Talk:Monty Hall problem/Archive 25

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How far have we come?

When I first came to this article it was in response to an RfC complaining that a group of editors were dominating the page by insisting that the simple solutions were incomplete or answered the wrong question and that this should appear prominently and early in the article.

Now we have Rick saying that the simple solutions are incomplete or answer the wrong question and that this should appear prominently and early in the article.

This assertion is simply not supported by the sources. There are very few, if any, reliable sources that clearly say that, in the event that the producer places the car uniformly, the player chooses uniformly, and the host chooses an unchosen goat-hiding door uniformly the simple solutions are wrong. Pretty well all the sources which criticise the simple solutions do so on the basis that the host chooses non-uniformly. Martin Hogbin (talk) 08:19, 17 May 2011 (UTC)

It is not "now" but rather "still the same". Many (published) forms of simple solution are still "incomplete" and/or still solve "another problem" depending what you pick as your "canonical problem" and various sources do state that explicitly. The only thing hasn't changed here, is your claim that wouldn't be so.

There is no need to mention a particular (detailed) criticism of the simple solution, there is however a need to point out the ambiguity of the problem, which causes it to still be a subject of (academic) debate and allowing for various "correct" solution (including some leading p=1/2 as the winning probability for switching). But even more importantly the lead needs to be a somewhat appropriate summary of the article (or its most important aspects, the criticism of the simple solution might not be among them, but the ambiguity clearly is).--Kmhkmh (talk) 09:47, 17 May 2011 (UTC)

Kmhkmh, yes of course we want to put in the lead something on ambiguity in the question the fact that there is criticism of the simple solutions but not Rick's own version of the truth as he has proposed above. Martin Hogbin (talk) 16:11, 17 May 2011 (UTC) What do you think of my latest suggestion above for the lead? Martin Hogbin (talk) 16:21, 17 May 2011 (UTC)

I am taking a wiki-break for the next three weeks and unfortunately I don't have time now to add my two dollars worth. Here are two cents: 1) I agree with Martin; 2) take a look at The Monty Hall Problem: Switching is Forced by the Strategic Thinking by Alexander Gnedin. MHP is a moving target. While people on wikipedia still struggle to relax the icy grip on the article in which Rick Block has been holding it for several years (an icy grip based on taking one particular academic viewpoint of MHP as dominating all others) the problem moves on, both in academic and popular literature. Richard Gill (talk) 08:34, 17 May 2011 (UTC)

PS Gnedin writes In this paper we elaborate details of the game-theoretic approach. Our main point is that the fundamental principle of eliminating the dominated strategies provides a convincing explanation of the advantage of the switching action to the man from the street, as compared with the more sophisticated arguments based on decision trees, conditional probabilities and Bayes' theorem. Every Contestant's policy "choose door Y and stay with it" is outperformed by a policy "choose door Y' different from Y then switch", no matter what Host does. Once the man from the street adopts strategic thinking and realizes that there is a two-step action, the comparison of alternatives becomes obvious and, moreover, free of any probability considerations.

PPS Rick writes the "academic" interpretation, which IMO (supported by Krauss and Wang, referenced in the article) matches how nearly everyone understands the problem whether they have the vocabulary to express it this way or not... Here he claims that nearly everyone understands the problem in the same way as he does. This is however manifestly untrue. Nearly every editor of the page on MHP does not understand the problem as Rick does, so that takes care of what the majority of interested laypersons think. As to whether the academic interpretation is what the majority of people in academia think ... this is also manifestly untrue. In one particular academic sub-world (the teachers of elementary probability and statistics), a certain formulation of MHP (as a problem to be solved using Bayes theorem) became traditional. In other academic subworlds, completely different formulations are adopted. And most recently, Gill - himself coming from the traditional world of the Bayes theorem pushers - has been arguing strongly that that kind of solution is an example of solution-driven science, and the kind of bad statistical thinking which leads to so many catastrophes in law and public debate. Richard Gill (talk) 08:48, 17 May 2011 (UTC)

We have no need for "manifestly untrue" based on personal observation (mine is different than yours in that regard btw.) There are some (psychological) publications that deal more or less explicitly with the question how the problem is perceived by people (like Krauss & Wang for instance).--Kmhkmh (talk) 09:55, 17 May 2011 (UTC)

Indeed, how we humans psychologically perceive the problem explains why most people initially get the answer wrong. How the problem is best solved by different kinds of ordinary people and in different academic communities (many different ways) is quite another matter. Krauss and Wang have their own bias, I have another, Rick and Kmhmh and Martin have other biases. I just point out that Rick's "nearly everyone" is not a helpful way to achieve concensus. I am not arguing against his person but against his position, which seems to me to be as unchanging solid as ever. Richard Gill (talk) 06:59, 18 May 2011 (UTC)

Can we please stick to reliable sources here rather than engage in further ad hominem personal attacks?

Martin - in response to your assertion above that I'm saying "that the simple solutions are incomplete or answer the wrong question and that this should appear prominently and early in the article" please remind me where I said this (in the form of a diff if you would), or is this how you're summarizing the suggested text at the end of this edit? If the latter, please explain since neither of the words "incomplete" or "wrong" (or any synonyms) are anywhere in this text. As far as what I am proposing, what I've said repeatedly (for example [1]) is that I want an accessible conditional solution to appear fairly early, presented in an editorially neutral fashion per WP:NPOV, because both simple and conditional solutions are prevalent in the literature.

Richard - If you're not reading Krauss and Wang to say that 97% of their test subjects initially created a conditional mental model of the problem, how are you reading what they're saying? Per Tijfo098's oft-repeated comments, K&W say other things as well (like training subjects in Bayes theorem does not increase the ability to correctly solve related problems), but this in no way contradicts the observation that nearly all people initially understand the problem to be about the (conditional) probabilities at the point the player has picked door 1 and has seen the host open door 3 (at least until they're bludgeoned into changing their mental model of the problem). Conjecturing how nearly every editor of the MHP article understands the problem is both pointless and irrelevant. What is relevant is what reliable sources say. Many reliable sources attempt to bludgeon the reader into changing their mental model (i.e. present simple solutions). Many reliable sources directly address the conditional probability. Some (a not insignificant number of) reliable sources specifically criticize sources presenting simple solutions. Some reliable sources present a game theory based approach. Our task as editors (per WP:NPOV) is to represent what all of these sources say "fairly, proportionately, and as far as possible without bias". Our own opinions about what these sources say (even Richard Gill's opinion) is irrelevant. -- Rick Block (talk) 16:05, 17 May 2011 (UTC)

Rick, how Krauss and Wang explain how most people see the problem and hence why they get it wrong does not tell us how the problem ought to be solved. It has to be solved by looking at it in another way, clearly. It is solved in many different ways, and different people prefer different approaches. It is not initially a well posed maths problem. It is a picture in our mind of a real world situation. How to mathematize it is not fixed by the problem. Richard Gill (talk) 07:02, 18 May 2011 (UTC)

Rick, your proposal for the lead says that the simple solutions do not answer the question asked. What about my latest suggestion above. Martin Hogbin (talk) 16:25, 17 May 2011 (UTC)

Speaking as someone with no dog in this fight (I neither agree with Rick or with Martin but instead remain neutral on the content dispute), I have an observation. If the consensus is as overwhelming as is claimed (my count does not confirm this), why then the reluctance to work together to create the two versions I discussed earlier as being a necessary step to resolving the content dispute? It is almost as if both one or parties aren't so sure that they have a case and are afraid that content dispute resolution will go against them. Instead I see arguing that goes nowhere. How is this different from all the past arguing that went nowhere? Why not give my plan (going through the steps of Wikipedia content dispute resolution ~~for the first time~~) a chance? Worst case it fails to work and you will be back where you are now ~~without me bugging you about never having tried content dispute resolution.~~ Guy Macon (talk) 17:17, 17 May 2011 (UTC)

I have no objection. What do you suggest is the next step? Martin Hogbin (talk) 19:27, 17 May 2011 (UTC)

See below. Guy Macon (talk) 10:53, 18 May 2011 (UTC)

