? Glkanter (talk) 15:37, 28 September 2010 (UTC)

What does "valid" mean in this context? You asked what values could be used, based on the sources. I'm telling you what values various sources use. In the context of the article, "the MHP" is not narrowly defined as Selvin's original problem, or the problem with his addendum, or the Whitaker/vos Savant/Parade version. There's a usual interpretation, according to sources like Barbeau, which matches Selvin + addendum. In this usual interpretation the host's choice between goats is 50/50. The article makes this clear. The "Solution" section (at least primarily) focuses on this interpretation. Many sources start with the Whitaker/vos Savant/Parade version (not even including vos Savant's later clarifications in Parade), rather than Selvin's. This version is not very specific. So, later authors have filled in the blanks however they like.

None of the sources I mentioned distinguish "what is" from "what the contestant knows" - i.e. they're addressing the question of what the probability is given the assumptions they specify. The question of what the probability is from the state of knowledge of the contestant requires explicitly saying or making further assumptions about what the contestant knows. Although we've discussed this issue on these pages (extensively), I don't recollect any source whose main approach explicitly focuses on "what the contestant knows" as opposed to "what the puzzle solver knows" (i.e. everything said in the problem statement). vos Savant alludes to this in one of her followup columns where she mentions that a "little green woman" emerging from a UFO at the point the contestant is deciding has a 50/50 chance of randomly picking the winning door because she (the alien) doesn't know the history of the setup. In this situation the state of knowledge of the alien explicitly differs from ours (the puzzle solver's, which presumably matches the contestant's). In their rejoinder to vos Savant's reply, Morgan et al. also allude to this: "One may argue that the information necessary to use the conditional solution is not available to the player" - however they go on to say "but this does not change the aforementioned fact [that Whitaker's question is a conditional probability problem]". -- Rick Block (talk) 17:00, 28 September 2010 (UTC)

All accurate, I suppose. All meaningless vis-a-vis the MHP, I'm certain. Totally void of Wikipedia policy and logic. And, of course, you failed to respond to a single question I posed. Glkanter (talk) 17:12, 28 September 2010 (UTC)

I think I responded to all your questions and that any reasonably intelligent third party reading this exchange would agree. -- Rick Block (talk) 18:43, 28 September 2010 (UTC)

I mainly agree with Rick, here. Moreover, reliable sources *do* add all kinds of assumptions to vos Savant/Whitaker, and disagree about which assumptions are natural, which are not. It's not forbidden. No-one has sole rights to the one and only true MHP, thank heavens. People who write books about MHP do discuss many different versions of MHP. The wikipedia page on MHP has to, as well. Fortunately we don't have to fight one another as soon as we understand that the more assumptions you make, the stronger the conclusions you can draw ... but that you thereby more and more limit your scope.

However the little green woman story is not relevant here, and Morgan et al's dogmatic opinion that MHP is a conditional probability problem is merely that: a dogmatic opinion. How do I know that? A) because other reliable sources have different opinions, and B) because Morgan et al give no argument for their opinion, they merely assert it, relying on a position of authority. Which they show they don't deserve, by making elementary mistakes in their algebra. Moreover the fact that their paper got published with mistakes and all, which no-one pointed out for 20 years till a couple of wikipedia editors noticed that, shows that peer review is not a guarantee of quality. Secondly, there are lots of reliable sources who do not fill in many blanks. Why? Because you don't have to make all kind of guesses in order to come up with a good justification for "switch". There is value in being able to advise the player in the situation when the player is not able to make all kinds of assumptions. Gill110951 (talk) 06:32, 30 September 2010 (UTC)

About a day late, and a dollar short, here, Richard. I quoted, and criticize, that exact passage just below. But, somehow, you managed to 'entirely agree with Rick'. And they call their opinion a 'fact'. Nice job. Glkanter (talk) 06:46, 30 September 2010 (UTC)

I agreed with the main thrust of Rick's criticism of Glkanter's claim that it has to be 50:50. I later noticed that Rick had slipped in a lot of side stuff which is irrelevant and moreover misleading so I changed "I totally agree" to "I mainly agree". Pythagoras proved Pythagoras' theorem (a squared plus be squared equals c squared). He didn't have to start giving three whole number values to a, b and c. Just as triangles can have all kinds of different length sides, so also can frequentist probabilities have all kinds of values. Thanks to algebra, we don't always need to know them in advance in order to draw rational decisions. Subjectivists are only interested in the internal consistency of their subjective beliefs. Frequentists are concerned with having to face up to various unknown realities, and want to do the best that is possible whatever the situation. Amusingly they often come to the same conclusion. That is because a frequentist minimax solution (minimizing the worst damage that can be done to you) is also a subectivist (Bayes) solution with a least favourable prior. Total lack of information is the least favourable situation for the subjectivist. In a symmetric problem least favourable distributions must satisfy the same symmetries, hence in a problem like this they have to be uniform. It's not a coincidence that everyone knows the answer is "switch" but people have totally different ideas about what probability means and hence what constitutes a solution to MHP. If only all editors of wikipedia MHP could get the logical facts straight in their mind and get accustomed to different ideas of what probability is all about... End of today's lecture. Gill110951 (talk) 13:23, 30 September 2010 (UTC)

Except, all I said was any discussion of host bias did not belong in the Solution section, as there are no such solutions by any reliable sources. I was also documenting why it IS NOT an NPOV violation if criticisms of the simple solutions do not accompany the simple solutions in the Solution section, which seems to be the only objection given. Do you agree or disagree with those 2 statements, Richard? Glkanter (talk) 15:16, 30 September 2010 (UTC)

When a word problem begins, "Suppose you're on a game show...", as both Selvin's and vos Savant's do, the only State of Knowledge that is of interest is the contestant's. And there are no premises given in the problem saying the host's bias is 0 & 1. Every conditional decision tree uses 50/50. Variants and academic examples are not the MHP. Even Morgan finally admitted that 2/3 & 1/3 is the answer. All that other stuff should be kept far from any Solution section. Glkanter (talk) 02:23, 29 September 2010 (UTC)

As both styles of solutions require the same 50/50 premise to work, and it is one of K&W's 'normalized' premises, and it was one of Selvin's premises, there is no difference in the host's behaviour between the published solutions when solving the MHP problem. Therefor, there is no reason, or need, to discuss a non-existent hypothetical difference (alleged flaw) in the Solutions section. It should come later in the article. The editor who is concerned that readers will be badly misled without the criticisms accompanying the simple solutions is relying on OR, not any reliable source. Still he insists we include his personal conclusions in the Solution section. Not only is that contrary to the reliable source requirement, it sounds like an NPOV violation to me. Glkanter (talk) 02:34, 29 September 2010 (UTC)

Yeah, here's another reason I think Morgan, et al are bums. You wrote that Morgan writes:

""One may argue that the information necessary to use the conditional solution is not available to the player" - however they go on to say "but this does not change the aforementioned fact [that Whitaker's question is a conditional probability problem]"".

What an unsupported conclusion. Contrary to every reliably sourced simple solution. A complete contradiction of Whitaker's word problem that they are criticizing which begins, "Suppose you are on a game show...". Hardly a prevailing notion worthy of prime attention in the article. Despite these facts, and even though Morgan acknowledges this flaw, you and other editors have taken a non-negotiable stance that Morgan's criticism based on this unsupported (flawed) conclusion must be prominent in the Solution section. Again, sounds like NPOV and UNDUE violations to me. Glkanter (talk) 02:57, 29 September 2010 (UTC)

Selvin, Morgan, and Subsequent Simple Solutions

This is Morgan's letter in 2010:

© 2010 American Statistical Association DOI: 10.1198/tast.2010.09227 The American Statistician, May 2010, Vol. 64, No. 2 193

Morgan, J. P., Chaganty, N. R., Dahiya, R. C., and Doviak, M. J. (1991),

“Let’s Make a Deal: The Player’s Dilemma,” The American Statistician,

45 (4), 284–287: Comment by Hogbin and Nijdam and Response

Response

[Begin Quote]"Our kind thanks to Mr. Hogbin and Dr. Nijdam for correcting our mistake. We will add that should the player have observed any previous plays of the game, those data, too, will modify the prior, and can produce posterior calculations other than 2/3 even with a symmetric prior. This, of course, is something else that we should have pursued. In any case, it should not distract from the essential fact that 1/(1+q) ≥ 1/2 regardless of q. Simply put, if the host must show a goat, the player should switch."

...

"We take this opportunity to address another issue related to our article, one that arose in vos Savant’s (1991) reply and in Bell’s (1992) letter, and has come up many times since. To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period."[End Quote]

And while Morgan may still not be aware of Selvin's 50/50 premise, we Wikipedia editors are. And there have been many simple solutions published since Morgan's 1991 paper, certainly not indicating agreement with Morgan's statement, as provided above by Rick Block:

""One may argue that the information necessary to use the conditional solution is not available to the player" - however they go on to say "but this does not change the aforementioned fact [that Whitaker's question is a conditional probability problem]"".

Morgan as a reliable source is not entitled to any UNDUE prominence in the article. Glkanter (talk) 03:18, 29 September 2010 (UTC)

In light of all of the above arguments in this "If it's not 50/50..." section, having a Solution section which presents the solutions in easy to complex order, and without criticisms or caveats about the simple solutions, can certainly not be considered an NPOV violation as claimed by some editors, and is, in fact, consistent with common publishing practices and the reliable sources, as per Wikipedia policies. Glkanter (talk) 03:41, 29 September 2010 (UTC)

Well, I've had my say on the NPOV issue. Maybe the opposing viewpoint would like to present their supporting arguments? Glkanter (talk) 04:41, 29 September 2010 (UTC)

Heck, go ahead and put Chun's decision tree as the first solution of a single solution section. But no criticisms or discussion about host bias, or anything but the 50/50 solution itself. And that second table, or whatever it is, gets eliminated. Why is it in the current article, anyways? Glkanter (talk) 07:02, 29 September 2010 (UTC)

Wikipedia can't report that there is only one way to answer Whitaker/vos Savant's verbal question ("suppose you're on a game show ... should you switch?"). Some sources go back to Selvin, some sources do not. Some sources like to solve it with subjective probability and they have equal reason to believe the host would open either door when he has a choice, so they use 50:50. They also have equal reason to believe the car is initially behind any of the three doors. So they initially choose Door 1 because 1 is their lucky number. Because of the symmetry of their knowledge the door numbers are irrelevant to them. For them, after Monty's opening of a door, it is two times more likely that the car is behind the "other door" than their initial choice, so they switch. The door numbers are irrelevant, just like the day of the week (they also don't condition on it's being Wednesday). I don't know if they would actually win the car 2/3 of the time in many real repetitions, whether on Tuesdays of Wednesdays. I actually rather doubt that they would win the car in exactly 2/3 of the situations when they had chosen Door 1 and Monty had opened 3, but they have totally no interest in those questions anyway. Other sources like to solve the problem with frequentist probability and for them the probability that Monty opens Door 3 when he can choose between 2 and 3 is typically unknown. They don't know how his brain works. Fortunately for them there is still an unequivocal answer to the question whether or not they should switch, provided that they know that the car was initially hidden at random, and it's "yes, switch". Their (correctly argued) reason being that all conditional probabilities favour switching. Not only does always switching give you the car with unconditional probability 2/3 in this situation, but we also see from inspection of the conditional probabilities that no other strategy (sometimes switching, sometimes not, depending on the door numbers) is better (Proof: by the Law of Total Probability). Some sources are only prepared to assume that the player's initial choice has probability 1/3 to hit the car. They correctly deduce that switching gives you the car with probability 2/3, hence they argue "switch". Given their assumption, that is all that they can can conclude. For some sources, that assumption is legitimate, because they imagine that the player chose his door at random to start with. Such sources don't want to make any assumptions at all about how the car is hidden and how Monty opens a door when he has a choice. Because they are using frequentist probability they don't know if the car was hidden completely at random. Vos Savant does not give this information.Gill110951 (talk) 16:05, 29 September 2010 (UTC)

I never expected Straw Man arguments from you, Richard. I have written countless times that all reliable sources, no matter how moronic, are to be included. The stuff I just bolded shows I don't even insist that the simple solutions come first. I object to stuff that isn't solutions being in the solution section. Read the current version's Conditional solution section. Of course that can't be the first solution given! Look at all that extraneous bs! Nothing to do with Chun's tree at all. The narrative should be one brief paragraph, not 5 or 6 paragraphs like it is now. And look at that mess of an OR table that follows. 7 images, comprising 21 doors, and all it really has is the same 4 numeric values as the tree. I'm trying very hard to make the the Wikipedia article that a reader will encounter a lot better, I'm not arguing some esoteric academic points here, Richard. We agree that Morgan's (and the opposing editors') claim that it must be solved conditionally is unsupported and contradicted by many reliable sources, yet you now 'mainly agree' with Rick? No, Richard, you don't. Glkanter (talk) 15:16, 30 September 2010 (UTC)

I agreed with Rick on the specific point you two were arguing about, not on his general point of view concerning how the MHP article should be written. Please let's stay focussed.

The claim that MHP *must* be solved with conditional probability is bs in the opinion of many sources. But it's a fact that many sources *do* solve it with conditional probability. Also the solutions where the host choice is not 50:50 are solutions to the problem "switch or stay?". They assume that all doors are initially equally likely to hide the car, and no more than that, and conclude that the posterior odds for Door 2 are 1:q, which supports switching even if you don't know q. The assumptions (and non-assumptions) are meaningful for people who use probability in a frequentist sense. This is not a common or very popular approach so of course the general reader needn't be botheredtoo much by it. In the symmmetric case you don't need to do conditional probability calculations to get the conditional answer 2/3, since symmetry does it for you with no sweat. In the non-symmetric case the smart way to do it is with Bayes theorem for the posterior odds, almost no sweat, and everyone can benefit from learning "posterior odds equals prior odds times likelihood ratio". Jason Rosenhouse and Jef Rosenthal (both have a book and an article) are pretty definitive sources nowadays. Rosenthal pushes the odds approach and shows how it beautifully handles many probability puzzles. We can have our cake and eat it if we focus on the facts and the sources and keep the math strictly to the minimum. The conditionalists have harmed their own cause by making it look so complicated, but it isn't complicated at all. Gill110951 (talk) 06:15, 1 October 2010 (UTC)

I guess you're saying I am wrong when I argue that every reliable source uses 50/50 when they use any values for the hosts method of choosing between two goats. Well, that's certainly how the Wikipedia article reads. The sources that say it could be any values p & q need to 1. ignore Selvin 2. ignore the rules of game shows (contestant's are not given hints as to the car's location, unless that is a rule of the game) and 3. ignore the principle of indifference/symmetry that they use 2 other times in the problem to do this. Well, we don't 'need' to ignore Selvin and the other 2 items, and shouldn't. According to Selvin, the problem relies on the 50/50 premise. So, sources that offer simple solutions are simply using Selvin's and K&W's premises. There's nothing to criticize there. Some sources do so, but they ignore Selvin and K&W. Put those sources later in the article. I suppose the p & q solution is in the current article's Conditional solution section, but that section is such a muddle, who can tell? The 50/50 criticism is equally valid (or invalid) for the conditional solutions. So any criticisms of the simple solutions for relying on that premise are equally valid criticisms for the conditional solution. Of course, Carlton's simple solution, where he conditions on the 100% likelihood of the host revealing a goat, shows the simple solutions, his, at least, don't require the 50/50 premise at all. And since no other 'host chooses between 2 goat doors' values are ever given as premises, there's nothing to criticize. Glkanter (talk) 06:56, 1 October 2010 (UTC)

Selvin's second letter computes a conditional probability and uses all the conventional equal probability premises. His first letter computes an unconditional probability and doesn't use the 50/50 premise, though he did write it down, for no reason at all. Instead he shows that in 6 out of 9 equally likely cases switching gives you the door. Apparently he takes the initial choice as random and the location of the car as random and simply uses the fact that switching gives you the car if staying would give you the goat and vice versa. The second letter is written as if he is giving the same solution again, but he isn't, he's giving the conditional solution. At the end of the note he credits Monty Hall himself with the shortest and best solution of all, namely that the chance you hit the car first time isn't changed by seeing him open a door and show a goat. This is the unconditional solution only using the fact that your initial choice has chance 1/3 to hit the car. In short, Selvin's two notes are quite a mess and contributed to the mess we are in today, here. He doesn't even realise that he is giving different solutions using different approaches and different assumptions.

I think your response is in the wrong place, but anyways... The paragraph that I'm challenging Selvin's attribution to is a criticism of the simple solutions as solving the wrong problem, and [paraphrasing] 'missing certain premises'. Selvin includes the premises. To suggest that Selvin is criticizing his own simple solution doesn't make sense. Plus, as you point out, he explicitly commends Monty Hall's simple solution in his 2nd letter. Selvin is not a critic of the simple solutions, as the article claims. Glkanter (talk) 09:53, 1 October 2010 (UTC)

If you don't use subjective probability you cannot use the principle of indifference to fix physical probabilities which you don't know. Anyway, the principle of indifference is much discredited these days even by dedicated subjectivists.

Let's be generous to Selvin and add him to the lists of sources who (sensibly, IMHO) see value in different approaches, simultaneously.Gill110951 (talk) 09:37, 1 October 2010 (UTC)

Further point of order

I thought we all agreed we wouldn't be editing the article without consensus here. Glkanter has removed a citation [11] (a change we've never talked about) and inserted the quoted version of Carlton's intuitive explanation as a solution [12] (which we've talked about but have not reached consensus). Rather than simply revert Nijdam subsequently changed the description of the Carlton quote to match what Carlton calls this explanation, but IMO both of the original edits are distinctly contrary to the spirit of the process we're engaged in.

I request Sunray to please comment on this. -- Rick Block (talk) 14:01, 29 September 2010 (UTC)

I have reverted to the status quo ante Glkanter's and Nijdam's edits, and applied for full protection, while we wait for the mediator to comment. glopk (talk) 15:27, 29 September 2010 (UTC)

The article is now locked until Oct 6, and I doubt the mediation will be done by then. Can we avoid another round when the lock expires, please-please-pretty-please-with-cherry-on-top? glopk (talk) 22:11, 29 September 2010 (UTC)

There is something of an assumption of bad faith in your request Glopk. I for one do not intend to make any controversial adits to the article. Provided we all operate a one-revert rule there should be no problem. Martin Hogbin (talk) 11:30, 1 October 2010 (UTC)

I requested a lock because I could clearly see coming another quote-vs-paraphrasis revertfest. The admin obviously agreed (after due consideration of the recent edit history, one hopes), and I note that Glkanter's SCREAMING REACTION TO THE REVERSAL supports my interpretation and preventive action. I have no idea of what this "something of an assumption of bad faith" might be, so I won't respond to it - if you have accusations to make, please state them clearly and avoid innuendo. glopk (talk) 15:20, 1 October 2010 (UTC)

I was referring to the assumption that, if the article was unlocked, someone would immediately begin editing in a controversial manner. Far better that we agree to all behave in a responsible way so that the article could be edited in an agreed way. Martin Hogbin (talk) 16:23, 1 October 2010 (UTC)

Agree that it'd be a far better way to proceed. In fact, come to think of it, I myself (implicitly) agreed to that when I entered the mediation, and have followed through. I suggest that, sooner than questioning something about my assumptions, you may instead direct your questions to those editors who have not. glopk (talk) 16:37, 1 October 2010 (UTC)

Article unlocking

I am proposing that we ask to have the article unlocked on the basis that everyone involved in this dispute agrees to work to a one-revert rule. That is to say, if an editor can make a change; if anyone reverts it, the article stays as it is. The same or a very similar change must not be made again unless everyone agrees.

Can we all agree to this? Martin Hogbin (talk) 16:45, 1 October 2010 (UTC)

• Agree, but with a caveat: only if a majority (at least) agrees as well, AND this majority includes both the other editors (Glkanter and Nijdam) involved in the edits immediately preceeding the lock. Otherwise I prefer to maintain the current no-edits de-facto rule, if necessary enforced with protection. glopk (talk) 17:10, 1 October 2010 (UTC)I

It's immaterial to me either way. My edits are always reverted, regardless of their validity. Glkanter (talk) 17:19, 1 October 2010 (UTC)

Glopk, I do not think we should be picking out individual editors. What I was proposing was essentially agreeing to lock the page against all controversial edits, but allowing those that we all agree to. Martin Hogbin (talk) 19:06, 1 October 2010 (UTC)

You seem to believe, with no factual basis that I can see, that all editors will respect the view of the majority and behave responsibly on an issue as simple and clear-cut as this one. Allow me to remind you what happened yesterday: a controversial paragraph (read: a previously edit-warred upon paragraph) was edited while the mediation is actively ongoing, when it is obvious to any reasonable person that no editorial consensus has been reached on that very paragraph, and edited in a way that didn't even try to be consensual. After the reversion, Glkanter wrote this, this and that. This represents in my view a clear disregard (I don't care if willful or ignorant) of the minimal amount of trust that's needed for this mediation to continue in any useful sense. Therefore, unless all the editors involved in yesterday's episode (myself included) are willing declare for the record that they understand and agree with the rule you are proposing, there is absolutely no point in pretending that such a rule would be in effect. glopk (talk) 20:16, 1 October 2010 (UTC)

Your level of righteous indignation is comical. The edits I made were valid, for the reasons I gave. The attributions deceive the reader, and are contradicted by the sources referenced. That's the only measure that should be applied. This mediation seemingly will never end, so applying your 'rules' regarding edits to the MHP article is the same as endorsing the current heavily POV-laden version of the article. Not me. Glkanter (talk) 20:39, 1 October 2010 (UTC)

I've already asked Sunray to comment on this incident - how about if we let him (? - come to think of it, I don't actually know whether Sunray is a he or a she) decide whether or on what terms the article protection should be lifted? -- Rick Block (talk) 21:07, 1 October 2010 (UTC)

The point is that if we all agree to adopt a one-revert rule there should be no problem. Let me start with the question in reverse. Is there anyone who is not prepared to accept a one-revert rule? Martin Hogbin (talk) 21:22, 1 October 2010 (UTC)

I see two points being raised here

1. The article should not be edited during the mediation unless there is consensus to do so.
2. The article could be edited if editors all agreed to adopt a one-revert rule.

We need to determine whether all participants would agree to a one-revert rule and in the meantime stick with the only edits by consensus rule. I will request that the lock be extended until there is consensus to unlock it under one of these two conditions. Sunray (talk) 21:45, 1 October 2010 (UTC)

Lemme put it this way: The next chance I get, I will remove Selvin and Grinstead & Snell as references for that conditional solution hogwash. Because they do not say those things in their sourced materials. And anybody willing to spend 3 minutes on this mediation can see that very clearly in the sections that follow this one. Glkanter (talk) 00:16, 2 October 2010 (UTC)

Sunray, given this last comment of Glkanter, implying lack of unanimity on rule (2) above, I ask that rule (1) above remain operative. Please request that the lock be extended for as long as needed.

Separately, and also based on his comment and recent behavior, I ask you and the other editors if user Glkanter can be considered as still participating to this mediation. I now believe it is a legitimate point to raise. glopk (talk) 00:38, 2 October 2010 (UTC)

Why, Glopk, I've been expecting you! Glkanter (talk) 00:47, 2 October 2010 (UTC)

Based on the above discussion, I think we shall just keep the article locked for now. Sunray (talk) 22:27, 2 October 2010 (UTC)

USING SELVIN AS A REFERENCE FOR MORGAN'S "IT IS A CONDITIONAL PROBLEM" IS INCORRECT AND GROSSLY MISLEADING TO THE READERS

Selvin should not be used as a reference, as he gives a simple solution following his problem statement. His name should be removed immediately. An edit request should be made without hesitation. Glkanter (talk) 09:21, 30 September 2010 (UTC)

The text in the article is "The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability". The reference is to (among others) Selvin's second letter, in which he says [13]:

The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random. An alternative solution to enumerating the mutually exclusive and equally likely outcomes is as follows:

A = event that keys are contained in box B

B = event that contestant chooses box B

C = event that Monty Hall opens box A

Then

P(keys in box B | contestant selects B and Monty opens A)

...

He's saying both that additional reasoning is required to justify his original (simple) answer, and (in the alternate solution) that the probability the keys are in the player's chosen box given which box the host opens is a conditional probability.

It seems to me using this source for the statement in the article is not incorrect, or grossly misleading, or worth shouting about. -- Rick Block (talk) 06:11, 1 October 2010 (UTC)

Yes, he says the 50/50 premise is required. Which should eliminate all the discussion in the solution section about any host bias. But he first solved his own word problem with a simple solution. Therefore, he is not saying 'it is a conditional probability'. That's the misleading part. He later offers a conditional solution, but doesn't in any way say his simple solution is incomplete or flawed. So, when it supports a certain POV (he gives a conditional solution in his 2nd letter), Selvin's letters are referenced, but when Selvin's letters don't support that POV (the 50/50 premise he states, he uses a simple solution) his letters are ignored. Using Selvin, who uses a simple solution to solve the problem he originated, to support 'it is a conditional probability' is incorrect. Glkanter (talk) 06:26, 1 October 2010 (UTC)

Selvin is an unfortunate source since his two short notes are full of inconsistencies. Additional reasoning is *not* required to justify Selvin's simple answer. However he does expound at length on the conditional solution so it isn't grossly incorrect to include him among those who think the conditional approach is important.

To justify that the unconditional probability that switching gives the car is 2/3, *all* that is needed is the assumption that the player's initial choice is correct with probability 1/3. Selvin confirms with approval Monty Hall's own observation of that fact.

Selvin's papers are important milestones in the history of MHP, but he's a bad source for sharp and clear logical writing and reasoning. We don't *have* to copy all the careless mistakes or sloppy arguments in the prehistory of MHP. We have to select and organise the material in accordance with WP policies, in a way that serves the interests of readers. Rick, I am so curious to know whether or not you agree with my "uno, due, tre" summary of Eternally True Facts, which I think provides a framework in which we can locate and correlate all the approaches to MHP which are important for the article. Strictly speaking wikipedia editors need not take any account of the logic of what they write, but it confuses readers when the text of a wikipedia article clearly includes logical fallacies. It provokes readers to become editors! That's why we are here now, because we have to cut the crap out of the article. Only then will it become stable. Gill110951 (talk) 06:54, 1 October 2010 (UTC)

I don't think a source (Selvin) that clearly states the 50/50 premise, offers a simple solution, and concludes from it that the odds are 2/3 & 1/3 should be used to support the statement "...but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability". I find that illogical. Otherwise, I have never before read on these talk pages or in any source that Selvin's letters are unreliable in any ways. I *have* read that of Morgan. Glkanter (talk) 07:06, 1 October 2010 (UTC)

Selvin's letters are brief, undogmatic, disorganised. In the first he presents an unconditional solution without making any use of the 50:50 assumption which he does write down for no good reason at all. In the second he presents the conditional solution in the fully symmetric case without commenting that it is a totally different approach but giving the reader the impression that he is repeating his earlier argument. At the end of that second note he also reproduces the short sharp simplest solution in which you only assume that your initial choice has chance 1/3 to be correct. Selvin is a reliable source for the peaceful coexistence of all solutions. No need to criticise any solution which is logically consistent. Gill110951 (talk) 10:59, 1 October 2010 (UTC)

Grinstead and Snell Only State That Whitaker's Is A Conditional problem

G&S describes these premises:

"We will assume that the car was put behind a door by rolling a three-sided die which made all three choices equally likely. Monty knows where the car is, and always opens a door with a goat behind it. Finally, we assume that if Monty has a choice of doors (i.e., the contestant has picked the door with the car behind it), he chooses each door with probability 1/2. Marilyn clearly expected her readers to assume that the game was played in this manner."

G&S offer their own 'before' problem statement, calling it a "simpler, related question." Then they give a solution similar to Carlton's simple solution, but without the host opening a door to reveal a goat. Then they say:

"This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3."

Then they offer a decision tree with every possible combination (not just doors 1 & 3), all giving the 2/3 & 1/3 results. In order to show these two solutions are different, they replace vos Savant's 50/50 premise with:

"Recall that we assumed in the original problem if the contestant chooses the door with the car, so that Monty has a choice of two doors, he chooses each of them with probability 1/2. Now suppose instead that in the case that he has a choice, he chooses the door with the larger number with probability 3/4."

These are the statements G&S are attributed to in the current article:

"The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability (Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b)."

The only part of that G&S discuss is that they call Whitaker's question a conditional problem. None of that other stuff is theirs. Glkanter (talk) 11:50, 1 October 2010 (UTC)

And once again this clearly shows that any "conditional approach", taking the door opened by the host as a new "condition", only can make any sense in case that some unthought whistle-blowing host is assumed who, by opening of one door, might have been giving additional secret information on the secret actual location of the car, unable to change it's respective location. Gerhardvalentin (talk) 12:23, 1 October 2010 (UTC)

If probability is in your information, yes, Gerhardvalentin. But if probability is in the physical world, e.g., we think of Monty's brain as a random number generator, then there is no reason at all to expect Monty's brain to generate perfect unbiased random choices. He can have a bias even if he intends to be unbiased. Gill110951 (talk) 13:27, 2 October 2010 (UTC)

Richard, I was referring to the aforementioned statement "Now suppose instead that in the case that he has a choice, he chooses the door with the larger number with probability 3/4." And I am referring to that existing 100 million protocol-list of an unbiased host. In checking this protocol, you can bet s.o. or another every now and then will be tracing to find patterns in droves that could be interpreted as to be "some host's bias". I'm even sure they will be tempted to "determine" this bias, but without any validity, because he isn't biased at all, not a bit. "Information out there" can be deluding, so you better be careful. Please beware, no predictions! Because the deception is already on the lookout.