Pictures

On request of Guy Macon, I explain here my concerns with the pictures next to the simple solution section. They also seem to provide a solution, but they don't. In the pictures a box is drawn around the two not chosen doors, and it is indicated the chance on the car to be in this box is 2/3. And so it is, not only because it is complementary to the chosen door, but equivalently because each of the doors inside the box has chance 1/3. Then in a subsequent picture door 3 is shown opened, and as it does not show the car, seemingly the conclusion is the chance on the car for this door is (now) 0, leading to the next conclusion that the car then must be with chance 2/3 behind the other door in the box. But why should this be? Although the chance for the box was 2/3 at the start, after the opening of door 3 the chances are changed. Clearly for door 3 the chance has changed from 1/3 to 0. It is not obvious that the chance for the box is again 2/3. This could have changed as well. A proof is needed to show this, or to show directly that the posterior chance for door 2 is 2/3, or equivalently 1/3 for door 1. The pictures give a fallacious image of a solution.81.167.87.82 (talk) 18:59, 10 May 2011 (UTC)

The picture need only illustrate the problem; it need not provide the proof. We know the big box will have the car 2/3 of the time. We also know that at least one of the doors in the box will have a goat. Monty is just pointing to a door with a goat. The a priori probability for the opened door did not change; we just know that the car is not behind that door. The picture is not fallacious. Glrx (talk) 19:14, 10 May 2011 (UTC)

Hi! Just to be clear, are you saying that it isn't better to switch (50/50 chance) and the picture incorrectly implies that it is? Or are you saying that it is better to switch (33/66 chance) and the picture fails to properly show this?Guy Macon (talk) 19:19, 10 May 2011 (UTC)

To whom are you directing your question? 81.167.87.82 or Glrx? I'm not saying the pix is a fallacious image, so I presume you mean 81.167.87.82. It is better to switch; the box still holds the 2/3 a priori probability. Glrx (talk) 19:32, 10 May 2011 (UTC)

@Guy: Okay, the last, as I already said so above. The posterior chance for door 2 on the car is indeed 2/3, but the pictures do not show (prove) this. I'm glad Glrx above shows in his comment where the problem lies. He, and many readers, will get from the pictures the wrong impression that the box has unalterably a chance of 2/3 on the car. 81.167.87.82 (talk) 19:39, 10 May 2011 (UTC)

Uh, I don't believe I have the wrong impression at all. Keep everything a priori and look at the information. A statement about something inside the box whose P{}=1 (and therefore carries no information) does not change the information outside the box. Glrx (talk) 21:20, 10 May 2011 (UTC)

You can't have it both ways. Either the box is showing the a priori probabilities, in which case both before and after the host opens "a door" the chances of each of the doors within the box remains 1/3 (you don't know which door the host opens, so you have no information with which to update the probability of either one) - or the "after" picture is showing posterior probabilities (where door 3's probability is now 0), in which case the (posterior) probability of the two doors within the box is not necessarily the same as the a prior probability. As it stands, the picture (confusingly) shows the posterior probability of the box of two doors after the P{}=1 event the host opens a door (which certainly remains 2/3, matching the a priori probability) but the posterior probability of door 3 after the event the host opens door 3 (which is certainly 0). These are different events. Pick one. Either show the posterior probabilities of door 2 and door 3 after the host opens "a door" (in which case the probability of the two doors together must remain 2/3 and the probability of each door must remain 1/3). Or show the posterior probabilities of door 2 and door 3 after the host opens door 3 (in which case the probability of door 3 must be 0, but the probabilities of the two doors do not necessarily remain 2/3). This image exemplifies the confusion Falk discusses concerning the difference between knowing "a door" will be (or has been) opened vs. knowing a specific door has been opened. These are not the same. -- Rick Block (talk) 04:23, 11 May 2011 (UTC)

A quick note to those who disagree with Rick. We get it. You don't agree. There is no need to disagree every time. Getting back to the above, Rick, what specific changes should be made to the picture or its caption to remove the confusion? Guy Macon (talk) 08:19, 11 May 2011 (UTC)

I take the numbers written above each illustration as a priori numbers. You don't have a problem with those. There's a P{}=1 event (unbiased Monty revealing a goat behind 2 or 3); the illustration doesn't literally show Monty's action a la Falk because it opened #3; that does not mean the pix is bad. An illustration with indistinguishable doors inside the box would probably be more confusing. If you can illustrate that better, then do so. The main bullet is the contestant gets to choose the box. If we accept P{}=1, then the posterior probs written below are reasonable. (If you argue that P{} isn't 1, then the probs are wrong, but that's not what is being illustrated.) Glrx (talk) 14:56, 11 May 2011 (UTC)

I think the caption definitely implies all the probabilities in the second image are posterior probabilities ("Player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance."). If this picture is showing a P{}=1 event, then the posterior probabilities shown below the figure are clearly incorrect since a P{}=1 event cannot change any probabilities - i.e. the posterior probabilities for door 2 and door 3 should both be 1/3 (even though door 3 is shown as open!). The point is that most people aren't thinking about the P{}=1 event, but the P{}≠1 event where the host opens door 3 (which is what the picture actually shows). With this event, the probability of the player's initially selected door is not necessarily fixed at 1/3, and the probability of the other two doors is not fixed at 2/3, so the probability of the open door being 0 does not (per Falk) imply the probability of the other unselected door is 2/3. Even though it turns out that the (conditional) probabilities are 1/3:2/3:0, it is not because of what the picture is showing. This argument is definitely presented in reliable sources, so has a place in this article, but (IMO) the place should not be at the start of a section purporting to present a "solution" to the problem. On the /Arguments page, Handy2000 suggests it might be presented as an example of an "incorrect solution". I think most sources presenting this "solution" explicitly present it as an aid to understanding, so moving this to such a section with appropriate qualifications would be OK with me. Presenting it as an example of an incorrect solution seems a little harsh, although it is equivalent to Morgan et al.'s F5 (which is presented in this paper as one of several incorrect solutions). Changing the caption of the second figure to "With the usual assumptions, the player's pick remains a 1/3 chance ..." would help, but this image in all likelihood leads to more misunderstanding than understanding. Again per Falk, there is no law of probability that says that the sum of the probability of any two doors together must be the same before and after the host opens a door. -- Rick Block (talk) 05:18, 12 May 2011 (UTC)

81.167.87.82, your comments sound extraordinarily familiar. Nothing to do with conditional probability, I suppose? Martin Hogbin (talk) 09:32, 11 May 2011 (UTC)

The above is a comment that should be placed on 81.167.87.82's user page, or possibly at WP:SPI depending on what you are implying. The Monty Hall Problem Talk Page is dedicated to suggested improvements to the Monty Hall Problem Page such as discussing whether the image should be modified. Guy Macon (talk) 08:05, 12 May 2011 (UTC)

Rick, you say:
"...the probability of the player's initially selected door is not necessarily fixed at 1/3, and the probability of the other two doors is not fixed at 2/3, so the probability of the open door being 0 does not (per Falk) imply the probability of the other unselected door is 2/3"
You are citing Falk, you say "(per Falk)", but you misinterpret Falk. Did you read Falk? She said quite another thing: The probability of the door selected by the guest does not change, unless there is a "given bias of the host in opening just a special door", if he has the choice, and that you "know" about that given bias. Otherwise not. Please don't misinterpret Falk. She says that the host's action "could" be giving you additional information on the actual location of the car, but just only under the "condition" that you exactly "know" about his given bias, otherwise not. But you don't know anything about such "known" bias, or do you? So anything "could" be, in effect. Yes, the probability of the other unselected door in any case is at least 1/2 or higher, and it "could be" even full 3/3, as per Falk. Repeat: "could". But as long as you don't exactly know the "known" host's bias, its extent and its direction, you are quite out of position to predict that "q" has another value than "1/2", to assign any other value than 1/2 to q. Trying to assign any other value to "q" is nothing more than "What can be, can be". Without relevance for the famous question, being a question not about "what could be, could be", but about the strict fact that staying never can be better than to switch, and that in each and every given case. And as long as no evidence about a given and known host's bias is provided, there is "no additional information" that could allow you to revise the odds on the door first selected by the guest. No chance. Gerhardvalentin (talk) 13:25, 13 May 2011 (UTC)

I've been away for some days, and notice that Rick Block has given a complete analysis. Nothing I can add. What remains is the question why these pictures has not long been removed or at least adjusted. 81.167.87.82 (talk) 21:27, 18 May 2011 (UTC)

Additional assumption concerning the host's choice

The value of the actual "conditional" probability of winning by switching can differ from the "overall" probability only in case that additional relevant info on the actual location of the car/objects is assumed to be revealed. Otherwise not.

Actually it reads:

A common variant of the problem, assumed by several academic authors as the canonical problem, does not make the simplifying assumption that the host must uniformly choose the door to open, but instead that he uses some other strategy. The confusion as to which formalization is authoritative has led to considerable acrimony, particularly because this variant makes proofs more involved without altering the optimality of the always-switch strategy for the player. In this variant, the player can have different probabilities of winning depending on the observed choice of the host, but in any case the probability of winning by switching is at least 1/2 (and can be as high as 1), while the overall probability of winning by switching is still exactly 2/3. The variants are sometimes presented in succession in textbooks and articles intended to teach the basics of probability theory and game theory. A considerable number of other generalizations have also been studied.

I suggest to add as follows:

A common variant of the problem, assumed by several academic authors as the canonical problem, does not make the simplifying assumption that the host must uniformly choose the door to open, but instead that he uses some other strategy. The confusion as to which formalization is authoritative has led to considerable acrimony, particularly because this variant makes proofs more involved without altering the optimality of the always-switch strategy for the player. In this variant, the player can have different probabilities of winning depending on the observed choice of the host, based on additional/closer information as to the actual location of the car. But in any case the assumed/observed "closer" probability of winning by switching will remain within the range of at least 1/2 (but never less), and can be as high as 1, while the overall probability of winning by switching of course remains unchanged still exactly 2/3. The variants are sometimes presented in succession in textbooks and articles intended to teach the basics of probability theory and game theory. A considerable number of other generalizations have also been studied.

Just to help improve perceivability of that aspect of the "host's choice". Regards, Gerhardvalentin (talk) 15:49, 11 May 2011 (UTC)

The paragraph needs work to be readable. Adding awkward technical phrases does not help. WP is an encyclopedia and not a journal article. Glrx (talk) 16:05, 11 May 2011 (UTC)

Right. But foremost it should say (not hide) what matters: Assumed "closer" info on the current/actual location of the objects. Gerhardvalentin (talk) 16:37, 11 May 2011 (UTC)

Actually, that's backwards. The lead paragraph needing to be easily understandable without technical phrases is a Wikipedia requirement. See WP:TECHNICAL for details. "Saying and not hiding what matters" is not a requirement for the lead. It can be explained in as detail using highly technical language lower in the article. The above paragraph is too technical for the lead. It either needs top be simplified or the suggested place to add it needs to be lower in the article. Guy Macon (talk) 08:33, 12 May 2011 (UTC)

Yes, exactly, from plain to more technical. Let's try to steadily say very clear what's up and what matters, never hiding behind misty obfuscating insinuations and innuendoes, even if sources sometimes should be hazy. Gerhardvalentin (talk) 16:16, 13 May 2011 (UTC)

Agree 100%. BTW, I like the way the article is going right now. I am seeing steady improvements.

I agree with Guy here. The above paragraph is far too technical and too detailed. Also we should point out that although some sources do not make the simplifying assumption that the host must uniformly choose the door to open they do, without giving any reason, make the simplifying assumption that the producer places the car uniformly behind the doors. This would be better in the 'Criticism' section. Martin Hogbin (talk) 22:10, 13 May 2011 (UTC)

How about:

In a common variant of the problem, assumed by several academic authors, the host is taken not to randomly choose which door to open, although the car is still placed randomly. In this variant the player still cannot lose by switching and will probably gain, having a 2/3 chance of winning on average if he switches. Martin Hogbin (talk) 22:17, 13 May 2011 (UTC)

Frankly, I'm confused and don't know what to say to these discussions. For this talk page section, Gerhard wasn't talking about a lead paragraph; he was talking about the last paragraph of the problem description section. In the proposed alternative paragraph, the first sentence is dense, overloaded with clauses, and loses the audience. The second sentence, in an attempt to be overly accurate (or deferential to other editors?) in a summary, sounds contradictory: a switching player "cannot lose" but has a 2/3 chance of winning? Consequently, I don't support the proposal. Glrx (talk) 15:23, 16 May 2011 (UTC)

If the intent is to address what we've been arguing about for multiple years, this completely misses the mark. What we've been talking about is NOT variants of the problem description, but two different interpretations of what probability is of interest. The "simple" solutions address the outcome of always switching as opposed to the outcome of always staying with the player's initial choice - without considering any impact of how the host chooses which specific door to open (only that the host will open "a door"). This corresponds to P(win by switching), or assuming the player has picked door 1, P(win by switching|player initially picked door 1). Both of these are 2/3. The "academic" interpretation, which IMO (supported by Krauss and Wang, referenced in the article) matches how nearly everyone understands the problem whether they have the vocabulary to express it this way or not, is that the probability of interest is the conditional probability for a player who has picked a specific door (say door 1) and has then seen the host open a specific door (say door 3). This corresponds to P(win by switching|player initially picked door 1 and host opened door 3).

People arrive at the 1/2 answer because they are thinking about the conditional situation (player having initially picked door 1 standing in front of a closed door 1, a closed door 2, and an open door 3 showing a goat). The probability of winning by switching in this situation averages 2/3 and is at least 1/2 and possibly as high as 1 depending on how the host chooses which door to open (per numerous academic references). The host has no choice if the player's initially selected door hides a goat (the host must open the only other door hiding a goat), however if the player's initially selected door hides the car the host chooses which "goat door" to open. If this choice is random, then the player's chance of winning by switching is 2/3 in all specific cases. If this choice might not be random (i.e. is not specified in the problem description to be random), then the player's chance of winning by switching is something in the range [1/2,1]. The "simple solutions" do not distinguish between these cases, because the outcome of always switching (or always staying with the initial choice) is not affected by this choice.

Given that nearly everyone interprets the problem to be about the conditional probability (again, whether everyone can express it this way is quite another issue), IMO the "variant interpretation" here is that of the "simple solutions". This interpretation requires the player to not know which door the host opens (the player is asked to decide whether to switch before the host opens a door, or the player can tell only that the host opened "a door" but not specifically which door), or an argument from more advanced probability that a player who does not know how the host chooses between two "goat doors" effectively has a 2/3 chance of winning (from this player's limited perspective).

If we're looking for wording to resolve the argument, IMO it should be something like: A common interpretation of the problem assumes the probability of interest is the probability of winning by switching in a specific case, such as the player picks Door 1 and the host then opens Door 3, as opposed to the probability of winning by always switching. These are mathematically different probabilities. The probability of winning by always switching is 2/3 if the host must open a door showing a goat and must make the offer to switch. However, with only these constraints the probability of winning by switching in any specific case might not be 2/3-it is at least 1/2 and perhaps as high as 1. To ensure a 2/3 chance of winning by switching in any specific case, the host must also be constrained to randomly choose which door to open if the player's initial choice happens to hide the car. -- Rick Block (talk) 17:43, 16 May 2011 (UTC)

I have no objection to the above wording. It does not violate WP:TECHNICAL and it is supported by many citations Having read the entire history of this talk page (which took many hours) I can say with confidence that Rick is correct when he says that what everyone has been arguing about for multiple years is based upon two different meanings of "probability." See Probability interpretations for details. Rick and Martin are not just talking about the same thing in different ways. There really are two different kinds of probability that we can use to describe / explain the Monty Hall Problem. And right now by my count the current consensus is two for one way of writing this page, two for the other way, one neutral (me) and several editors who have not expressed an opinion on the subject.