So you have no choice but to invent your own biased host, if you like him to be biased. Then you can determine how biased "your host" is, then you will know exactly about his given bias. Ruma Falk says "if he is biased and you KNOW of this bias." But if he isn't biased or you do not really KNOW of this bias, i.e. without "inventing your own host" you better take the "weekday" likewise, instead of the door he opened, as a "condition" in teaching conditional probability. Just in order not to cause confusion. You should not confuse your students. That's why textbooks have to invent their own host, and never can be using the host of the famous MHP-question. Gerhardvalentin (talk) 13:53, 2 October 2010 (UTC)

Richard, but so might the producer, who decides how to initially place the cars actually have a bias. If we work on this premise the problem is rather boringly insoluble. Martin Hogbin (talk) 17:41, 2 October 2010 (UTC)

Who is supposed to know about such producer's bias in locating the car? If such producer's bias isn't known, it can't matter. Relevant starting point: The player, initially selecting her door, had "no knowledge at all" on the location of the car. Other evidence? Kind regards, Gerhardvalentin (talk) 18:09, 2 October 2010 (UTC)

Gentlemen, please! I'm trying to remove unsupported bs from the article! Enough of your OR and other musings! Is it appropriate to attribute these statements:

"The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability (Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b)."

to Selvin, G & S, and Gill? Glkanter (talk) 18:17, 2 October 2010 (UTC)

Yes Garry, that's exactly what we all expect the article to be: Less confusing weasel wordings but clear presentation of the 50:50 vs. 2/3 dilemma, see The MHP in teaching conditional probability, The question asks for a decision, Conflict textbook-conditionalists, improve readability and so on. Gerhardvalentin (talk) 18:47, 2 October 2010 (UTC)

Well, one way to reach these goals is to show that the offending statements in the article are not supported by the claimed reliable sources, and are being given UNDUE WEIGHT. Then the statements can be removed, or relocated, as per Wikipedia policy. Do Selvin, G &S, or Gill support those statements? I think it's clear they do not. Glkanter (talk) 18:58, 2 October 2010 (UTC)

More to the point, gentlemen, are Grinstead and Snell being used properly in the article by being attributed to this:

"The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability (Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b)."

Is "Gill 2009b"'s attribution here appropriate? Glkanter (talk) 13:44, 2 October 2010 (UTC)

Actually, all of that "other stuff" is theirs. Their simple analysis is an analysis of "always switching" which they show is 2/3 (supporting "The popular solutions correctly show that the probability of winning for a player who always switches is 2/3"). They then immediately say what you've quoted above "This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3." supporting ", but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability." They even provide a counter example where the probability of winning is 2/3 for all players who switch but NOT 2/3 given which door the host has opened. What of this do you think they're not saying? -- Rick Block (talk) 13:59, 1 October 2010 (UTC)

All they say is it's a conditional problem. They say it's 50/50 per vos Savant as one of their premises for the MHP. They mention no 'additional reasoning'. Only when they change the problem from the MHP to something different with "Now suppose instead that in the case that he has a choice, he chooses the door with the larger number with probability 3/4." do the two solutions differ. They say nothing about the MHP being dependent on which doors are selected or opened. Their full blown tree shows every combination, as per their interpretation of vos Savant, has the same probabilities. Glkanter (talk) 14:56, 1 October 2010 (UTC)

This is what they say makes it conditional: "Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3." Not all that other stuff. Glkanter (talk) 15:01, 1 October 2010 (UTC)

THE PARAPHRASING OF CARLTON'S SIMPLE SOLUTION DELETES HIS 'AFTER THE HOST REVEALS A GOAT' STATEMENT AND MISLEADS THE READER

Carlton is clear that he is offering a solution to the 'after a host has revealed a goat' problem. The sources who claim the simple solutions are solving a different problem, one where the contestant decides 'before the host has revealed a goat' are contradicted by this. Omitting this portion of Carlton's solution is inappropriate, and dishonestly favors a POV. An edit request should be made immediately. Glkanter (talk) 09:26, 30 September 2010 (UTC)

Who supports putting the Conditional Solution section before the Simple solution section?

Just the way they are in the article today, let's just go ahead and switch the 2 solution sections. The reader will be better served this way, right? And the NPOV violations will no longer exist, right? Who's with me? Glkanter (talk) 15:57, 30 September 2010 (UTC)

Comments on a proposed solution section

Parked pending outcome of moderator discussion

Richard and Glkanter have commented on my talk page about the solution section I have proposed. To facilitate discussion about this I've moved the thread from my talk page to Wikipedia talk:Requests for mediation/Monty Hall problem/A proposed solution section, with slight formatting changes intended to make discussion of this proposal easier to follow. Sunray has asked above (#Question 9) if folks are willing to begin working collaboratively on article sections. There are now multiple new sections on this page. I think rather than each of us introducing whatever topics we want as new threads, Sunray should carefully control the spread of topics to keep the discussion focused. If we're going to proceed to discuss the text of a section, the solution section is a likely candidate. Please think about whether the format I've modeled might be useful. I'm not suggesting starting yet another thread, but if (when) we get to the point of talking about a section of the article I think this might be an approach we could follow. Sunray - can you suggest a way to proceed from where we are now? In particular, please suggest the topic or topics you'd like us to focus on and how you would like us to deal the recent proliferation of threads on this page. -- Rick Block (talk) 18:48, 30 September 2010 (UTC) Thanks Rick.Gill110951 (talk) 06:20, 1 October 2010 (UTC) Thanks indeed!. glopk (talk) 15:54, 1 October 2010 (UTC) This looks promising. I like the way Richard has made notes on the draft. A further development might be to do collaborative editing. One way to do that is to use different color fonts for different participants. A moderator is a good idea and I would suggest that it be one of the participants. As to suggesting topics: Perhaps that would be something a moderator could do? Sunray (talk) 22:15, 1 October 2010 (UTC) There's nothing 'promising' here, I'm afraid. Rick trotted out this NPOV violating Trojan Horse months ago. I think the recent comments by another editor support that viewpoint, only in a much more polite way. Glkanter (talk) 01:15, 2 October 2010 (UTC) Although some folks are commenting on this draft, I do not want to continue this discussion at this point unless we're all focused on it. I suggest we park this discussion pending the outcome of the moderator discussion. And, I'll go ahead and boldly park it. -- Rick Block (talk) 18:50, 2 October 2010 (UTC)

Moderator?

I suspect there's little chance we could agree on any of the participants being a moderator. The only one I can think of who might possibly be acceptable to everyone is Kmhkmh. I guess the first step is to ask whether Kmhkmh would be willing, or if there's anyone who would object to Kmhkmh acting as moderator. Unless everyone agrees on Kmhkmh I think we'll have to look elsewhere. Isn't this typically a role a mediator serves? If not, we could perhaps try to find an uninvolved admin who might be willing, or possibly a volunteer from WP:WikiProject Mathematics (although I highly doubt anyone there not already participating would be willing, and even if they were they'd in all likelihood bring their own baggage as well). -- Rick Block (talk) 01:08, 2 October 2010 (UTC)

I think Martin would be a good moderator. He keeps calm and polite always, is thoughtful and sensitive, writes clearly and concisely, is amenable to compromise. Gill110951 (talk) 08:47, 2 October 2010 (UTC)

Thanks for that glowing reference Gill, I will remember you when I am looking for a job ;-) I actually think that both Kmhkmh and myself are too partisan to be good moderators. Maybe we could have some kind of joint role? Martin Hogbin (talk) 12:49, 2 October 2010 (UTC)

Yes, Kmhkmh or Martin likewise, both able to classify the aspects of sources, taking care of their various perspectives. Gerhardvalentin (talk) 09:45, 2 October 2010 (UTC)

Honestly, I don't see why a moderator is needed at all. How about agreeing to follow some simple debating rules, e.g.: (1) talking stick - you get one chance to intervene, then shut up until everyone else has responded, declined to respond or done nothing for a day or two. (2) 300 characters limit for all responses, as a way to force everybody to stay concise focused, (3)... ? glopk (talk) 18:25, 2 October 2010 (UTC)

I think this could work combined with explicitly parking side issues that might come up. 300 characters is a rather draconian limit - maybe 300 words? -- Rick Block (talk) 19:00, 2 October 2010 (UTC)

I was referring to "talk", not "article edits". A similar "rule" I'd gladly follow: 300 char/comment plus half the words edited. Direct source quotes don't count. This favors focus and edits above chatter. No magic in the 300 char limits, let's just pick a number that's about 2-3 text lines tops. glopk (talk) 00:03, 3 October 2010 (UTC)

There we go - 298 chars :-) glopk (talk) 00:03, 3 October 2010 (UTC)

Well thanks for the confidence that some express towards me, but I don't want to work as moderator/mediator here. First of all I've been involved at least partially as party in the confliczt for quite a while and second I have no particular experience with/knowledge of the mediation process in WP. In other words aside from being considered biased or partisan by some I also would be prone to commit procedural mistakes or errors. --Kmhkmh (talk) 19:33, 2 October 2010 (UTC)

I didn't mean to imply that we needed another mediator. By moderator, I intended to mean (though realize I wasn't clear on this) a discussion leader. The role would be to assist in choosing topics, checking consensus, and keeping the discussion moving. This role would be well-suited to two participants. Kmhkmh and Martin were proposed. Would you consider two be willing to co-moderate? Sunray (talk) 21:58, 2 October 2010 (UTC)

Oh I see, I may have partially misunderstood the suggestion, but i have to decline anyway. However I'd like to suggest somebody else who might be suited for the task: User talk:Tsirel. He's a distinguished math prof., who shortly participated in the discussion without taking sides or pushing POV. He might be ideally suited to be a moderator provided he has time/interest to help out.--Kmhkmh (talk) 22:27, 2 October 2010 (UTC)

I should add per glopk's comment. Continuing with the talking stick approach makes sense and a mediator is not obligatory, especially if participants are making an effort to keep the discussion focused. Sunray (talk) 22:01, 2 October 2010 (UTC)

The way I read things, the 'moderator' concept is not in step with what many editors have in mind. Maybe you could address my recent edits showing that much (all ?) of the Conditional solution section is not supported by the sources being attributed to it? The object is to edit the article in a consensus manner to make it more readable and better reflect the sources. I think bringing the article in line with what the sources actually, and clearly, say will accomplish this, based simply on Wikipedia policies and the sources. Maybe even bringing this mediation to a conclusion. Glkanter (talk) 22:06, 2 October 2010 (UTC)

I agree with you that the article needs to be better brought in line with sources. And also that we can do this based on WP policies. That is why I've suggested that we go through the article on a subpage of this page. Would you be willing to try that approach? Sunray (talk) 21:54, 4 October 2010 (UTC)

Yes, simply removing from the current article the items that should not be there is, imho, 99.9% of what needs to be done to bring the article in line with the sources and Wikipedia policy. I welcome such a mediated discussion and consensus editing. Glkanter (talk) 22:02, 4 October 2010 (UTC)

I see that the "participant as moderator" idea isn't going to fly. Therefore, by default, I will attempt to moderate discussions. I should have more time this week and I am also hoping that another mediator will assist - Will, if he is free, or maybe someone else, if not. However, I do think it is best if participants choose the topics and we can proceed as long as there is consensus. Sunray (talk) 21:54, 4 October 2010 (UTC)

Gillman and Morgan

I have to wonder if the statement:

"The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability (Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b)."

is supported by any of the attributed sources other than the self-criticizing Morgan, et al. Which leaves Gillman, which is not available online. Given the unsupported and erroneous use of Selvin and Grinstead and Snell for attribution, I think it is reasonable to question whether Gillman is appropriately attributed as well. Could anyone provide the text which supports the use of Gillman as an attributed source to the above?

Also, in view of Morgan's letter of 2010, it appears even they no longer support that statement being attributed to them:

© 2010 American Statistical Association DOI: 10.1198/tast.2010.09227 The American Statistician, May 2010, Vol. 64, No. 2 193

Morgan, J. P., Chaganty, N. R., Dahiya, R. C., and Doviak, M. J. (1991),

“Let’s Make a Deal: The Player’s Dilemma,” The American Statistician, 45 (4), 284–287: Comment by Hogbin and Nijdam and Response

Response

[Begin Quote]"Our kind thanks to Mr. Hogbin and Dr. Nijdam for correcting our mistake. We will add that should the player have observed any previous plays of the game, those data, too, will modify the prior, and can produce posterior calculations other than 2/3 even with a symmetric prior. This, of course, is something else that we should have pursued. In any case, it should not distract from the essential fact that 1/(1+q) ≥ 1/2 regardless of q. Simply put, if the host must show a goat, the player should switch."

Meaning: if the host must show a goat, the relevant conditional probability is at least 1/2. Nijdam (talk) 20:25, 10 October 2010 (UTC)

I agree with Nijdam here. Gill110951 (talk) 09:38, 11 October 2010 (UTC)

...

"We take this opportunity to address another issue related to our article, one that arose in vos Savant’s (1991) reply and in Bell’s (1992) letter, and has come up many times since. To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period.[End Quote]

Meaning: the relevant conditional probability is 2/3. Nijdam (talk) 20:25, 10 October 2010 (UTC)

I agree with Nijdam here. Gill110951 (talk) 09:38, 11 October 2010 (UTC)

No, they've always said its a conditional probability. They used to say you couldn't give the probability a numeric value without using a non-existent premise of a 50/50 host bias, thus their "false solution #6". Now they weasel word it, call it a 'conditions implicit in the problem', then acknowledge its '2/3, period'. Like everybody else always knew. Glkanter (talk) 09:55, 11 October 2010 (UTC)

I don't see why some people say it is implicit in the problem that it's 50/50. You can argue that if your notion of probability is subjectivist and all you have to go on are vos Savant's words, then it's 50/50, etc. For you, the three doors are initially equally likely to hide the car, and for you, it is equally likely that the host will open Door 2 or Door 3 if the car happens to be behind the door you have chosen, Door 1. The 50/50 is not a matter of host bias or no host bias. It is a matter of your lack of knowledge of any bias, and more specifically, that your opinions about bias are not changed by renumbering the doors in any way. For any admitted likelihood of any bias in one direction you would admit of an equal likelihood of equal bias in the other direction. If the game were in reality repeated nine thousand times with the same person as host, you could expect to see significant bias in the host's choices. The host probably thinks he spontaneously opens a door "at random" each time he has a choice, but human beings are not good randomizers, that's a known fact. You would also see significant divergence from complete randomness in the location of the car too, for the same reason (unless the car was hidden using a good, physical, randomizer).

Exactly, the location of the car, and the host's choice as well, if in 1/3 he should have a choice, may "seem to show some bias", even given that there's no bias at all. You have to "know" such a bias or you are free to "assume it". But you never can be "sure to have detected it without a doubt". Gerhardvalentin (talk) 10:47, 12 October 2010 (UTC)

The usual uniform at random assumptions are only "implicit in the problem" for those who use probability in the subjective sense. Their conclusion is also only a subjective probability. Given that they were initially prepared to bet at equal odds that the car is behind any of the three doors, and because they were initially prepared to bet at equal odds that the host would open either door if he has a choice, rationally they should bet at odds 2 to 1 that the car would be behind the other door after the host has opened a door revealing a goat. But what would actually happen in many many repetitions of the game? How often would they win? How often would they win, for each configuration of door chosen and door opened? No way to tell. The only way to be sure of anything is for the player to choose a door at random and thereafter switch. He'll get the car 2/3 of the time, that's for sure. I'm not sure he'll also get the car 2/3 of the times that he chose Door 1 and the host opened Door 3, but (if I was the player), I wouldn't care either Gill110951 (talk) 11:57, 11 October 2010 (UTC)

There's no support given that the Conditional solution section, and various other sections of the MHP article should be dominated by this questionably sourced statement and POV. Glkanter (talk) 18:48, 2 October 2010 (UTC)

So should we start with the "dilemma": The MHP in teaching conditional probability, Question asks for a decision, Conflict textbook-conditionalists, Improve readability etc. ? (Apparently looking as various topics, but in reality and effectively all of this being just aspects of one single topic only):

"Is an extra lesson in lecturing how to teach and how to learn conditional probability by means of the very popular MHP-example" inescapably to dominate the WP MHP article, despite all of this never can be any helpful at all for readers to really understand the 50:50 vs. 2/3-dilemma.