What I have been advocating is that we work together to come up with two versions that only differ where the two ways of writing this page differ. Once we have that, other editors will be able to make a choice between the two versions, consensus can be attained, and we will be done with this multi-year content dispute. Having the page reflect one position and not the other without consensus is not acceptable. Arguing about it for a couple more years is not acceptable. We need to work together as a team - a team with strong good-faith disagreements as to what the content of the MHP page should be, but a team nonetheless - and resolve this content dispute. I have presented a plan for doing so. Does anyone have a better plan? Guy Macon (talk) 19:59, 16 May 2011 (UTC)

Unfortunately Guy your plan misunderstands the situation completely. You still seem to imagine that this is a just disagreement between myself and Rick. It is not, it is a disagreement between the original 'page owners', including Rick, and nearly every other editor who has shown an interest in the subject. The current consensus structure and content does not represent one view but a compromise between the views of most editors and those of the page owners.

If you want to try to reach a compromise we need to start with the two original positions.—Preceding unsigned comment added by Martin Hogbin (talk • contribs) 08:19, 17 May 2011

As a positive suggestion, how about:

Some sources criticise the simple solutions because they because they fail for a variant of the problem in which the car is still placed randomly but the host is taken not to randomly choose between which of the two doors he may open under the game rules. In this variant the player still cannot lose by switching and will probably gain, having a 2/3 chance of winning on average if he switches. Martin Hogbin (talk) 16:19, 17 May 2011 (UTC)

This does not capture the point that solutions based on conditional probability have equal (arguably more) wp:weight in the literature than "simple" solutions. We're not mentioning anything about other approaches (like game theory) as well. How about:

Many sources show the chances of winning by switching are 2/3 by examining all possible outcomes where players switch and where players stay with their original choice. Assuming any specific case is as likely as any other, this means the probability of winning by switching if the player picks Door 1 and the host opens Door 3 must be 2/3 as well. Many other sources directly show the probability of winning by switching in the case the player picks Door 1 and the host opens Door 3 is 2/3 using an analysis based on conditional probability. The literature includes solutions to many variations of the problem based on these other approaches such as game theory.

Rick Block (talk) 15:02, 18 May 2011 (UTC)

Longstanding Content Dispute Resolution Plan Version II

There exists a longstanding good-faith content dispute, with two different visions for how the Monty Hall problem page should be written.

The two most vocal proponents of each vision are Martin Hogbin and Rick Block, but each vision has other editors who support it.

This is a plan for resolving the content dispute.

STEP ZERO (ALREADY DONE): Solve problem of misbehaving editors. This was accomplished by arbitration committee, and nobody is misbehaving now. Although a bit of frustration sometimes pops up (watch it, people!) everyone appears to be working together in good faith. It should be noted that this was done by following the steps of Wikipedia user behavior dispute resolution, and that we have not tried Wikipedia content dispute resolution, except for the early steps that are the same for both.

STEP ONE: Martin, with Rick and everyone else helping, makes this the best article he can following Martin's vision.

STEP TWO: Rick steps in and, with Martin and everyone else helping, makes the minimum changes needed to make it the best article he can following his vision.

STEP THREE: We then all work toward making the two versions as close as possible, differing only where the editors actually disagree about the content.

At this point we will have two versions that are almost identical except where there is a content dispute. This is a necessary step before we can ask anyone who is uninvolved to help resolve the longstanding good-faith content dispute - nobody is going to crawl through the endless arguments on this talk page. I suggest a subpage with the two versions on it and a succinct set of arguments in favor of each.

I am purposely leaving the question of which version stays up while we follow the next steps undefined. It needs to be discussed (in another, section, please!) but does not matter as far as resolving the content dispute goes.

STEP FOUR: We start following the steps listed in Wikipedia:Dispute resolution ("Resolving content disputes" section). Somewhere along the way, a consensus will be formed as to which vision we will follow. At that point, someone will have to let it go and accept the fact that the consensus was against them. If they refuse, we will be back to a behavioral problem, which we will solve in the usual way.

STEP FIVE: We request a featured article review.

As always, any plan we make as a group is subject to consensus, so if anyone disagrees with the above plan, please feel free to discuss your concerns. Guy Macon (talk) 02:53, 18 May 2011 (UTC)

Excellent plan. Richard Gill (talk) 06:55, 18 May 2011 (UTC)

I think a step has been missed out. We need someone from the opposing POV from Rick to also edit the current page according to their opinion. I cannot do this as my proposal was intended to be a way that both POVs could be properly represented without conflict. If others do not see it that way then I suggest that another editor, Gerhard or Lambiam maybe, steps in to edit the article according to his POV, then we look to see how we can reconcile the two POVs. Martin Hogbin (talk) 08:37, 18 May 2011 (UTC)

Alternatively, would could jump straight to step four (again), which I thought was Guy's suggestion above. Martin Hogbin (talk) 08:39, 18 May 2011 (UTC)

Alas, I don't believe that Rick agrees that the current version is an acceptable compromise. Nor do I believe that Rick could create a version that you (Martin) would find to be an acceptable compromise. If it were that easy, someone would have done it long ago.

It is unreasonable to expect anyone offering outside help to read lengthy talk page arguments, which they would have to do if we jumped right to step four. We really do need to encapsulate the essence of the content dispute in a short document first. I have explained this in detail several times already, so I am puzzled as to how you could have misunderstood it - I will assume that I wasn't clear enough in my previous explanations. I am agnostic as to how we arrive at the above-mentioned document, just as long as at the end everyone agrees that their position is properly represented.

Regarding your use of "(again)" above, repeatedly asserting that we tried content dispute resolution doesn't make it true. Either provide diffs showing that the steps "Ask at a subject-specific Wikipedia:WikiProject talk page" and "Ask for help at the content noticeboard" were completed or stop making the claim.See below Guy Macon (talk) 10:43, 18 May 2011 (UTC)

Guy - all steps of WP:DR have been previously followed, up to and including formal mediation:

third opinion - it was not a dispute between only two editors, so this step was skipped
subject specific page - see [2]
NPOV noticeboard - see [3]
informal mediation, see [4]
formal mediation, see [5]

This definitely doesn't mean that we can't or shouldn't try again, but Martin's claim that we've tried content dispute resolution is completely correct. -- Rick Block (talk) 15:14, 18 May 2011 (UTC)

My apologies to Martin. I was wrong. I did read the entire archive of this talk page, but somehow missed that. I also went to Wikipedia:Content noticeboard and searched the archives for "Monty" and got no results, yet there they are. Sorry about that. Guy Macon (talk) 16:46, 18 May 2011 (UTC)

I don't like the plan (whether the principals like it or not). It appears to cede ownership of this article to two editors. Abstractly, I would not expect a good result from telling advocates from opposite universes to sit down and compromise. Practically, it seems that every suggested change, no matter how narrowly it is focused, is criticized because it does not encompass some larger aspect. I see confusion/disagreement about whether something is a problem variation or an interpretation. Seemingly simple statements get long responses. The result is editors propose complex, tortured, sentences that are incomprehensible to the target audience.

Reasonable editors have been scared off this page. Reasonable editors don't like the idea of getting sucked into a lame discussion. The label is not a badge of honor.

Instead of straw polls on an issue, there's a period of silence and then editors claim that they hold proxies for this, that, and the other thing. Instead of clear statements of the issues, there are references to so-and-so's position that are buried in a record so confused that even the speaker doesn't have a clear understanding.

The major problem with the plan is that it has form but does not address content. There's an elephant in the room, but there's not a clear idea of what that elephant is. It sounds like there is broad agreement that some issue should be covered, but the disagreement seems to be one of prominence: where does the coverage start or how much text is devoted to it.

What I want to see is a clear statement of that controversy. And I mean the content issue -- not that person X doesn't like the article and if it is changed then person Y won't like it.

Glrx (talk) 18:29, 18 May 2011 (UTC)

I agree that this is not a dispute between myself and Rick, that is why I suggested above that Gerhard or Lambiam should give their views also.

There is some confusion over the content issue so I will explain below. Martin Hogbin (talk) 21:50, 18 May 2011 (UTC)

I also agree that it is not a dispute between Martin and Rick. It is very likely that they just happen to have a bit more free time to write about it, but nobody wants to exclude any editor's views. Besides it being a matter of basic fairness, I really don't want to see a concise description send through content dispute resolution, a consensus achieved and then have someone pop up and claim their views were ignored, followed by a couple of more years of debating the issue. Guy Macon (talk) 22:12, 18 May 2011 (UTC)

The content dispute.

Martin's statement

For the benefit of newer editors let me explain what the content dispute is about. Standard rules that the host must always open an unchosen door to reveal a goat and must always offer the swap apply throughout

The dispute is about how valid the simple solutions to the MHP are and how any weakness in them should be presented in the article.

Undisputed facts

1) If Whitaker's question is taken to mean that the car is initially hidden uniformly, then the player picks door 1, then the host opens door 3 to reveal a goat and then the player has to decide whether to swap or not (generally called the conditional formulation here), and if the host is known not to choose a door uniformly when he has a choice (say we know he prefers to open door 3 when it is allowed), then the simple solutions fail. They give the wrong answer.

2) If Whitaker's question is taken to be asking a more general question, with the numbers being added purely to make the game rules clear (they were in fact added for this very purpose by vos Savant). In other words the question is taken to ask, 'Is it generally better to swap or not, and by how much' (generally called the unconditional formulation here) then the simple solutions are fine and the answer (probability of winning the car by switching)is exactly 2/3.

Disputed

3) As 1 above but the host chooses uniformly at random between doors when he has a choice (generally taken to be the 'standard' formulation of the problem). Here it is not disputed that the answer is 2/3 but some people believe that the simple solutions are not correct in this case.

What we say in the article

Those who believe that the simple solutions are seriously incorrect in case 3 want either to make the more complex solutions that involve conditional probability more prominent in the article or want to alert the reader early on to the fact that the simple solution may be deficient in some way or that there are other ways to solve the problem than the simple solutions.

Those who believe that the simple solutions to case 3 above are totally correct or correct but for extreme pedantry (or that Whitaker probably intended to ask version 1 anyway) consider that references to a more complex solutions or warning that they are wrong early on in the article hamper understanding of the, already notoriously difficult, solution to the problem. They also believe that version 2) is not the real MHP but an academic extension or variant of the problem which should be given only minor prominence in the article along with other variants. Martin Hogbin (talk) 22:25, 18 May 2011 (UTC)

Both sides claim that the sources support their POV. Martin Hogbin (talk) 22:27, 18 May 2011 (UTC)

Is that the issue?

Thank you for the summary. I'd characterize the heading "What we say in the article" to be more in line with "How the dispute affects the article", but your statements are clear. Does anyone dispute Martin's characterization or have something to add? Glrx (talk) 03:58, 19 May 2011 (UTC)

Rick's response

I object to Martin's characterization. Here's mine. I know it's longer - but the conflict is not simple (or it would have been resolved long ago). -- Rick Block (talk) 05:28, 19 May 2011 (UTC)

Undisputed facts

1) The usual presentation of the problem includes something like the following (this is from the version published in Parade) You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

2) Many sources (essentially all popular sources but academic sources as well but generally from fields other than math/probability) show the chances of winning by switching are 2/3 by examining all possible outcomes where players switch as opposed to outcomes where players stay with their original choice (we're calling these "simple solutions" here). Although these sources generally (never?) say this, assuming any specific case is as likely as any other means the probability of winning by switching if the player picks Door 1 and the host opens Door 3 must be 2/3 as well.

By "any case as likely as any other" do you mean invariance under relabelling the doors? The assumption which has to be made is that the host's two choices are equally likely, if he has a choice. This implies symmetry and that implies, indeed, irrelevance of specific door numbers. Richard Gill (talk) 12:01, 11 June 2011 (UTC)

3) Many other sources (essentially all introductory probability textbooks but numerous math/probability papers as well) directly show the probability of winning by switching in the case the player picks Door 1 and the host opens Door 3 is 2/3 using an analysis based on conditional probability.

4) The literature includes solutions to both the original problem and many variations based on these and other approaches such as game theory.

10) Krauss & Wang provide a source for an undisputed claim that most people (they don't say how many) encountering the Parade version of the problem make additional assumptions (the car is initially uniformly located behind the 3 doors and if the host has to choose which of two goat doors to open the host chooses randomly) that force the conditional probability the car is behind door 2 to be the same as the probability of winning for a fixed strategy of switching.

Possibly disputed

11) The probability of winning for a fixed strategy of switching (or a fixed strategy of staying) is mathematically different from the conditional probability the car is behind door 2 (or door 1) if the player has picked door 1 and the host has opened door 3 - whether or not these two probabilities have the same numeric value (in exactly the same sense as 2/3 of a meter is not the same as 2/3 of a kilometer).

Disputed

5) The literature includes a significant number of sources that to one degree or another criticize sources presenting "simple solutions" on the grounds that these solutions fail to address the specific case mentioned in the usual presentation of the problem which is (by definition) a conditional probability.

6) Presenting several sections discussing "simple solutions" in detail first in the article without any mention that other solutions even exist creates a structural POV, per WP:STRUCTURE, in violation of WP:NPOV.

7) Saying anything about conditional probability in the article, in particular contrasting the "simple solutions" with solutions based on conditional probability, constitutes criticism of the simple solutions essentially endorsing the POV of the sources from #5.

8) Saying anything about conditional probability in any early section of the article violates WP:TECHNICAL.

9) The appropriate WP:WEIGHT the "simple solutions" should get in the article far exceeds that of solutions based on conditional probability.

12) Krauss & Wang provide a source for an undisputed claim that most people (nearly everyone, 97% of their sample) initially encountering the Parade version of the problem interpret the probability of interest to be the conditional probability the car is behind door 2 given the player has picked door 1 and the host has opened door 3, not the probability of winning for a fixed strategy of switching - and changing people's focus from the former to the latter is difficult.

Discussion

I've added three additional items, including one that I'm not sure is disputed. As Guy mentions I'm focusing these explicitly on relevant Wikipedia content policies. I think it's worth noting that WP:NPOV is one of the three core content policies that "cannot be superseded by other policies or guidelines, or by editors' consensus" - which I interpret to mean that if there's a theoretical conflict between WP:NPOV and, say, WP:TECHNICAL that we must choose to satisfy NPOV first. -- Rick Block (talk) 15:58, 19 May 2011 (UTC)

I think we can remove number 8. An early version that mentioned conditional probability in any early section dis indeed violates WP:TECHNICAL, but Rick later wrote a version that mentioned conditional probability in any early section without any violation of WP:TECHNICAL. As far as I can tell, the WP:TECHNICAL issue is gone. Guy Macon (talk) 09:32, 19 May 2011 (UTC)

Re: "if there's a theoretical conflict between WP:NPOV and, say, WP:TECHNICAL that we must choose to satisfy NPOV first" this is true unless WP:NPOV itself tells you differently. For example, WP:NPOV says:

"Segregation of text or other content into different regions or subsections, based solely on the apparent POV of the content itself [may] create an apparent hierarchy of fact where details in the main passage appear "true" and "undisputed", whereas other, segregated material is deemed "controversial", and therefore more likely to be false."

The phrase "based solely on the apparent POV of the content" means that segregation of content into different subsections for other reasons is allowed. WP:TECHNICAL is another reason, and thus segregation of content into different subsections based on WP:TECHNICAL is not a violation of WP:NPOV, not being based solely on the apparent POV of the content.

This, of course, became a moot point when Rick provided an example that had his preferred structure without being too technical, but it is not true that if there is a conflict between the particular paragraph of WP:NPOV I quoted above and WP:TECHNICAL that we must choose to satisfy WP:NPOV first. In that particular case, we must choose to satisfy WP:TECHNICAL - because that's what WP:NPOV tells us to do. Guy Macon (talk) 17:22, 19 May 2011 (UTC)

Solicit other disputes

Thank you for those comments; I know it takes time and effort. There are some parallels with Martin's, but also more focus on WP criteria. Are there other disputes that Martin or Rick missed? Or other objections? Glrx (talk) 06:07, 19 May 2011 (UTC)

Glrx, that is principally what the two year dispute has been about. Perhaps you could give us your opinion. Is the aricle in its present form biased and, if so, in what way? Martin Hogbin (talk) 20:05, 19 May 2011 (UTC)

That's a fair question given that I've been asking everybody for input, but my concerns are different. I think the dispute has contorted the article and serves neither side well.