Some strange historical excessive peripheral proliferation (esp. host's behavior e.g.) may then, if ever, be mentioned at the end of the article. Regards, Gerhardvalentin (talk) 22:35, 2 October 2010 (UTC)

My preference is that we simply delete the stuff in the current article that violates Wikipedia policy by being OR, but is falsely attributed to reliable sources. Glkanter (talk) 23:00, 2 October 2010 (UTC)

Since you're such a fan of direct quotes, here are several:

• "The basis to my solution is that Monty Hall knows which box contains the keys and when he can open either of two boxes without exposing the keys, he chooses between them at random" (Selvin 1975b)
• "We begin by enumerating and discussing the most appealing of the false solutions. ... The player has chosen door 1, the host has then revealed a goat behind door 3, and the player is now offered the option to switch. Thus is the player having been given additional information, faced with a conditional probability problem." (Morgan et al. 1991)
• "[vos Savant's solution] does not address the problem posed, in which the host has shown you a goat at #3. ... This is a conditional probability, which takes account of this extra condition." (Gillman 1992 - italics in the original)
• "This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3. " (Grinstead and Snell 2006:137)

We could, of course, replace the text you object to with direct quotes, or (slightly better) put the quotes in a footnote leaving some paraphrase in place. However, your continued insistence that the sources don't say what they say is well past tiresome at this point. -- Rick Block (talk) 23:16, 2 October 2010 (UTC)

Thank you, Rick. Those accurate quotes in no way support anything related to the conditional solution section other than Grinstead and Snell and Gillman saying it's a conditional problem. The idea of using Selvin, who champions 2 simple solutions to support that OR bs currently so prominent in the article is laughable. Tiresome? It's not of my doing. Glkanter (talk) 00:46, 3 October 2010 (UTC)

And there can be no question that dominating the Conditional solution section, and criticizing the simple solutions throughout the article, reflecting OR and the writings of Morgan alone is an UNDUE wight violations and a clear NPOV violation. 19 years later, Morgan has acknowledged an outright error in their paper (severely undercutting their so-called 'proof'), and as quoted above, now say 'Simply put, if the host must show a goat, the player should switch.' and 'the answer is 2/3, period'. Let's put those 2 together:

If the host must show a goat, it's 2/3, and the player should switch.

Not 'after the host has shown a goat', but 'if the host must show a goat'.

Not much different than the simple solutions, really. Not enough at all to support the bias and UNDUE weighting in the current article. Glkanter (talk) 03:07, 3 October 2010 (UTC)

"The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens. That probability is a conditional probability" is a completely true statement. The popular solutions derive, in correct ways, the unconditional probability, and of course don't need many assumptions in order to get it. If you want a conditional probability you'll have to do more work. (If you are in the completly symmetric case you'll instinctively feel that you don't need the conditional probabilities). Gill110951 (talk) 16:02, 3 October 2010 (UTC)

There are no sources who say it's true given the (K & W) premises of the puzzle as stated earlier in the article. Not even Morgan, anymore. The 50/50 host bias, the host always reveals a goat, and the host always offers the switch have all been stated unambiguously in the article. G & S state them, and say that's what vos Savant intended, and yet they are used for attribution. Selvin states them, and compliments Monty, yet he is used for attribution. Selvin does not say anywhere, 'That probability is a conditional probability'. That whole paragraph is given as the intro to lengthy, rambling, undecipherable criticisms of the simple solutions. It does not come from Selvin or G&S or Gillman or Gill (?). Only the 'That probability is a conditional probability' portion comes from G&S and Gillman. And with those premises, even Morgan doesn't argue that the solution is incorrect anymore. Unless there is "additional reasoning" that you know of, that I am overlooking? Glkanter (talk) 17:01, 3 October 2010 (UTC)

The addition reasoning required if you choose to answer the symmetrical conditional problem is simply to notice an obvious and intuitive symmetry, which immediately tells you that the door number opened by the host makes no difference to the answer. Martin Hogbin (talk) 18:27, 3 October 2010 (UTC)

The simple solutions answer a problem with the exact same premises as the conditional solutions. I don't understand what 'extra reasoning' is needed to solve the puzzle using either method. All those symmetries (required or not) are stated outright. Looks like 'additional reasoning' are some sort of weasel words, anyways. If an additional premise is required, call it a 'premise'. What else could 'reasoning' apply to? That's a heck of a way to open the Conditional solution section, isn't it? Again, is that something that the sources being attributed have said?

There is no requirement that any solution reference every premise. That the simple solutions rely on each of the premises (and do not contradict them) is not addressed or disputed in the Simple solution section. Also, there are no premises to the puzzle (K & W, G & S) that say the host must open door 3. Glkanter (talk) 18:56, 3 October 2010 (UTC)

The disputed sentence says that the probability of winning given the player has chosen door 1 and the host has opened door 3 is called a conditional probability. It seems to me that that is a totally uncontroversial statement. For those who ever followed a class in probability, it should come across as highly recognisable. It simply expresses what as far as I know is a universal convention when talking about probability in the English language. The sentence doesn't say you *have* to compute a conditional probability in order to solve MHP. It's not a normative statement. The sources listed there merely confirm that probability A given B is usually called a conditional probability and some of the sources also mention that it is universally agreed that the probability of A given B, by definition, is equal to the probability of A and B, divided by the probability of B.

The question which bothers students of MHP is whether the vos Savant's question can or ought to be solved by investigating a conditional probability. The sources (and the editors) disagree. We have to agree that that seems to be a matter of taste, convention, context, ... Gill110951 (talk) 16:06, 4 October 2010 (UTC)

Suggested discussion topic - the key issue

Attempts nibble round the edge of the subject and find an issue that we can agree on seem to have failed. That, I think, is because most editors have in the back of their minds the central disagreement here, which is between the vos Savant's solution and the archetypal conditional solution by Morgan et al, loosely referred to here as the unconditional and conditional approaches respectively.

It is clear that there is never going to be agreement between editors on this subject thus we need to find a way to work together whilst agreeing to disagree on 'The Truth'.

Most sources do not address this conflict directly. Many sources just give vos Savants solution (or other 'unconditional' solutions) whilst others just give conditional solutions (often with comments about deficiencies in the simple solutions). There is, however, one tertiary source which does address the issue of vos Savant vs Morgan directly and that is Rosenthal.It is a recent reliable tertiary source that discusses the arguments between vos Savant and Morgan at some length. It says of the simple solution to the fully symmetrical problem:

It is actually correct but I consider it "shaky" because it fails for slight variants of the problem.

I know that we can still ague about exactly what it means but can we all agree to accept this as a valid opinion of the status of the simple solution to the fully symmetric (K & W explicit statement) version of the problem? Martin Hogbin (talk) 10:34, 3 October 2010 (UTC)

'Slight variants' = 'Not the MHP', right? Grinstead and Snell say vos Savant clearly intended the 50/50 host, then say clearly they are changing that premise. There's really only one 'shaky' source that makes the whole thing out to be a conflict, anyways. In no way making that the prevalent viewpoint of the reliable sources, and making the current article biased and suffering greatly from UNDUE weight & NPOV violations. Glkanter (talk) 10:46, 3 October 2010 (UTC)

I am not sure if you are agreeing or disagreeing with using the Rosenthal statement as a reasonable description of the status of the simple solutions. Martin Hogbin (talk) 14:17, 3 October 2010 (UTC)

Oh, I agree. I've been focused on all the false attributions in the article that say the opposite, that's all. Say, I seem to recall you and I went around on this exact quote with another of the active editors some time back. My recollection is that other editor was arguing that Rosenthal's 'intent' was the opposite of "It is actually correct". I'm a little wary of going down that path again! Glkanter (talk) 14:52, 3 October 2010 (UTC)

I guess we should take the words to have their normal English meaning this time. Martin Hogbin (talk) 15:41, 3 October 2010 (UTC)

Rosenthal says, correctly, that we knew the host was going to open a door and reveal a goat, so the probability the initial choice hid the car remains unaltered at 1/3. Conditioning on a certain event does not change any probabilities. So far, he hasn't talked at all about the host behaviour and not raised the question whether we should be iterested in a conditional or unconditional probability. Then he says that in the Monty Fall case, where Monty slips and knocks open a door at random and reveals a goat, the first argument does not work. No indeed, it doesn't work: in this case we would be conditioning on an event of probability 2/3, not 1. This "slight variant" is not the MHP. His third example, Monty Crawl, is the situation with extreme host bias. When the player chooses Door 1, the host opens Door 2 if it hides a goat, but Door 3 if Door 2 hides a car, since he crawls from his position near Door 1 and wants to crawl the shortest distance possible. Apparently the player knows this, and therefore switches when Monty crawls all the way to 3, but stays where he is when Monty opened Door 2 (though he might as well have switched in that case also). Rosenthal then shows how you can use Bayes theorem in the odds form, especially simple when initial odds are even, to compute conditional probabilities in all three problems. They are all different. He doesn't remark that his unconditional solution also applies to Monty Crawl. He doesn't say what is wrong with his short solution to the original problem. He doesn't say that he considers it wrong because he actually wants to know the conditional probability, not the unconditional probability. At the end of his paper he says that his shaky solution is wrong because it needs the assumption that the host chooses 50:50 when he has a choice, but didn't make use of this assumption. That's not true. If you only want an unconditional probability you don't have to say anything about bias.

In summary, Rosenthal gives a correct argument for finding the unconditional probability. He calls it "shaky" because apparently he thinks you really want to know the conditional probability, but he doesn't say that till later in the paper, and only then does he point out that you will then need a further assumption.

Selvin on the other hand gives his approval both to conditional and unconditional solutions. He praises Monty Hall for writing down the short simple unconditional solution in an as elegant and concise a way as possible.

The main disagreement between editors is on whether one should solve the problem with conditional or unconditional probability. There is a similar disagreement among the sources. We have to leave that for what it is: a matter of opinion. We can all agree on the mathematical truth, which is that Rosenthal's "shaky solution" is a correct argument for deriving the unconditional probability both in Monty Hall and Monty Crawl, and it is simply not applicable to Monty Fall. Gill110951 (talk) 15:51, 3 October 2010 (UTC)

My point was not about Rosenthal in general but about whether we could all accept the his statement above as a reasonable description of the simple solutions, that is to say, correct but shaky. Martin Hogbin (talk) 16:11, 3 October 2010 (UTC)

Yes Martin, I think so. But with some other aspect. Rosenthal (Crawl, Fall etc.) is speaking about "slight variants of the problem". What does that mean? Whitaker's question does not imply such "slight variants". Such "slight variants" are not the MHP. Neither Monty Crawl nor Monty Fall nor Monty xyz. So what is he about to say? My view: Rosenthal is speaking of "shaky variants". But such shaky variants of shaky sources do not address Whitaker's question, they do address "other" game shows with different rules and different hosts that may blab additional information on the actual (secret) location of the car, or may be crippled and senile...
Non of such "slight variant" is the MHP. Gerhardvalentin (talk) 18:51, 3 October 2010 (UTC)

Sorry Martin, the point of my long story was to say that I disagree that the short solutions are "shaky". If you are satisfied with an unconditional probability, then they are generally speaking rock-solid. Trying to be charitable to Rosenthal I would say that he introduces his Monty Crawl problem -- in which always switching also has overall success probability 2/3 and cannot be beaten by any mixed strategy! -- as an argument why you might be more interested, in general, in computing the conditional probability. Though it is a bad example since the conclusion is still "switch", and what he calls a shaky argument is still applicable and valid and shows you that always switching gives success chance 2/3, for Monty Crawl just as for Monty Hall. The example *suggests* that the conditional probabilities *might* conceivably tell a different story than unconditional, though they don't, and as we know, they can't, ever, since all doors were initially assumed equally likely. Monty Fall is a different problem altogether. The short solution doesn't apply to it because ... it doesn't apply to it ... and indeed the answer (given that Monty slipped and coincidentally showed you a goat, not a car) is not 2/3, it's 1/2. You still have symmetry, and all the conditional probalities, given which particular door you first chose, and which door Monty accidentally knocked open, are also all 1/2 (when a goat is revealed). Anyway, Rosenthal goes ahead and does the conditional probability in the standard Monty Hall and finds by calculation that the conditional probabilities are all 2/3. He wants to show you that you needn't think much, just depend on a single reliable approach which will deal with all problems: Bayes in the odds form, with initially equal odds, hence especially easy. OK, so it turns out that the advice whether you should stay or switch does not depend on the actual door numbers. He might have saved himself the bother of doing the calculations since in the "standard version" you have full symmetry so you can see in advance that you might as well ignore the door numbers. You know in advance that you'll not be interested in the conditional probability since you know in advance that it won't be different from the unconditional.

There is another advantage of the simplest of simple solutions. If you take the point of view that the player chooses a door at random and knows nothing about how the game show works, then we know he initially gets the right door first time 1 time in 3, and that switching gives him the car 2 times in 3. Not assuming anything about the host's behaviour and not being a subjectivist, that's all he can assume and all he can conclude. He doesn't compute a conditional probability since he doesn't, can't know it. This is also rock-solid, and quite simply *different* from the other solutions. A totally different assumption. Gill110951 (talk) 15:36, 4 October 2010 (UTC)

I think the central issue here is that there are three different types of solutions that address three different populations of players, or equivalently three different points of deciding whether to switch, or equivalently three different formal probabilities and that some editors either are confused about which type of solution is which or refuse to admit that these three types of solutions are indeed different (no matter how many reliable sources say so) or want to avoid characterizing solutions in this way (again, no matter how many reliable sources do). This is not the subjectivist vs. frequentist distinction that Richard keeps bringing up, but actually different interpretations of the problem. The three types of solutions are Richard's uno, due, tre (see talk:Monty Hall problem#Consistency), which also directly correspond to the three subsections in my proposed solution section. That there are three types of solutions is the Truth (according to Richard), but more importantly is supported by many reliable sources that distinguish them. There are indeed many sources that do not self-categorize their solution as one of these, but there are plenty of tertiary sources that do. The three types are:

1. Solutions addressing all players (900 players out of 900 broadcasts of the show). These solutions address the probability of winning by switching using a pre-selected strategy of switching, i.e. deciding to switch before even picking a door. They address the formal probability

   
   

   $P({\text{win by switching}})\,$

   
   Rick, that's not exact enough. Deciding after "a door" will have been selected, because whether #1 or #2 or #3 makes no difference at all, and also after the host will have opened one of his two other doors, independent of which one, because that simply just will not make any difference at all for the answer to the question "should you switch, yes or no". Gerhardvalentin (talk) 21:50, 4 October 2010 (UTC)

   Actually, what I'm talking about here is what sources say. Several sources (e.g. Carlton, and Grinstead and Snell) present explanations where they make it very clear they're talking about players using a pre-selected strategy - i.e. the players have decided what they are doing before they have initially picked a door. The probability pertains to a sample set consisting of all players who pre-select the same strategy. What these solutions are literally saying is that if you decide before even picking a door that you're going to switch, then your probability of winning is 2/3 (so long as your initial chance of picking the car is 1/3). I've never seen a solution like this that says anything whatsoever about the probability of winning if you've initially picked any specific door, or if you've picked a specific door and the host has opened some other specific door. These approaches don't say that choosing #1 or #2 or #3 makes no difference, but rather something completely different - specifically they say that on the average players who pre-decide to switch will win 2/3 of the time. A similar consideration pertains to your other comment. -- Rick Block (talk) 05:45, 5 October 2010 (UTC)

   
   

   No, that's incorrect on both counts. This is Carlton's simple solution, Rick:

   "As long as you initially pick a goat prize, you can’t lose: Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car. In fact, the only way you can lose is if you guessed the car’s location correctly in the first place and then switched away. Hence, whether the strategy works just depends on whether you initially picked a goat (2 chances out of 3) or the car (1 chance out of 3)."