Generally, the article is strangely organized. Some of that results from the above dispute. I did not expect the bifurcation of "Simple solutions" and "Other solutions". (Both headings sound dismissive.) Dividing the two solutions section with the intervening "Sources of confusion" and "Criticism of the simple solutions" breaks up the presentation. The "Sources of confusion" is not confusion about the simple solutions, but rather how the problem is interpreted; it's a problem variation issue that affects any solution -- not just the simple ones. Although the "Criticism of the simple solutions" section has been intertwined with "Other solutions", it is also a variation issue: what question is being asked? Then we have an awkward "Alternative derivations" section (symmetry and irrelevance) that ties into some of the Simple solutions; Simple and Other are therefore confused. Naively, I'd expect all the solutions to be grouped together. (Perhaps this is Rick's issue 6.) In a way, showing the decision tree and Bayes early would set the stage for the problem interpretation / assumption / criticism issue because it forces the choice of q. After the solutions have been covered, then I'd expect problem variations and criticisms. Maybe there should be a separate psych section; part of the fascination is why people have trouble with the MHP.

I am confused about how divisive the door quantization issue is (one specific case versus switch in general strategy), so I'd like to seek comment on that. Does it just need mention somewhere in the article or does it need prominence? Krauss and Wang, for example, suggest that the question is clear, but that many test subjects misapply the say-door-3 in their reasoning. That's a different kind of confusion than solving a different problem. I'm unsure how much prominence is wanted for this issue.

I am ignorant of the literature that covers criticism of the simple solutions.

Looking past the dispute, many of the items in the disputed lists are not presented well. Topics are diffuse rather than concise. The variation issue, for example, is raised in at least four sections (Problem description, Sources of Confusion, Criticism of the simple, and Other host behaviors). The Problem description has a lot of detail before the reader can appreciate its importance. (Previously, I wanted to move the Problem description down.) Even undisputed material is diffuse. Bayes is covered twice; once in symbols and once in words. N doors is treated twice.

My criticisms are about the article and not the dispute. I have some other things to say, but the above comments indicate my position.

Glrx (talk) 15:50, 20 May 2011 (UTC)

Literature that "criticizes" the "simple solution" or explicitly points out the subtle differences about the exact interpretations of the problem and which probabilities are to be considered comprises among others: Morgan at al, Eisenhower,Rosenhouse, Rosenthal,Behrens.

I agree that the 2-3 years of battling over the article (mostly between 2 faction, crudely speaking Martin versus Rick) has resulted in a "bad" article regarding certain aspects, though it is still rather informative imho.--Kmhkmh (talk) 16:04, 20 May 2011 (UTC)

Kmhkmh, if you think that I am having too strong an influence over the content of this article why not give us more of your own input? I will happily take a step back for a while. Martin Hogbin (talk) 08:15, 21 May 2011 (UTC)

I assume this was just a simple copy/cut & paste error, nevertheless please be more careful with that next time. As far as your stepping back is concerned, imho it might definitely help the article if you and Rick step down for a while or rather for good even. If a conflict apparently cannot be resolved over the course of 3 years, the only way out might be to drop it and leave it to other editors. In that regard I like Gill's suggestion of leaving the overhaul to new trustworthy uninvolved editors and in doubt simply accepting their version. However I have no plans in stepping in such a function, as I'm not really a new editor to this page and I was partially involved in the past conflicts. Moreover I prefer to invest my time in writing or improving other articles, where from my perspective the result is much more productive and hence my time better spent.--Kmhkmh (talk) 10:12, 21 May 2011 (UTC)

I cannot follow your 'cut and paste' complaint, the diff seems not to work. Whatever I did wrong was a mistake.

You suggestion that I step down for good is impolite, uncalled for, and against the spirit of WP. We have just had an arbcom decision on user behaviour on this article and it has not criticised my editing or stance against page ownership. I am trying to find a way that all editors here can work together to improve the article. I was hoping that you might be prepared to help by contributing in some way. Maybe some more new editors will come here and I for one will welcome them. Martin Hogbin (talk) 17:49, 21 May 2011 (UTC)

Well the diff link is working well, but to spell it out explicitly. You modified my posting and (accidentally) removed 2 sources I was listing for Glrx. As far as the "uncalled for" is concerned. You explicitly asked me and suggested yourself stepping back for a while, so I answered. In other words I was merely answering your question. I'm happy to contribute when I'm convinced that a content contribution is not a waste of time until then I prefer to contribute content elsewhere, where it is more productive and possible more urgently needed.--Kmhkmh (talk) 00:06, 22 May 2011 (UTC)

Yes, I see. I accidentally cut-and-pasted rather than copied two sources. Sorry. Martin Hogbin (talk) 08:29, 22 May 2011 (UTC)

This article would be of much lower quality if not for Martin Hogbin's work on it. Although I naturally tend to comment when I disagree about something, there is a lot of his work that I don't comment on because it is really quite good. Guy Macon (talk) 19:19, 21 May 2011 (UTC)

Glrx, it was really Morgan et al that started the criticism of the simple solutions, though it should be mentioned that Seymann was, in the same publication, somewhat critical of their criticism. Secondary/tertiary sources, such as Rosenhouse, Rosenthal discuss the issue but are not uniformly critical of the simple solutions. It is the views expressed by these sources that should inform us on the subject regarding this article. Martin Hogbin (talk) 08:20, 21 May 2011 (UTC)

Comment from passer-by

I confess I haven't read all the voluminous discussion, and I may not have exactly grasped the nature of the dispute. However, FWIW, and from someone uninvolved with any of the recent argument (though a minor contributor to this article several few years ago), I thought I'd share my 2 cents. In my opinion, the problem, as generally understood, is that the numbering of the doors is irrelevant, that the problem is supposed to be entirely "symmetrical", that the host opens a door at random, and all other obvious and sensible assumptions are made, such that the standard answer of 2/3 is exactly and unequivocally correct. I think the article should lead with this interpretation and this solution. The exploration of all the other unintended possibilities and quirks and quibbles of the problem should not be over-emphasised but can be mentioned later for hard-core readers.

My general impression of the article at the moment is that it makes the problem, in its (as I believe) generally understood form seem more complicated and elaborate than it actually is. The whole actual problem, as I understand it, really can be dealt with in a handful of paragraphs. Then there could maybe be a stronger indication to the reader that "the rest of the article is nit-picking for hard-core pedants". 86.181.201.59 (talk) 13:11, 27 May 2011 (UTC)

I completely agree with you. However a long term contingent of editors has tried to insist that what you call "nit-picking for hard-core pedants" should be a prominent part of the article. The current structure, in which the simple probability puzzle that everyone gets wrong is dealt with first and the nits are picked later is intended as a compromise. Martin Hogbin (talk) 18:10, 27 May 2011 (UTC)

While I do think it was worth trying, ~~I don't think the compromise worked.~~ (expanded and clarified below,) That's why I want to settle the longstanding content dispute once and for all, with one unified vision as to which kind of article this is and with one group of editors being told that consensus is against them and that they simply are not going to get their way.

Alas, nobody has been willing to step up to the plate and work together to create the two versions I discussed earlier, and nobody is going to be willing to read all the voluminous discussion required if we fail to create those two versions. I must conclude from this that my plan has failed, that nobody has come up with an alternate plan, and that the argument will continue forever, with some extra spice thrown in as the bans on certain editors expire and they rejoin the argument. I am seriously considering unwatching this talk page with a request that someone notify me if things change in such a way that my plan for resolving the longstanding content dispute can be restarted. Guy Macon (talk) 19:41, 27 May 2011 (UTC)

Guy, on what basis do you say that the compromise has not worked? I have seen only one complaint. Martin Hogbin (talk) 22:21, 27 May 2011 (UTC)

I apologize for being unclear. I did not mean to criticize or endorse the current structure and content. Let me try again; the compromise did not achieve the goal of satisfying all parties and thus ending the argument. It appears likely that some of the parties will never be satisfied except by a solution that is totally unacceptable to some of the other parties. In other words, it looks like the content dispute is intractable and no compromise is possible. That's why I want to settle the longstanding content dispute once and for all, with one unified vision as to which kind of article this is and with one group of editors being told that consensus is against them and that they simply are not going to get their way. For that, I need help. Guy Macon (talk) 23:04, 27 May 2011 (UTC)

Well one important point of the problem (partially responsible for much of the academic debate) is exactly the ambiguity of the problem and that it is not necessarily obvious that the problem is to be understood symmetrically.--Kmhkmh (talk) 20:00, 27 May 2011 (UTC)

Although of interest to some, I think that is a kind of "second-level" problem or "meta-problem". The problem that, IMO, most punters coming here will want to read about is the simple "symmetrical" one, which I will refer to as the "standard" problem. I don't see any problem in dealing with the other interpretational complexities later in the article, and, indeed, the current organisation does largely seem to try to follow the "explain standard problem first, pick nits later" structure. However, there is IMO rather too much up-front agonising over interpretation quibbles. All of this can be dismissed in a few simple sentences at this stage, with a forward reference as necessary. Other simple tweaks might also help. For example, flicking through the headings "Simple solutions" ... "Vos Savant's solution" ... "Other simple solutions" ... "Other solutions" ... "Criticism of the simple solutions" ... "Conditional probability solutions" ... gives the impression that there are a plethora of different answers (presumably disputed) to the standard problem. In fact, AFAIK there is only one answer to the standard problem, and that answer is undisputed. 86.176.215.139 (talk) 20:55, 27 May 2011 (UTC)

Regrettably even that answer is disputed by what you call the "hard-core pedants". Martin Hogbin (talk) 22:28, 27 May 2011 (UTC)

Yes 86.181.201.59, you are right. The MHP is necessarily obvious to be understood as symmetrical. That any asymmetry eventually could be suspected is of no relevance for the MHP and for the decision to switch. And even to exactly know about even the most extreme asymmetry, its degree and its direction, does not change that decision.

Important: As long as no current and fixed *additional knowledge* about the direction and the degree of a known asymmetry is available, the odds on the door first selected by the guest remain 1/3. Period.

And even if you definitely should have additional information about the actual location of the car (but, for the MHP, you don't!), the odds on the door first selected by the guest could at highest move from 1/3 to zero (but never less) resp. from 1/3 to 1/2 (but never more). Irrelevant for the decision to be made. So that decision does not change.

But as long as you do not definitely have fixed *additional* information on the actual location of the car, it is completely useless to speculate about "which door" the host just has opened. Full stop.

But: On the other hand, the MHP is a welcome example in teaching conditional probability calculi, without affecting the MHP. Gerhardvalentin (talk) 22:56, 27 May 2011 (UTC)

"Additional knowledge" is not just "real" additional knowledge, but simply differences of how you judge/assess the situation and the host. And depending on that you might or might not find symmetry convincing, more importantly those different assessments (without real additional knowledge) do influence the probability calculation and they might even influence your decision (this is completely independent of Morgan's argument btw, see for instance Otto Georgii or Steinbach)--Kmhkmh (talk) 23:30, 27 May 2011 (UTC)

Thank you, Kmhkmh. You wrote ... Not just "real" additional knowledge, but simply differences of how you judge ... the situation ...
Yes, you are right. Exactly, that's it: You, and I, and anyone other are free to assume whatever one may like to assume. And free to do conditional probability calculi for any assumption one makes. "Without real additional knowledge" anyone is totally free to assume that the car is *not* behind the door first chosen by the guest, and on that base do his calculus on the probability to win by switching. Or, anyone is also free to assume (without any "real" additional knowledge) that the odds on the door first selected by the guest may have changed from 1/3 to 2/5. Totally free. And to do adequate calculi to get the respective odds on the door offered to switch on.

But this fact is solely a matter of conditional probability calculi only, never affecting the MHP and the decision asked for. No matter what any misinterpretation of any mathematical source may announce. Confusing?
No matter of how interesting and exciting the results of mathematical calculations may be, and regardless of how useful the MHP example may be in practicing mathematical skills.

Confusing? This confusion must first be over for now. Then we'll see. – As you wrote above: "Additional knowledge" is not just "real" additional knowledge, but simply differences of how you judge ... the situation ..."

Without *given additional knowledge* all of that does not affect the famous fallacy known as "The Monty hall paradoxon" nor does it affect the simple answer asked for. Are we closer than expected? Gerhardvalentin (talk) 09:32, 28 May 2011 (UTC)

Well all of those probability calculi are a part of what is called the "The Monty hall paradoxon" in literature. And in some cases various considerations do influence the decision for switching (and hence the answer to the actual question) and not just the exact probabilities. A particular example for that would be Steinbach, who suggests a randomized strategy (a coin toss decides whether you switch or not) rather than switching always.--Kmhkmh (talk) 09:45, 28 May 2011 (UTC)

What counts to find the correct answer is what you "know". Probability calculi based on any private assumptions may be part of one's "private MHP", but are not "part of *the* MHP". – However, the MHP really is part in teaching conditional probability calculi. And so it's part of the "MHP in literature", especially in textbooks. That's a fact. For the MHP, you don't have any additional "information" as to the actual location of the car. Never. And if you like, you may assume "anything you like". Solving the paradoxon?
Brilliant solution: tossing a coin. That reinforces my confidence, I see we are much closer than expected. Regards, Gerhardvalentin (talk) 10:19, 28 May 2011 (UTC)

There is no "private MHP" and "the MHP" or rather "the MHP" is entirety of the reputable literature published on it. To emphasize that again we don't define what "the" MHP is, but the reputable literature on it does. What we privately think or assume is largely of no consequence for WP.--Kmhkmh (talk) 10:26, 28 May 2011 (UTC)

Yes, and WP is for the reader. That the MHP is a welcome example in teaching conditional probability theory and in reputable math textbooks, showing there "necessary" different (unknown) assumptions just for practice only, should never confuse the article anymore. We have come a long way forward, thank you. Gerhardvalentin (talk) 11:14, 28 May 2011 (UTC)

Actually reading your comment I somewhat doubt it.--Kmhkmh (talk) 11:49, 28 May 2011 (UTC)

The doubt is all mine, Kmhkmh. I cannot believe what you just try to make me believe. Yes, you say “all of those probability calculi are a part of what is called the "The Monty hall paradox" in literature”. But here we are just talking about relevant academic literature concerning and addressing the famous MHP, and not about just only mere "textbooks for teaching and demonstrating and for learning conditional probability theory by means of some 'MHP examples', for students of conditional probability theory", such countless textbooks just representing another issue, quite another point of view.

Yes, in lessons the teacher can and will present all sorts of full baskets of underlying assumptions that never are to be given, but of course the teacher can, and is free to, just for training purpose. Not addressing the famous paradox in any way, but addressing the success of teaching conditional probability theory. And just to know that the chance to win by switching will always remain within the fixed range of (max) 1 to (min) 1/2, i.e. exactly 2/3 on average (Morgan et al., 1991), you never need those textbooks nor those who teach conditional probability theory in high schools. And actual reliable sources say us even more: Regardless of the average probability to win the car by switching of 2/3, even the chance of winning the car by switching in any specific game the player might be in, having chosen just any door whatsoever (1, 2 or 3), and the host afterwards has opened another door, is also 2/3. So even the specific case that the player should have chosen door 1 and host should have opened door 3 is also 2/3. That's the folklore of up to date's science and up to date's sources.

Yes, what we privately think or assume is of no consequence for WP. But you say "there is no 'private' MHP. Really?