   That's obviously not a situation where the contestant has switched before a door has been opened to reveal a goat.

   Grinstead and Snell purposely create their own game with different rules, they are not referring to the simple solution any other source offers:

   "We begin by describing a simpler, related question. We say that a contestant is using the "stay" strategy if he picks a door, and, if offered a chance to switch to another door, declines to do so (i.e., he stays with his original choice). Similarly, we say that the contestant is using the "switch" strategy if he picks a door, and, if offered a chance to switch to another door, takes the offer. Now suppose that a contestant decides in advance to play the "stay" strategy. His only action in this case is to pick a door (and decline an invitation to switch, if one is offered)."

   This is obviously a different puzzle that any simple solution source is solving, and is not the MHP. Glkanter (talk) 06:06, 5 October 2010 (UTC)

   
   

2. Solutions addressing all players who pick a specific door, say door 1 (assuming players are making their initial pick uniformly randomly, about 300 players out of 900 broadcasts of the show). These solutions address the probability of winning if deciding to switch after choosing an initial door, but before the host has opened a door. They address the formal probability

   
   

   $P({\text{win by switching AND player selects door 1}})\,$

   
   Rick, that's not exact enough. Deciding after having chosen an initial door, yes, but also after the host will have opened one of his two doors, #2 or #3, because this will make no difference at all for the answer to the question "should you switch, yes or no". Gerhardvalentin (talk) 21:50, 4 October 2010 (UTC)

   
   

3. Solutions addressing players who pick a specific door, say door 1, and see the host open a specific door, say door 3 (assuming players are making their initial pick uniformly randomly and the host chooses evenly between two goats if given the chance, about 150 players out of 900 broadcasts of the show). These solutions address the conditional probability of winning if deciding to switch after choosing a door and seeing which door the host opens. They address the formal probability

   
   

   $P({\text{win by switching AND player selects door 1 and host opens door 3}})\,$

The first is "unconditional" and the last two are "conditional". What I think many editors are saying is that the article should distinguish these types of solutions from each other, and should accurately describe the problem these solutions each address. IMO categorizing a solution which does not categorize itself using tertiary sources is a perfectly valid (and NPOV) approach. -- Rick Block (talk) 20:25, 3 October 2010 (UTC)

Lengthy OR exposition like the one above, and those provided by other editors, only serve to obscure the main point:

There are unsourced (and erroneously sourced) OR statements prominent in the article that are related to a point of view that is not prevalent in the reliably sourced materials.

These are violations of Wikipedia policies regarding OR, UNDUE weighting and NPOV. Glkanter (talk) 21:00, 3 October 2010 (UTC)

Sorry Glkanter, I disagree. That exposition by Rick was no more OR than drawing "new" decision trees. It was an attempt to overview what is "out there in the sources". The sources do not agree. — Preceding unsigned comment added by Gill110951 (talk • contribs)

Thank you Rick, for this clear presentation of the three different types of starting points for an approach to the answer to the question "should you switch?".
*added:* But it's not exact enough, above I've added two remarks. Gerhardvalentin (talk) 21:50, 4 October 2010 (UTC)
In determining a probability, it makes sense to use all those data which may influence the probability sought after.
[Just a word to semantics: Data (i.e. known or maybe supposed facts) can represent an "effect size" whatsoever, or they don't. In everyday life, the word "condition" however is used in the sense of "mandatory requirement" and "indispensable prerequisite" with striking effect. Note: calling just any data "a condition", as in mathematics, is a little bit confusing for the average citizen like me. Using "additional data without any effect size", whether reasonable or not, and calling them "conditions", should be translated for me to "using additional data with no or unknown effect size".]
Now, apart from the version of the secret traitor, the use of any "neutral data without any effect size" that never do influence the probability sought, as "additional data" is not recommended and doesn't make sense. If you do use them yet, I bet you have some reason to use them. And the sources clearly show why they are using such additional data: originally for any known host's bias to be included. But – using "conditional probability with q=1/2" makes no sense. This is an unnecessary laborious task. You can do it, but this is without additional nutritional value to get the answer to the question "should you switch".
As to me: Why did you forget to mention that explicitly above to item 2 and to item 3.
For - in case the guest has selected "say, door 1", it makes no difference whether the "MHP-host" has opened door 2 or has opened door 3. (Note: I'm not talking about "other hosts in other game shows"). Making this distinction "additional data without effect size" vs. "additional condition" explicit, could be helpful indeed for the readability of the article. Regards, Gerhardvalentin (talk) 18:36, 4 October 2010 (UTC)

It was just a simple question

Can we take this statement by Rosenthal to be a good description of the status of the simple solution?

It is actually correct but I consider it "shaky" because it fails for slight variants of the problem. Martin Hogbin (talk) 20:38, 3 October 2010 (UTC)

It is no better and no worse than the description from any other reliable tertiary source. For example, Grinstead and Snell say "though correct, does not quite solve the problem that Craig posed". Gillman says "does not address the problem posed, in which the host has shown you a goat at #3". Morgan et al, says "false". Eisenhauer [14] says "unconvincing and misleading". -- Rick Block (talk) 01:13, 4 October 2010 (UTC)

Eisenhauer simply quotes Morgan et al's denunciation of vos Savant's simple solutions, in a kind of orgy of self-satisfaction. Grinstead and Snell don't even know the problem that Craig posed, it was somewhat different from Marilyns "quotation" of it. Gillman says "does not address the problem posed" but doesn't explain why vos Savant's words imply that you *have* to come up with a conditional probability. Look: if you can compute a conditional probability because you have enough information to do so, then go ahead and do so. If you can't, don't. All these authors merely use their authority as maths teachers, or the authority of earlier writers who did the same, to say "you *must* compute a conditional probability". No one ever gave an argument! It's rather embarassing. They say: you must because I say so. MHP started with Selvin, and Selvin in his first letter was completely satisfied with an unconditional solution, while at the end of his second letter he again reproduced, with approval another, simple unconditional solution. Nalebuff, a big guy in mathematical economics and game theory, gave the simple solution. It seems to me that the only conclusion can be that it is a matter of taste, how to solve MHP. Careful mathematical analysis shows moreover that the simple solution relies on less input that the full solution, hence gives a weaker conclusion which can be more widely applied. Both solutions have an independent right to existence. Both tells you something valuable which the other one doesn't tell you. Why is this so hard to grasp? Gill110951 (talk) 16:24, 4 October 2010 (UTC)

There is a potential explanation, but it's not that nice, so i hope it isn't the true after all. That explantion is simply, that much of the (somewhat pointless and repeating) exchange here, is a continutation or proxy of the old (mostly pointless) vos Savant versus Morgan exchange. That also means it's somewhat ego driven rather than technical.--Kmhkmh (talk) 16:43, 5 October 2010 (UTC)

Richard, you say immediately above:

"Both solutions have an independent right to existence. Both tells you something valuable which the other one doesn't tell you. Why is this so hard to grasp?"

How well does the current article reflect that conclusion? Do you find the commentary in the Conditioanal solution section consistent with your conclusion? Glkanter (talk) 14:43, 5 October 2010 (UTC)

Yes, some sources answer it with a conditional decision tree, and explain why they do so. In no way supporting all that erroneously 'sourced' OR bs in the Conditional solution section and throughout the article. And all that stuff is obviously not the prevailing opinion of the sources. But Ill have to refresh myself on the whole Eisenhauer paper. It rings a bell that you were using him improperly as well. Glkanter (talk) 01:26, 4 October 2010 (UTC)

No. The status of the simple solution is that it is a rock-solid solution to a simple problem. There is a difference of opinion among the reliable sources concerning which problem you ought to be answering. Rosenthal goes for the conditional but he doesn't explain why. Gill110951 (talk) 15:50, 4 October 2010 (UTC)

Lemme be clear: I do not say one interpretation is better than the other. I say the POV of the article that one interpretation is better than the other is erroneous and should be immediately removed. I also say the attributions of sources are contrary to what the sources say. You and Rick offer the exact same quotations that I do, yet interpret them in some unfathomable way. There is no point to the introductory paragraph of the Conditional solution section, other than to cast doubt on the simple solutions. Using Selvin, and the others (except for the fact that they have chosen to offer a conditional solution) to imply there is something better about the conditional solutions is wrong. Intentionally wrong. What 'additional reasoning' (simply: premises - without the weasel wording) is not covered by K& W as per the earlier part of the article? Or by Selvin? Let's stick to the accuracy of the article, shall we? By the way, are 'pre-publication' articles considered reliable sources? Glkanter (talk) 16:37, 4 October 2010 (UTC)

Richard, would the article be any less accurate, or any more difficult for the reader, if all those simple-solution-bashing & conditional-solution-building-up paragraphs in the narrative in the Conditional SOLUTION section were removed? I don't think so. I think the article would be on its way to being less verbose, more reflective of the reliable sources, and more accessible. I suggested a couple of days ago that the conditional solution could be offered first, just without all that bs that currently accompanies it. And without that 2nd image that for a long time was the visual aid for the simple solution. How can that be, huh? Why should there be any more narrative than the paragraph that is right next to the conditional decision tree? That is the only place a reference to Selvin or Grinstead and Snell is appropriate. Not as support for Morgan, as the section now reads. Glkanter (talk) 17:20, 4 October 2010 (UTC)

Good arguments for the conditional solution can be retained and weak arguments removed. The most simple of simple solutions (that given by Monty Hall himself, as quoted by Selvin, with approval, in his second letter to the editor) makes the weakest premise of all, namely that your initial choice is right with probability 1/3, and correctly deduces from this premise that the "always-switcher" gets the car with probability 2/3. The only good reason I can think of for looking at the conditional probability would be if this might uncover specific situations where you can know that the conditional probability is less than 1/2. In that situation there would be a better strategy still than "always switching", namely "always switch except when the conditional probability that switching gives the car is known to be less than 1/2". In order to say something about the conditional probabilities, you'll have to make stronger premises, thereby make your analysis more restrictive. You will be able to say more, but only about less situations. Going whole hog, the exercise becomes self-defeating, since in the fully specified, fully symmetric case there is no reason to look at the conditional probabilities since we can see in advance that they'll all be the same as the unconditional. The intermediate case where the car is hidden by the quiz-team uniformly at random, but where we don't assume anything about the host behaviour (fairly reasonable if we are frequentist rather than subjectivist in our understanding of probability) seems to me to be the only situation warranting computation of conditional probabilities. It is not completely far-fetched and it provides a nice illustration of the power and ease of the prior/posterior odds formulation of Bayes' theorem. As we know, it delivers the result that all the conditional probabilities are at least 1/2, so also then, nothing beats always switching. And that's the whole story.Gill110951 (talk) 18:50, 4 October 2010 (UTC)

Is there a reliable source for the 'intermediate case' where the host acts in a fashion other than 50/50? Do they provide a solution, or call it unsolvable? Do they criticize the conditional solutions that require 50/50 for the solution? Do they criticize the simple solutions that use 50/50 as a premise? Glkanter (talk) 14:27, 5 October 2010 (UTC)

I guess that is a 'no' then everybody. I was just trying to find one thing that we could agree on. Martin Hogbin (talk) 19:28, 4 October 2010 (UTC)

But why do we have to agree with that statement? As Rick Block has repeated many many times, it is a POV issue - the POV of the sources, not ours, and the only way out is report on all well-sourced POVs in an NPOV way. As applied to the statement cited at the top of this section, why not just report on it, along with all the others pointed out by Rick Block above? Would it be acceptable to describe the simple solution(s) and then say "some sources regard this solution as "shaky" because it fails for slight variants of the problem [ref here], whereas others reject it as answering the wrong question [more refs]? . glopk (talk) 20:00, 4 October 2010 (UTC)

Any of the sourced statements are worthy of discussion as to their appropriateness in the article as per Wikipedia's policies on prevalence, NPOV, UNDUE Weight, etc. Unfortunately, there is an improperly sourced OR manifesto taking up the entire Conditional solution section and much of the rest of the article. Glkanter (talk) 21:07, 4 October 2010 (UTC)

As the article stands now, it would be equally as 'fair' and 'beneficial to the reader' and 'within Wikipedia policy' to offer the conditional decision tree solution first, then extrapolate from the various simple sources an entire section (or more) on why the conditional solution is a waste of time and wrong. Glkanter (talk) 21:54, 4 October 2010 (UTC)

@Everyone. I was just trying to find a statement that we could all accept as a compromise position on the status of the simple solution as as solution to the full symmetric conditional problem. I was not asking everyone to agree with it fully, I know that Glkanter, Richard, Gerhard, and myself consider the simple solution not to be shaky at all, and that Rick, Kmhmkh, Nijdam, and Glopk consider it to be rather worse than shaky, maybe even wrong, but I was hoping that, if we could all accept this middle-of-the-road description of the situation by a reliable source, we might make some progress. Martin Hogbin (talk) 10:14, 5 October 2010 (UTC)

If we just put what the sources actually say in the article, as per their prevalence, none of that other stuff would need to be discussed. Glkanter (talk) 14:18, 5 October 2010 (UTC)

Rosenthal

@Martin - I consider Rosenthal's "shaky" solution to be of type #1 above. It is a rock solid solution to the question "what is your chance of winning if you decide to switch before even selecting a door" or (equivalently) "what is the overall chance of winning for all players who switch" or (equivalently) "what is P(win by switching)". Per my comment to Gerhard above, solutions like this (at least typically) say nothing whatsoever about a player who has picked, say door 1, and has then seen the host open, say door 3, and is deciding whether to switch at this point. Rick Block (talk) 14:53, 5 October 2010 (UTC)

I made clear what problem I was talking about, it was the 'fully symmetric conditional problem'. I know you do not consider the simple solutions as valid for this case but I was hoping that you might accept Rosenthal's statement as a reasonable compromise position. Glkanter seems willing to do this and so would I be. Martin Hogbin (talk) 17:59, 5 October 2010 (UTC)

@Martin, perhaps you should re-read the solution that Rosenthal regards as "shaky": it makes no explicit distinction between conditional and unconditional problem: if anything the "you knew the host..." sentence may (very weakly) indicate that the player decides whether to stick or switch in advance. However, Rosenthal is obviously aware of the distinction (the Monty Crawl variant makes no sense unless the doors are equally identified in the host's and player's SOK). So my question is: if we accepted your view, and reported Rosenthal's "shaky" solution as "correct but shaky" for the 'fully symmetrical conditional problem", would we be faithfully reporting to the readers what the source actually says? I don't think so, at a minimum we would be putting in Rosenthal mouth something he does not say. This is the only thing that matters: what I, you and Glkanter "seem willing" to do doesn't count a fig. glopk (talk) 19:13, 5 October 2010 (UTC)

It would be a curious comment for anyone to make about the unconditional problem since we all agree that there is nothing whatever shaky about the simple solutions when applied to the unconditional problem. What slight variants to the unconditional problem do you suppose that Rosenthal had in mind?

In any case, you have completely missed the point. I was looking for a statement that we might all agree to accept as a compromise, not 'the truth'. Although you have often expressed a distaste for it, you seem set on continuing the same arguments we have all had for years. I am trying to find a way forwards. Martin Hogbin (talk) 22:05, 5 October 2010 (UTC)

I get the point: you are trying to make progress. So am I. Are you questioning my good faith in participating in this mediation, based on my disagreeing with your view that a certain statement by Rosenthal is a fair summary of the value of the simple solution as applied to the fully symmetric conditional problem? glopk (talk) 23:32, 6 October 2010 (UTC)

I am not questioning your good faith, i am responding to your suggestion above that Rosenthal may have been referring to the unconditional problem. It is clear to me that this is not the case for two reasons:

• Rosenthal refers to the simple solution as 'shaky'. We all agree that there is nothing shaky about the simple solution to the unconditional case.
• Rosenthal refers to 'slight variants' of the problem', for which the simple solution fails. What slight variants could these possibly be, we all know the simple solutions are fine for the unconditional problem regardless of the host goat-door-choice policy.

My question to you is simply whether you would be willing accept Rosenthal's statement about the status of the simple solution to the fully symmetric conditional problem as a reasonable compromise? If not, we will have to look for some other point on which we might all be able to agree. Martin Hogbin (talk) 08:28, 7 October 2010 (UTC)

Some curious logic on display up there. Let me parse this syllogism: Major:Rosenthal calls simple solutions "shaky", Minor: We all agree there is nothing shaky about them in the unconditional case. Conclusion: Rosenthal's talking about the conditional case. Unfortunately this kind of reasoning does not hold much water. Regarding your question, specifically, I support Rick Block's answer above, and note also that "shaky" is a rather weasely way state a judgment about the correctness of a proof. glopk (talk) 01:24, 9 October 2010 (UTC)

I think I understand Rosenthal quite well. Rosenthal calls the simple solution "shaky" (as reasoning for "switch") because he correctly believes that you should use conditional probability if you can. He gives the example of Monty Crawl to demonstrate that conditioning can sometimes alter the conclusion which you would get from not conditioning. Note that in Monty Crawl the simple solution again says, correctly, that switching gives you the car with probability 2/3 hence seems to say "switch". Yet in one of the cases which might occur, the conditional probability is 50/50 hence you might as well stay. Now in the completely symmetric case you can see in advance that the conditional probabilities are all the same as the unconditional so you don't bother to calculate the conditional probability the long way round. The general rule is: "condition on everything you can, which might change the probability, don't bother to condition on anything which won't change the probability anyway". Gill110951 (talk) 06:41, 9 October 2010 (UTC)

Carlton

@Glkanter - you've quoted Carlton's explanation but you keep omitting the part that makes it clear he's talking about ALL players INTENDING to switch, not the players who have picked, say door 1, and have then seen the host open, say door 3. The omitted part is "Imagine you plan to play Let’s Make a Deal and employ the “switching strategy.” As long as you initially pick a goat prize, you can’t lose: ". The player isn't switching before the host opens a door, but has decided to switch before the host opens a door. This is the difference between a type 1 or type 2 solution (above) and a type 3 solution. Type 1 and type 2 solutions produce the same probability of winning as a type 3 solution, but if and only if the problem is fully symmetrical. Carlton's explanation says nothing about symmetry, and nothing whatsoever about the probability in the example case given in the usual problem statement. IMO, this is precisely why he calls it an "intuitive explanation" and not a solution. Grinstead and Snell make it clear what problem their type 1 solution addresses - if you don't consider this "the MHP" then you're agreeing that these solutions don't exactly address the problem as stated. -- Rick Block (talk) 14:53, 5 October 2010 (UTC)

There you go again, deciding 'intent' on behalf of the sources. Carlton's player plays the exact same game as everyone else, making the decision after a door has been opened to reveal a goat: "Monty Hall must reveal the location of the other goat, and you switch to the remaining door - the car." Those are his words.

Grinstead & Snells say they are describing a different game, as they write: "We begin by describing a simpler, related question." Those are their words.

In any case, an editor's disputed interpretation of what a source is saying does not constitute a prevalence among the reliable sources worthy of the multiple NPOV violtions, UNDUE Weight violations, and OR violations in the current MHP article. Glkanter (talk) 15:14, 5 October 2010 (UTC)

I'm not deciding "intent" on behalf of sources, I'm comprehending what the sources say. The article cannot include full quotes in full context from a significant number of sources, but instead needs to paraphrase what they say - paying attention to NPOV, UNDUE, and OR. For example, G&S indeed say "We being by describing a simpler, related question." They proceed to present the EXACT SAME "solution" that Carlton calls an "intuitive explanation". This is precisely the same as the solution Morgan et al. explicitly criticizes as applying to a different problem, which is the same problem G&S say is a "simpler, related question". It is not OR or POV or misleading in any way for the article to present the POV of these sources that this solution does not quite address the problem as stated, or to use the G&S quote about this, i.e. "This very simple analysis, though correct, does not quite solve the problem that Craig posed". We can quibble about the exact wording to use to introduce this quote in a way that makes it clear this is the POV of some sources, or whether we should use a quote or a paraphrase, but this quote is summarizing what all three of these sources are ALL saying. They don't use exactly the same words. They don't say it in exactly the same way. But they are all saying the same thing. Understanding this requires comprehending what they're saying. I really can't tell if you simply disagree with what they're saying or are not comprehending what they're saying, but for the article to say what they say is not an NPOV violation, or OR. UNDUE is a much trickier topic that we collectively have never talked about. You admit above [15] you haven't extensively read the sources. IMO, this means you are in no position to judge UNDUE. -- Rick Block (talk) 16:56, 5 October 2010 (UTC)

"You admit above [16] you haven't extensively read the sources. IMO, this means you are in no position to judge UNDUE. -- Rick Block (talk) 16:56, 5 October 2010 (UTC)"

Really? This is what I 'admit' in that diff:

"How much have I read? Not as much as you. More than you think, probably. Enough that I understand there are conflicting sources that deserve proper representation in the article. My main disagreement with you, at least on this issue, is how to interpret Wikipedia's policy on NPOV and reflect that policy properly in the article."

I wonder if that grossly unfounded personal attack of my knowledge of the sources of the MHP is based on what I've been arguing, and the support I provide for my arguments, or something else. Glkanter (talk) 17:25, 5 October 2010 (UTC)

It's quite clear that you have drawn an incorrect conclusion from that diff. And I ought to know, I wrote it. And it's quite clear, isn't it?

"How much have I read? Not as much as you. More than you think, probably.'

For you to then write your conclusion and attribution of:

"you admit above you haven't extensively read the sources" is false and misleading.

I assume this is intentional, as it is such a clear passage, and your criticism so erroneous. This is no different than the misapplied techniques you use to support your positions regarding the sources that are misused in the article. Glkanter (talk) 17:40, 5 October 2010 (UTC)

I've certainly read the sources that are erroneously given attribution in the Conditional solution section, and more. That you insist a POV be inappropriately dominant throughout the article is why we're in mediation. Glkanter (talk) 17:01, 5 October 2010 (UTC)

Which do you deny? That your favored POV is dominant in the article? Or that the dominance is inappropriate? Or both? Glkanter (talk) 17:03, 5 October 2010 (UTC)

My favored POV is NPOV, and IMO the current version of the article favors the "simple solutions are correct" POV (because of the placement of the "Aids to understanding" section) and (also IMO) this presentation does not "fairly, proportionately, and as far as possible without bias, [represent] all significant views that have been published by reliable sources" (i.e. is not NPOV). So, yes, I deny that either my actual POV or what you think is my favored POV (presumably that "simple solutions are incorrect") is dominant in the article. I asked before and you didn't exactly answer. Please tell us a rough approximation of how many sources you've read. Is it more like 10 or less, more like 50, or more like 100 or more? As I've said before, I've personally read at least 50 including all the references currently in the article and (demonstrating I'm not cherry picking and only reading sources that have some particular POV) as many as I could locate that are in Barbeau's 1993 and 2000 literature summaries (which are in turn referenced in the article - and if you look, you'll find that many of Barbeau's references are also referenced in the article). And, please spare me the lecture about personal attacks. I'm sure if what I said can be construed as a personal attack Sunray will comment. In fact, why don't we simply remove any doubt.

Sunray - would you please comment on whether what I said above is a personal attack? --Rick Block (talk) 19:52, 5 October 2010 (UTC)

The statement is contrary to the agreed upon guidelines in that it is a personal comment. It actually goes farther and makes a judgement about an individual. It is a good example of the kind of comment we should avoid on this page. I would remove it, but since it has been repeated several times by each of you, I will leave it. Rick: would you be able to find ways to discuss topics without making personal comments about other participants? Sunray (talk) 21:55, 5 October 2010 (UTC)

Sunray - Would you agree Glkanter's comment above "There you go again, deciding 'intent' on behalf of the sources." is another example? I apologize for responding in kind. I do think we need to talk about UNDUE, which I think requires more than a cursory familiarity with the literature. I would be happy for you to lead such a discussion. -- Rick Block (talk) 22:46, 5 October 2010 (UTC)

Not to interfere with any response from Sunray, but the focus should be on the article, not my qualifications to edit a Wikipedia article. I have provided documentation, and others have agreed, that the sourced attributions in the Conditional solution section do not meet Wikipedia's standards. Glkanter (talk) 20:14, 5 October 2010 (UTC)

If you're going to keep claiming UNDUE, please answer the question about how many sources you've read. -- Rick Block (talk) 21:11, 5 October 2010 (UTC)

I've probably responded to every question you've ever posed to me. I even answered your original question on this topic. You tell a total falsehood about my response, then act like nothing happened. That's bs. The question and answer are of no consequence here, unless you plan on establishing a minimum reading requirement for each editor. Let's stick to discussing the NPOV violations, etc. in the article. Glkanter (talk) 21:20, 5 October 2010 (UTC)

Why

... am i having such a difficult time differentiating this page from this page? Isn't there supposed to be some structure here rather than "Hey, who's with me" exclamations? This mediation seems to be overrun with positioning by one editor who thinks that this is the talk page linked above. There is no merit to anyone's argument by being repetitive (IMHO) - it just seems like an attempt to invoke the squeaky wheel principle. hydnjo (talk) 02:54, 4 October 2010 (UTC)

I agree, it is messy and not helping. But to be fair it has been like that for years on the discussion page and related threads elsewhere and there's a reason why the first moderator has given up. The discussion is extremely hard to control and tends to lose focus. i.e. there's a constant raising of new issues before even a single old one gets actually settled. Another negative side effect is, that simply too much noise is created making it harder for everybody to follow the overall discussion.--Kmhkmh (talk) 16:38, 5 October 2010 (UTC)

Why not? The problems with the article haven't changed. My factual statements about the false attributions are met with avoidance and dissembling. The mediator is looking for a moderator. This has been going on for what, 9 months? All I see is that my efforts (edits, discussion, supporting quotes, logical arguments) to improve the article so that it meets Wikipedia policy are beaten down, contrary to Wikipedia policy. Why should I stop pointing out obvious errors, and policy violations because other editors would prefer an improper POV is maintained? Unless you're criticizing the other guys, in which case i support you 100%. Glkanter (talk) 03:11, 4 October 2010 (UTC)

I agree that the discussion has not remained focused and take a large measure of the responsibility for that. I was looking for a moderator, but that is not in the cards. So, I will have to take a stronger role, and with the help of participants and more time to put in, (at least this week), that should be easier for me than it was before.

If suggested collaboratively editing the article and there seems to be general support for that approach. Would someone propose a section of the article to start on? Sunray (talk) 22:36, 4 October 2010 (UTC)

I think you have to start with the underlying disagreement over the validity of the simple solutions, because everyone's opinion, suggestions for changes, and interpretation of the sources will be dominated by their opinion on this point. Martin Hogbin (talk) 22:12, 5 October 2010 (UTC)

That may be true, and it shows one of the biggest problems of the endless discussions and this mediation. It's all about the sources and their prevalence, not our opinions of any solution. Glkanter (talk) 22:17, 5 October 2010 (UTC)

Picking a section to edit

I propose we start with the Conditional solution section as per my stated concerns in various sections of this mediation talk page. Glkanter (talk) 23:54, 4 October 2010 (UTC)

I think we need to talk about whether it is appropriate to separate the "simple" solutions from the conditional solution with a lengthy "Aids to understanding" section. We started to get into this with Question 10 above, but the discussion quickly grew completely out of control. We could also roll all the way back to the Carlton discussion (which I think we nearly closed on), or resume the discussion on my proposed solution section (that I've marked as "parked" above, it now has comments from Richard, Glkanter, and Martin that I haven't responded to). We've been at this for nearly two months without (as far as I can tell) making any tangible progress. I don't know about anyone else but I'm finding this extremely frustrating. I have a pretty good idea what the problem is and I wouldn't be averse to taking it to arbcom at this point. I'm also willing to keep trying here. -- Rick Block (talk) 06:14, 5 October 2010 (UTC)

But Rick, I have proposed, more than once, making the Conditional solution the first in the single Solution section. That would be Chun's decision tree accompanied by a narrative about the tree. I don't see how bringing up 'Aids to understanding' has anything to do with the progress Sunray noted today. It almost seems like a diversion, no? Likewise, I have pointed out how the paraphrasing in the article omits critical information about Carlton's simple solution, and why a full quote of his very brief simple solution is more appropriate. Arbitration is not for content disputes. None of the editors has shown any support for your proposed OR Solution section, it's not on the table anymore. No, we need to keep a laser-like focus on the article, insuring proper attribution of the sources, removing the NPOV violations, and better adherence to Wikipedia Policies regarding UNDUE weight. I'm sure with Sunray's (plus another mediator's) enhanced availability, we'll make great progress very quickly. Glkanter (talk) 07:20, 5 October 2010 (UTC)

I suggest we let others weigh in on the best place to begin. Sunray (talk) 21:57, 5 October 2010 (UTC)

See my comment above. Everything else is dependent on addressing the main point at issue, as is demonstrated here. Martin Hogbin (talk) 22:30, 5 October 2010 (UTC)

The disputed NPOV material begins (for the most part) in the Condition solution section. I have pointed out what I believe are numerous erroneous attributions to sources, as well as NPOV and UNDUE weight violations in that section. Rick has stated he does not agree with my claims at all. Why start anywhere else? I would perceive trying to leave this topic as another delaying tactic. Glkanter (talk) 22:07, 5 October 2010 (UTC)

I do agree that that section tends to state as fact what is actually just the opinion of some editors here. Martin Hogbin (talk) 22:32, 5 October 2010 (UTC)

I think a likely outcome is that once all attributions are corrected or unsupported statements are removed, we'll find that the 'prevalence' argument has been much overstated. Glkanter (talk) 22:40, 5 October 2010 (UTC)

I think it would be fun to tackle the Conditional solution section. It's a fact that some editors (and some sources) disagree that you should compute a conditional probability, but plenty of sources do think you should, and the MHP page has to include a section on it. The section states that some sources thank that Craig Whitaker is asking for a conditional probability. It's true that a lot of influential sources apparently think so. I don't see any incorrect attribution or unsupported statement in the section, at present. It says that it takes more work to find the conditional probability than the unconditional (compare Selvin's computation in his second letter, with the words he cites of Monty Hall; note that Selvin's computation uses all the premises he had listed before, while Monty Hall's argument only needs the premise that the initial choice is correct one third of the time). The section gives some ways to do the computation. It could of course use Boris Tsirelson's symmetry argument to show that computation can be replaced by logical argument in the fully symmetric case. It could follow Rosenthal's influential paper and book and use Bayes theorem in the odds formulation rather than going to first principles. I think the fact that the numerical answer is the same confuses some editors and no doubt would confuse many readers. I ponder on the question, why *should* you compute the conditional probability? I don't think the section gives a good answer to it. If you agree that we are asking for an action, not for a probability, then the purpose of probability calculation is to find out if the probability is at least 1/2 that the other door hides the car. Then switching is justified. The question is, in view of the unconditional probability being already 2/3, could we have any reason whatsoever to suspect that the conditional probability might be less than 1/2? I think the reason so many sources are completely satisfied with the unconditional solution is because they find it inconceivable that further information could alter the balance of the odds. And in the fully symmetric case they know by instinct that it cannot change the odds at all. Gill110951 (talk) 13:09, 6 October 2010 (UTC)

Yes, doing this section has the advantage that we will need to tackle many of the disputed issues. As Richard says, perhaps we should start with the question of why the conditional probability required at all. Martin Hogbin (talk) 16:17, 6 October 2010 (UTC)

Richard says many things. What he does not say is whether there are reputable sources addressing the question of why the conditional probability required at all. Unless someone can point at some such sources, this question is perfectly irrelevant to the present discussion, and indeed to the WP article on the MHP. Focus, please, focus... glopk (talk) 17:57, 6 October 2010 (UTC)

There may be such sources, that is one of the things we could discuss. Martin Hogbin (talk) 19:28, 6 October 2010 (UTC)

OK, by all means, let's discuss such alleged source, in the context of a unified Solution section, comprising of both simple and conditional solutions, to be placed before the Aids To Understanding section. Deal? glopk (talk) 23:22, 6 October 2010 (UTC)

The proposal was to discuss the conditional solution section. There is no unified solution section and there is no consensus to have one. Martin Hogbin (talk) 08:42, 7 October 2010 (UTC)

Jeffrey Rosenthal's (2008) "Monty Hall, Monty Fall, Monty Crawl" is a source specifically written to address the question "why we must use conditional probability". Rosenthal first gives the simple unconditional solution to the standard MHP. He says that he thinks it's "shaky" and his paper is going to show us why. He introduces Monty Crawl: if the player chooses Door 1 and then Monty crawls over the stage from Door 1 to the first door which he can open in order to reveal a goat, and this happens to be Door 2, then the player needn't bother to switch. Rosenthal shows that if we "fill in the gaps" in vos Savant's verbal formulation of MHP a different way from the conventional way, conditional probabilities can give us a different answer from the unconditional. He does this precisely in order to argue that you "must" also think about the conditional probability in the standard version of MHP: because it *might* give a different advice from the unconditional probability. If you always switch you win with unconditional probability 2/3, but without doing some extra work, he's saying, we can't know that we couldn't maybe do better still: there might be situations which favour staying. Always switching is good, but maybe sometimes staying is better. When deciding on a course of action in an uncertain situation, you should take everything known into account which you can, which might influence the chance of success. The argument between the conditionalists and unconditionalists would be easily solved in favour of the conditionalists, if the standard assumptions were such that sometimes switching is better, sometimes not switching is better, depending on the specific door numbers. Unfortunately, under the standard assumptions, the conditionalists do not sell their own logically completely sound position very well: they insist on doing a laborious calculation which only shows what many people intuitively know for sure, namely that the specific door numbers don't matter. Fortunately, one can do a simple and illuminating calculation: the odds form of Bayes, which becomes what Rosenthal and Rosenhouse call the proportionality principle in the situation with uniform prior odds. Or one can argue from symmetry that the conditional probabilities must be all equal hence equal to the unconditional (Boris Tsirelson).

In summary: it's not "why we must compute the conditional probability", but it's "why we must ask ourselves the question if we need a conditional probability". My answer is: yes, we ought to ask ourselves this latter question, and we should dispose of it as quickly as possible, as far as the main part of the article is concerned. Note that if you don't buy all the standard premises you may not even be able to compute a conditional probability. The "active" solution: "choose a door uniformly at random and thereafter switch" doesn't make any premises at all. You don't compute a conditional probability because you can't. Gill110951 (talk) 09:10, 7 October 2010 (UTC)

Or, one could argue, "If it's from a reliable source, it should be considered for inclusion in the article. Just maybe not in any 'Solutions' sections." And the corollary, "If it's not from a reliable source, it doesn't go in the article anywhere.'" Glkanter (talk) 10:04, 7 October 2010 (UTC)

Does Anyone Know Specifically What 'Additional Reasoning' Refers To?

This is the 1st sentence of the 2nd paragraph of the Conditional solution section:

"In particular, Morgan et al. (1991) state that many popular solutions are incomplete because they do not explicitly address their interpretation of vos Savant's rewording of Whitaker's original question (Seymann). The popular solutions correctly show that the probability of winning for a player who always switches is 2/3, but without additional reasoning this does not necessarily mean the probability of winning by switching is 2/3 given which door the player has chosen and which door the host opens."

K & W and Selvin list out all the premises of the MHP. By my count the full list of premises is mentioned at least 3 times before the Solution section. What 'additional reasoning' is required? Glkanter (talk) 03:36, 6 October 2010 (UTC)

Listing premises is not the same as logical reasoning. A number of sources think that vos Savant is asking for a conditional probability, not an unconditional probability. It's much easier to find the unconditional probability (less premises and less reasoning needed) than to find the conditional probability. Gill110951 (talk) 12:07, 6 October 2010 (UTC)

Thank you for the response. The various sources that say the problem is a conditional problem are prominent in the first paragraph of the Conditional solution section, although there are no attributions. I don't see how that second paragraph is necessary, and I think it just muddles things up. I just re-read Morgan's paper, and that first sentence "In particular, Morgan et al. (1991) state that many popular solutions are incomplete because they do not explicitly address their interpretation of vos Savant's rewording of Whitaker's original question (Seymann)." doesn't come from them. Or from Seymann. Morgan simply say 4 of the 6 false solutions answer the wrong question, the unconditional question. They do not say 'incomplete'. As the 50/50 premise is in the article, the rest of that paragraph regarding the probability of each door is unnecessary, and in fact is discussing some sort of different problem, not the MHP in the article. The various sources I've read say certain simple solutions are answering the wrong problem, nothing about 'additional reasoning'. Glkanter (talk) 12:50, 6 October 2010 (UTC)

Morgan's 5th false solution comes up with the result of 50/50. The 6th is a verbal rendering of the Conditional decision tree in the Conditional solution section. Morgan calls this false because it relies on the not-stated 50/50 host bias. Glkanter (talk) 12:57, 6 October 2010 (UTC)

The essential MHP

I summarized what I have learnt about MHP from these discussions in [17]. An earlier writing of mine was accepted for publication but I am going to offer this completely rewritten text as "final revision". Comments are welcome either on my talk page or by personal email. Gill110951 (talk) 17:38, 7 October 2010 (UTC)

What to focus on

Can we do a multi-vote to decide what topic we want to focus on? Please indicate below which of the topics you're willing for us to focus on next (multiple is fine, in fact preferred). I suggest the topic getting the most votes is what we should focus on. And by "focus" I mean we talk about a single topic, in a structured way, mediated/moderated by Sunray. I would suggest any side topics be acknowledged and explicitly parked, like #Comments on a proposed solution section (above).

1) Collaborative editing of the existing "Conditional solution" section, at Wikipedia talk:Requests for mediation/Monty Hall problem/Conditional probability solution

• Rick Block (talk)

• Gill110951 (talk) 20:24, 7 October 2010 (UTC)

• Martin Hogbin (talk) 22:42, 7 October 2010 (UTC)

• [Including the title?] Nijdam (talk) 12:12, 8 October 2010 (UTC)
  • Nijdam, I'm inquisitive, what title would you suggest for this section? Gill110951 (talk) 16:03, 8 October 2010 (UTC)
    • • I want to avoid that computing the conditional probability as a basis for the decision is just seen as one of more possibilities, a variant, in stead of being the correct way of solving the MHP (as a probability puzzle).Nijdam (talk) 11:10, 10 October 2010 (UTC)

"Solving the MHP"? In 1/3 staying wins, and in 2/3 switching wins. Probability? – Possible extreme case: In max. 2/3 it can be 50:50 and in max. 1/3 it can be 100:0, and you never can get any "closer". You know that for sure. And so you know that you should switch. Without ever having to "compute the conditional probability", because it's given that it offers no additional nutritional value. But you can have a link to Bayes' Monty Hall Example. Gerhardvalentin (talk) 10:47, 12 October 2010 (UTC)

• • • Could we agree that the title would also be a subject for discussion if this topic is picked and avoid further comment right now? Sunray (talk) 21:30, 8 October 2010 (UTC)
• glopk (talk) 21:21, 8 October 2010 (UTC)

2) What constitutes NPOV, in particular due and undue weight

• Rick Block (talk)

3) Collaborative editing of the solution section proposed at Wikipedia talk:Requests for mediation/Monty Hall problem/A proposed solution section

• Rick Block (talk)
• glopk (talk) 21:21, 8 October 2010 (UTC)

4) Wording of Carlton's intuitive explanation, in the existing "Simple solutions" section

• Rick Block (talk)

5) Placement of the "Aids to understanding" section

• Rick Block (talk)
• glopk (talk) 21:21, 8 October 2010 (UTC)

If I omitted whatever you want to focus on, please feel free to add it. I voted for all, by which I mean I don't care what we focus on. On the other hand, I very much want us to focus on ONE topic. -- Rick Block (talk) 18:49, 7 October 2010 (UTC)

Thanks for taking this initiative. It should give a clear indication of what people would like to focus on. I've put the "Conditional probability solution" section up as a subpage and could add any other section that participants would like to work on in the same section. Sunray (talk) 21:47, 7 October 2010 (UTC)

Where is the subpage please?

Item #1 in the table of contents. Glkanter (talk) 22:55, 7 October 2010 (UTC)

I'm expecting Sunray to "officially" declare the topic once the front runner is clear. -- Rick Block (talk) 00:15, 8 October 2010 (UTC)

There are a few more participants to hear from. Let's give them a couple of days to weigh in. Will Beback is available once more and has agreed to moderate this. I will be around, but am still on the road, so will not always be able to be online. Sunray (talk) 21:35, 8 October 2010 (UTC)

I will be moderating the /Conditional probability solution discussion. I expect editors to work towards consensus, to follow the agreed-upon groundrules, and to be concise in their postings.   Will Beback  talk  07:05, 9 October 2010 (UTC)

I have no particular opinion on the issue we should focus on first, but I'd really appreciate if in fact we actually on do focus on whatever we pick and settle it before digressing to other subjects.--Kmhkmh (talk) 13:26, 9 October 2010 (UTC)

Edits to article

Just so nobody is blindsided, I plan on making the same edits to the article that I made last week. In addition, I will be removing both Morgan and Seymann as attributed sources for the 1st sentence of the 2nd paragraph on the Conditional solution section. I have demonstrated unambiguously on this talk page that all those attributions are erroneous, and that critical information is omitted from Carlton's simple solution. I'm going to delete the 2nd image, too. It originally was in the article as the visual aid for the so-called simple solution (!?), has only the 4 same numeric values that the decision tree has, is confusing to decipher, and is probably OR. This is a good example of 'less is more'. If the article remains unprotected, I expect to make these changes by lunchtime on Friday. Glkanter (talk) 00:58, 8 October 2010 (UTC)

Sunray - can you please comment about this? IMO, none of these are consensus changes. -- Rick Block (talk) 01:10, 8 October 2010 (UTC)

Rick, I've been re-reading Morgan's paper and a rejoinder. I think the criticisms in the Wikipedia MHP article are, in general, a work of fiction, mostly unsupported or mis-attributed OR and POV. Martin's lengthy criticisms of Morgan actually leave out significant things from Morgan. Like how they call the conditional decision tree solution 'false'. Or how they were unaware of Selvin until notified by another American Statistician reader. And then, in a rejoinder, they horribly butcher Monty Hall's correspondence with Selvin, ignoring, much as you do, Selvin's praise for Monty's Carlton-like simple solution, of course to the benefit of whatever point they were making. Morgan simply says the simple solutions are false because they're indifferent to *which* door is open, that they don't the 'use the information in the number of the door shown." Which, of course, all the sources (yes, even Morgan) agree doesn't exist in the 50/50 MHP. And the 50/50 premise, all the the sources, like Selvin, Grinstead & Snell, and Krauss & Wong, pretty much everybody but Morgan, (although Morgan does question themselves, "But then, it may be that only an academician, and no one connected with a game show, would ever consider p <> q.") say is a premise of the MHP. All that other stuff in the Wikipedia article is made up. Glkanter (talk) 03:04, 8 October 2010 (UTC)

It seems to me that the first three sentences of the second paragraph of The Conditional Solution state neutral facts that everyone can agree on. Morgan et al. (1991) think that Whitaker wants to know P(A|B), not P(A), where A is "switching gives car" and B is "Player chose Door 1 and host opened Door 3". It's a fact that P(A|B), read "probability of A given B", is called a conditional probability and the cited references say so. If you want to know P(A|B) then either you sit down and compute it directly in one way another, or you compute P(A) and give reasons why P(A|B) does not depend on B (probabilists say: "A is independent of B"). Glkanter calls these distinctions academic but that is a matter of opinion. Read Rosenhouse's wide-ranging well researched and carefully written (2009) book "The Monty Hall Problem". He spends a lot of time on this issue. Like it or not, the Monty Hall Problem has moved on since Selvin's 1975 letters to the American Statistician, vos Savant's 1990 article in Parade, and Morgan et al's 1991 emotional and agressive reaction to vos Savant's solutions. There are reliable sources, old and new, who don't agree with Morgan et al. 1991 that you *have* to compute a conditional probability, but no reliable source claims that proving the conditional probability is 2/3 is the same as proving the unconditional probability is 2/3. The numbers 2/3 are the same but the meanings of the statements P(A)=2/3 and P(A|B)=2/3 are different, like it or not. Gill110951 (talk) 07:29, 8 October 2010 (UTC)

Well, Richard, there is at least one source who disagrees with your statement:

"...but no reliable source claims that proving the conditional probability is 2/3 is the same as proving the unconditional probability is 2/3. The numbers 2/3 are the same but the meanings of the statements P(A)=2/3 and P(A|B)=2/3 are different, like it or not."

"Certainly the condition p = q = 1/2 should have been put on via a randomization device at this point. It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one. But then, it may be that only an academician, and no one connected with a game show, would ever consider p <> q."

[Let's Make a Deal: The Player's Dilemma]: Rejoinder

Author(s): J. P. Morgan, N. R. Chaganty, R. C. Dahiya, M. J. Doviak

Source: The American Statistician, Vol. 45, No. 4 (Nov., 1991), p. 289

Published by: American Statistical Association

But that's not a point I'm much interested in, anyways. Otherwise, I'd be happy to discuss the 1st three sentences of the 2nd paragraph and document that the statements are not supported by the sources attributed to them. Regardless of your opinion of what the facts are, or which editors agree with them. Glkanter (talk) 07:45, 8 October 2010 (UTC)

Morgan et al. (1991b) say exactly what I say: if p=q then one can use symmetry to argue that P(A) and P(A|B) are the same. Hence one can solve the conditional problem by computing the unconditional probability and then remarking on the symmetry. I too am rereading Morgan et al. (1991a,b, 2010) and appreciating their work more and more. Gill110951 (talk) 08:35, 8 October 2010 (UTC)

There are three sentences and there are sources associated with the first and the third. The third sentence says that the probability of A given B is called a conditional probability and associates sources with that claim. The first sentence says that Morgan et al (1991a) interpret Whitaker's question as asking for a conditional probability. Seymann agrees. Morgan et al are their own reliable source here, that is exactly what that paper states: Whitaker is asking for a conditional probability. Editors and other sources may or may not agree with Morgan et al, and editors and other sources might find their paper unimportant, but it can't be denied that Morgan et al say that we should compute a conditional probability. Their later writings 1991b and 2010 do not adopt a different point of view. They still think that Whitaker's question must be answered by computing a conditional probability. It's particularly easy under the full uniform-random assumptions, by symmetry. That's what they say. Gill110951 (talk) 09:16, 8 October 2010 (UTC)

I stand by my earlier criticisms of the 2nd paragraph. Neither Morgan or Seymann use or imply the word 'incomplete'. To use Selvin (or Seymann) to support the contention that the MHP is exclusively to be interpreted as a conditional problem is inconceivable, given Selvin's own simple solution and his praise for Monty Hall's simple solution. Selvin, Morgan and Seymann say nothing of 'incomplete', or 'additional reasoning'. Grinstead and Snell *do* say it's a conditional problem, but not that it's incomplete or requires 'additional reasoning', all as I have earlier documented. Given that the full statement of premises precedes this section in the article 3 times, there is no 'additional reasoning' required, nor has a source been presented who makes that claim. Glkanter (talk) 10:17, 8 October 2010 (UTC)

Morgan 1991a,b, 2010 consistently say Whitaker needs to know the conditional probability, and all their works gives additional reasoning (beyond what is needed to get the unconditional probability) to derive the conditional probability. In particular in their rejoinder 1991b they explicitly give the additional reasoning of symmetry to go from the unconditional to the conditional probability. In 1991a they give full reasoning to get the conditional probability from first principles. Writing down the premises is irrelevant here. Indeed Selvin does not say that MHP is to be exclusively considered as a problem in conditional probability. Grinstead and Snell do say it is and they give full reasoning to get the conditional probability, so they manifestly don't think that the simple solution does the job. Seymann doesn't say much beyond remarking that you can't solve vos Savant's problem without making additional assumptions. Gill110951 (talk) 10:41, 8 October 2010 (UTC)

I can't tell for sure, are you now agreeing that those statements in the Wikipedia article are improperly attributed as to Selvin, Seymann, Morgan, and Grinstead and Snell (except Morgan and G & S do say it must be conditional)? The symmetry is already a premise that the reader has been provided, nothing additional is required. It already exists. There's no need for any of that. Besides, Morgan and G & S don't say the symmetry argument is required, they say the MHP has to be viewed as a conditional problem. Heck, Morgan calls the decision tree solution 'false' because it relies on the 50/50 premise. How could they support the argument that it's required 'additional reasoning' for the 'false' simple solutions? It doesn't make sense! Why not just present the conditional decision tree at this point in the article, and save the meaningless mumbo jumbo for later? You are right, the MHP *has* evolved beyond Morgan. All the way back to 1975, when Selvin clearly stated each of the 'missing' premises Morgan makes an issue of in their paper, in what they call 'the vos Savant Scenario'. Glkanter (talk) 12:05, 8 October 2010 (UTC)

No, I'm not agreeing. I not only think that the three sentences report objectively true facts, I also think that the citations are relevant, with the sole exception of Seymann (1991). The statement "That probability is a conditional probability" can correctly be supported by citing Selvin 1975b; Morgan et al. 1991; Gillman 1992; Grinstead and Snell 2006:137; Gill 2009b. The point of view that MHP must be solved by giving a conditional probability is the point of view of Morgan et al. and they never departed from that point of view. They also have always acknowledged that more work has to be done to deduce that the conditional probability is 2/3, than to deduce that the unconditional probability is 2/3. They actually state that the simple solutions are wrong, not incomplete. By wrong, they mean that the argumentation given in the simple solution does not prove that the conditional probability is also 2/3. I think "incomplete" is a very reasonable word to use in this context to describe their standpoint. Especially since their 1991b rejoinder does show how an appeal to symmetry indeed completes the proof. The only reference which is out of place is Seymann, I think. Gill110951 (talk) 12:44, 8 October 2010 (UTC)

I have reverted, will someone (not me, I've been there already) please request that the article be locked again? Sunray, will you please ask Glkanter whether he is seriously interested in pursuing this mediation? glopk (talk) 15:45, 8 October 2010 (UTC)

The article has been protected once again. As to Glkanter's participation: He hasn't always acceded to my requests and has made some statements that have indicated less than full commitment to positive participation in the mediation. I would be willing to AGF regarding his participation if I hear a positive response from him on this, though. Garry would you be willing to reaffirm your commitment to participate collaboratively in this mediation and support the groundrules? Sunray (talk) 21:27, 8 October 2010 (UTC)

Well, Sunray, I don't agree with the sentiment and conclusion that I am not committed to a positive participation. As we are in mediation, it's obvious there are conflicting goals and approaches to reaching consensus, editing the article, and even whether the article needs to be changed. I'm not aware of any groundrule violations that I've made since you told me to stop making accusations against the other editors. I think I'm bringing up very relevant arguments as to why the article needs to be changed. Others don't see it that way. My arguments for the most part are ignored, or I'm told I'm wrong. I don't think so. So, to the best of my ability, I am trying to change the article via this mediation. I edited the article this morning. Some other editors didn't like that. I was not sneaky, I had discussed the changes at great length, I got reverted. Earlier this week I was personally attacked by an editor who grossly misquoted me in order to make a horrible accusation. I don't understand what I'm doing wrong, other than advocating an unpopular viewpoint that the article suffers from NPOV, OR and UNDUE weight violations. What are my alternatives, should I give up? Glkanter (talk) 22:13, 8 October 2010 (UTC)

I agree that you did respond to my request to stop making accusations. I also agree that you have been discussing various aspects of the article reasonably. I was unhappy that you edited the article absent consensus and contrary to my request. However, that is now in the past, so I'm happy to move on. As to my not responding to points you raise about the article: I don't see that as my role. Mediators are not supposed to act as subject matter experts. We are here to facilitate discussion and identify points of consensus. Sunray (talk) 15:24, 9 October 2010 (UTC)

Thank you for the response. I guess that means if an editor makes sourced, relevant edits to an article in mediation that are always reverted by other editors, this is not an 'anomaly' within the mediator's purview. I see a problem with that, but I guess now I better understand the mediator's role. Glkanter (talk) 19:27, 9 October 2010 (UTC)

The arguments I make are based on the sources, Wikipedia policies, and sound logic. Other than one brief comment a few days ago by you, Sunray, I'm not sure that you read my arguments supporting the changes I am calling for. For example, in today's discussion with Richard, I show that Morgan criticizes every solution presented in the article, but only the simple solutions reflect that, at great length in the article. I see that as an NPOV violation. Seems pretty straight forward. Glkanter (talk) 22:36, 8 October 2010 (UTC)

Here's another one. Selvin originates the 'game show' puzzle in a letter to a professional journal. He has the host revealing a particular empty box (revealing a goat behind door 3). He offers a simple, unconditional solution. He receives correspondence from his colleagues, and specifies 3 premises in a subsequent letter. In that second letter, he concludes with praise for Monty Hall's very simple unconditional Carlton-like solution (Selvin and Hall had corresponded). He names his puzzle 'The Monty Hall Problem'. He also provides a conditional solution, maybe for those who had written him about this type of solution. There is no way this source should be given an attribution to support the section criticizing the simple solutions as not solving the right problem. To do so, as the article does, is illogical, and simply not consistency with the source. Glkanter (talk) 23:42, 8 October 2010 (UTC)

The last point mentioned by Glkanter point illustrates what is going on. I think there is no problem with good faith, but there is a problem with understanding of written texts. Whether this is because editors don't read carefully, or their minds or clouded by emotion, or they are not aware of technical issues of probability theory, I don't know. Glkanter thinks that Selvin is cited where he was cited in order to support a point of view that conditioning is good, not conditioning is wrong. I think that Selvin is cited where he was cited in order to refer to a place where conditional probabilities are computed for MHP. The text is like this:

Morgan et al are conditionalists [cite Morgan et al.].

The probability of A given B is a conditional probability [no citation].

Computing probability of A given B is a different job from computing probability of A [citation to lots of people who compute conditional probabilities, including Selvin].

We understand that Glkanter has a problem with the point of view of the editors who push the personal opinion that you have to solve MHP with conditional probability. He pushes the personal opinion that you shouldn't. However the words themselves, to my mind, are totally neutral. I don't understand Glkanter's difficulty with them. Gill110951 (talk) 06:56, 9 October 2010 (UTC)

With the exception of the paragraph that describes the decision tree, most everything else in the Conditional solution section is there to promote the idea that the simple solutions are flawed. Selvin is not one of the sources that advocates that. Neither is Seymann. Many other sources simply state their belief that the simple solutions are correct, but do not address their (conditional) interpretation of the puzzle. There are many more issues with that section and its source attributions. I will follow Will BeBack's instruction to be brief, and not bring them up (again) at this time. Glkanter (talk) 11:47, 9 October 2010 (UTC)

And please, Richard, don't attempt to speak for me, as you do in 2 of the 3 paragraphs above. Glkanter (talk) 11:52, 9 October 2010 (UTC)

The Elusive Consensus

If you don't mind, Will BeBack, I'd like to share an observation with you.

Some editors in this mediation argue primarily from the literal words of the sources.

Some editors argue primarily on the sources' intent.

Some editors argue primarily from the Truth.

Worse than herding cats. Until we are all using the same rulebook, nothing's going to change. Glkanter (talk) 21:35, 9 October 2010 (UTC)

I'd like to discourage the creation for fresh threads. Let's stick to the task at hand: /Conditional probability solution   Will Beback  talk  21:45, 9 October 2010 (UTC)

I will not start a new thread but I'd like to make a general observation which I think might help us. MHP is a puzzle, a paradox. Usual intuition gives the wrong answer. One has to take a step back, think clearly and think out of the box in order to see that the player ought to switch. This means that at some point we have to suspend our direct intuitions and switch to careful conscious logical thinking. Later, we may develop an intuition which helps us give the right answer in similar situations in the future, but for newcomers to MHP that intuition is not there.

When we change mode and start using conscious rational thought instead of immediate intuition, we must distinguish in the original real world story between those elements which are important to keep in mind, and those which are merely ornamentation. I think that the disagreement between conditionalists and unconditionalists is actually not a disagreement between using conditional probability formalism or not, but is a disagreement whether the specific door-numbers are ornamentation or essential. Afterall, both are interested in the situation after a door has been opened, and clearly the probability that there is a car behind door 3 (or after "a door") after it is opened revealing a goat has indeed dramatically changed.

In a sense the disagreement is harmless, because under the usual supplementary assumptions about equal probabities all over the place, people who take the door-numbers on board, and work carefully through the more complicated problem do discover, after going to all that trouble, that whether or not you should switch does not depend on the specific door numbers in hand. However, what if we had discovered that the door numbers were relevant??? Admittedly you would have to change the problem a lot to make them relevant! See the paper by Jef Rosenthal. But please remember that a priori, the first time people hear about MHP, almost no-one thinks it is of any relevance that of the two doors closed after the host has opened Door 3 and revealed a goat: one was selected by the Player and the other (the other door left closed) was selected by the Host. To say it differently: newcomers to MHP see at this stage of the game two closed doors with numbers 1 and 2 on them and one opened door, Door 3, with a goat standing there; they don't "see" that the two closed doors got to be in that state of being still closed in a very asymmetric way. It's not the numbers and the open/closed which counts, it is the specific history which led up to that situation. It's important that Player chose 1 and Host left 2 closed, rather than Player chose 2 and Host left 1 closed. In fact: it's only important that Player chose A_DOOR, Host left AN_OTHER_DOOR closed.

Could we agree that the big question is not conditional or unconditional, but whether the door numers are taken care of outside or inside the formal analysis? The advantage of keeping them outside, in the informal, pre-processing stage, is that newcomers to MHP are much more easily taken on board, and that the essence of MHP comes across to anyone. You don't have to have a PhD to understand that its smart to switch. The point to taking them inside is that this does make the solution "more complete". Because, as I have said before, by taking explicit note in the logic/mathematics of the specific door numbers, one does rule out the possibility that some mixed strategy of switching or staying depending on the door numbers in the case at hand, is better still.

The big first step in enjoying MHP is to understand that switching beats staying. For the connoisseurs it is important to realise that these two are not the only options and that we can with just a bit more work say something better still. Clearly the editors are divided in the accent which needs to be put on the stuff for the connoisseurs. There is at least one editor who thinks the connoisseurs ought to be totally ignored, there is at least one editor who thinks that MHP is really only for the connoisseurs. Personally I find both of those two points of view not conducive to constructive editing of an important wikipedia page.

I realise that all this is a personal opinion. However it is a personal opinion which I have come to out of the challenge to find a synthesis of the different positions held by different editors here, which respects them all, does them all justice, and results in something richer.

Moreover it is strongly influenced by my professional life-long experience consulting in statistics, with clients from law and medicine, psychology and linguistics, computer science and quantum physics. This has taught me that solving real world problems with mathematics and logic is an art. One must make choices which are a matter of taste. One has to balance degree of relevance to the actual problem, and amenability to formal solution. A solution to a problem in medicine or law which is absolutely correct from a scientific point of view but which one can't explain to a medical doctor or a lawyer is deeply unsatisfactory. We always have to compromise, and finding a beautiful compromise is a creative process. "The solution" depends not just on the problem but also on who asked the problem.

Wikipedia is an encyclopaedia, and people come from many backgrounds to learn about MHP. We want most everyone to be able to find "their" solution. That means that we do have to present different solutions in an unprejudiced way. I think that means that all editors working on the page have to work hard to try to understand all the main points of view which are out there. Richard Gill (talk) 12:06, 29 October 2010 (UTC)

The Elusive Consensus, 2

I think all editors, except myself, want the main part of the article to take on board all the standard assumptions of equal probabilities all over the place. I'm happy to abide with the majority on this as long as it is said *explicitly* near the beginning of the article that that is the point of view of many sources, but not all. That resolves my difference of opinion with everyone else: alternative assumption sets come later in specialist sections.

1. Can everyone agree that there is a conceptual difference between the statements: "2/3 of the time, the car is behind the other closed door", and "2/3 of the times that the player chose Door 1 and the host opened Door 3, the car is behind Door 2"?

2. Can everyone agree that whatever vos Savant and Whitaker might have intended, and whatever all the other sources might think or claim they are doing, the simple solutions (I refer to the reasoning in these solutions, not to their bottom line "switch") only directly tell us that the first statement is true, wherease the more complex conditional solutions directly tell us that the second statement is true?

3. Can everyone agree that for most people it is intuitively obvious that the relative frequency with which the car is behind the other closed door won't change when considered separately for any particular combination of initial door chosen and door opened by host? (I take it everyone does agree that this is true - what I am looking for consensus on, is the idea that most people will take this for granted).

4. Can everyone agree that if (counterfactually!) those relative frequencies could have been different, then we could have been more interested in the in the conditional probability than the unconditional probability, that switching gives the car?

5. Can everyone agree that because of the full symmetry of the fully specified standard problem, the second (conditional probability) statement follows logically from the first (unconditional)?

If so, then the difference between the "completeness" of the simple solutions and conditional solutions to the full or standard MHP, is merely in whether or not the "details" I have listed here as points 1 to 5 are mentioned explicitly or not. Some editors think strongly that a solution is not complete if you have not dealt explicitly with these issues, others think that bothering about those details is a total waste of time. But more importantly, some sources implicitly have the one opinion, others explicitly have the other.

It would be good if we could agree on the facts of the matter, so that finally all that remains is to agree on relative weight paid to different subtopics. Richard Gill (talk)