What counts for the paradox is what you know (the MHP), and not a "non-MHP" with a host who offers the option to switch only if the guest by chance should have chosen the car. Academic sources showing such horse-dealer tricks do it to free the famous paradox of such scenarios. And what counts is reputable sources. Not baskets of examples for pupils in teaching and learning conditional probability theory. Gerhardvalentin (talk) 00:33, 12 June 2011 (UTC)

Well there is no argument that the article's structure and the naming of the sections could/should be improved.

However your notion that there is only one answer to the "standard problem" is not quite correct, there isn't even an agreement in literature what exactly the "standard problem" and depending how you resolve the disambiguation of the original problem you get indeed different solutions. There are good reasons to assume symmetry (being one way to resolve ambiguities) but there are also good reasons not to do so.

There's nothing wrong with describing the "simple solution" first and potential caveats, differing interpretations and problem variations later. I think most editors (maybe even all) agree to that.

Nevertheless the lead of an article is not simply the first chapter, but it also to provide a short summary of the article and describe its most important aspects. Since the ambiguity is of the central characteristics of this problem it needs to mentioned in the lead. If a reader reads only the lead (or maybe an additional chapter or two) and then leaves thinking the 1/2 versus 2/3 is essentially all that there is to MHP and not even being aware that there are ambiguities, then we've done a rather lousy job of informing him.--Kmhkmh (talk) 23:19, 27 May 2011 (UTC)

Kmhkmh, I have added a comment to the lead about criticism of the simple solutions (I forgot to log in). Does this address some of your points? Martin Hogbin (talk) 09:57, 28 May 2011 (UTC)

Personally I still prefer Tijfo098's version, but your edit or a variation of it might be sufficient at least.--Kmhkmh (talk) 10:14, 28 May 2011 (UTC)

Perhaps we can work together to improve things then. What about Tijfo098's version do you like? Martin Hogbin (talk) 11:27, 28 May 2011 (UTC)

I think it is clear that the most often discussed version of MHP is the one with the natural probability assumptions - if probability represents knowledge (the so-called subjectivist interpretation) - that initially all doors are equally likely to hide the car and that if the host has a choice, he's equally likely, as far as we are concerned, to make either choice. (The man in the street is a subjectivist in his understanding of "probability". As Kraus and Wang's finding confirms!) Everyone is agreed that the answer to Whitaker's question is "switch". Everyone is agreed that the chance that you'll get the car by switching is 2/3, and this is not only true overall but also given the specific numbers written on the door chosen by the player and the door opened by the host. Admittedly, some people don't or won't see or make the distinction, but they'll agree when it's pointed out to them: it's 2/3 and the specific numbers are irrelevant. Both intuitively and in the technical sense, the specific number on the door which gets opened is independent of whether or not the initial door hides a goat.

Switching gives the car if and only if you initially pick a goat. The chance you initially pick a goat is 2/3. So the chance of winning the car by switching is 2/3. The chance can't depend on the specific numbers of the doors concerned in a specific case (e.g., Door 1; Door 3) by symmetry. So the chance of winning the car by switching in the specific case that player chose 1 and host opened 3 is also 2/3.

I fail to understand why the article can't contain such a complete and at the same time elementary and non-technical solution - integrating a correct "simple" solution and a correct "conditional" solution - as early as possible, even in the lead. Synthesis! Avoid the technical concept "conditional probability"; use the intuitive concept "independent".

Two years ago some of Rick's supporters objected that doing it this way was OR ("own research") - the symmetry argument had been pointed out by editor Boris Tsirelson. Well: later Rick found versions of the symmetry argument in the discussion contributions to Morgan et al; Georgii in his text-book uses symmetry to discard door numbers "in advance"; Persi Diaconis promoted symmetry and supported Vos Savant; and last but not least, yours truly wrote this all up in two encyclopedia articles and a journal publication, specifically as "service to the community" - service to you my dear fellow Wikipedia editors.

The solution is not to polarize, by segregating solutions artificially into two "opposing" camps. The solution is to merge, to present the bigger picture in which the simple and conditional solutions are in synergy with one another; both are simple; neither is technical. Richard Gill (talk) 13:05, 11 June 2011 (UTC)

86.181.201.59, your help in improving this article would be greatly appreciated. Martin Hogbin (talk) 09:57, 28 May 2011 (UTC)

The academic debate

Kmhkmh has mentioned 'the academic debate' on this subject. Is there a current debate on the subject and, if so what is it and where is it, or are we just talking about Morgan et al and successors and the more popular books on the subject? Martin Hogbin (talk) 11:58, 28 May 2011 (UTC)

There is probably no debate in the sense of a famous political os science debate, but there is a still increasing number of publications looking at different aspects and angles of MHP and occasionally criticizing other/older approaches. Some of those are somewhat related to Morgan's argument or even a copy (Eisenhower,Rosenthal, Gillman, partially Rosenhouse) others completely independent like Georgii (arguing from "true" Bayesian perspective and suggesting 2 possible solutions) and Steinbach (German) (treating it as decision theory problem and suggesting to randomize the decision (flip a coin)). In particular Georgii reviewed some of the sources on MHP and states:

"These sources present a number of different approaches to the problem, which shows that a general agreement has not yet been reached"

--Kmhkmh (talk) 14:52, 28 May 2011 (UTC)

Georgii and Steinbach are both ten years behind, in the early days of the Parade article, not knowing if is Monty certain to open a door and reveal a goat, or does he only open a door sometimes... ? Thus they address an ambiguity in Vos Savant's original wording, which she resolved in her very next column, and which is *not* the issue for us, at all.

For Georgii there are two different problems: "our" MHP and a non-MHP. He gives a solution to each. There is nothing Bayesian about his work except that his equal probability assumptions come from indifference. His solution to MHP is to ignore door numbers at the start by symmetry (gives independence) and then do a simple solution. This is the solution which I believe is implicit in Diaconis work (and he personally has confirmed that) but was disputed here, "OR" said Kmhkmh. Yet it's in his very own favourite Georgii!

Steinbach solves, using game theory (minimax), the amusing problem when we don't know if the host is playing Game 1 (MHP) or Game 2 (non MHP: Monty only shows a goat when we have chosen the car!). The answer is to randomize, 50-50. It's fun. Diaconis had remarked that you can't solve the problem if it's still ambiguous. Steinbach shows you can (in a sense). This is a lovely variant, well worth adding to one of the advanced sections. Richard Gill (talk) 13:46, 11 June 2011 (UTC)

As far as I can see the Georgii is nothing but a rehash of Morgan and it makes exactly the same mistake. I guess we must regard it as a reliable source of some kind

One other point. Which sources clearly criticise the simple solutions in the specific case that the host is taken to choose a legal door uniformly? Martin Hogbin (talk) 15:42, 28 May 2011 (UTC)

No offense but I answered you question, so as far as I'm concerned take it or leave it. I have zero interest in starting yet another endless and pointless arguments debate. Maybe except Steinbach all of that has been discussed before, Tijfo098 pointed out Georgii's argument and you've been arguing with Rick particular now for almost 3 years over the "clear criticism". Also note that my comment above wasn't even referring to the "criticism of the simple solution" in particular, but simply stating that there is a "academic debate" of the problem in general and that there is no universally accepted "best" treatment of the problem yet.--Kmhkmh (talk) 16:19, 28 May 2011 (UTC)

Yes, thank you for answering my question. I was just noting that there is nothing new. I am amazed by the poor standard of academic publication on the subject. It seems everyone want to get some clever new angle on the subject to get something published. Of course, I accept that it is not my opinion that counts but that of published sources. Martin Hogbin (talk) 16:38, 28 May 2011 (UTC)

If Steinbach uses randomization in the "stay or switch" stage it can only be because he is not certain the host would have opened a door anyway. So that reference is irrelevant. Georgii is a nice example of an academic probability writer who pushes a simple solution, ignoring door numbers from the start, by symmetry (equivalently, by ignorance). I suggest the editors here study Gill (2011: Statistica Neerlandica) for a succinct and comprehensive and almost state of the art overview.

I just read Steinbach. It's a 12 year old unpublished technical report. Did it appear in print anywhere? He considers the problem in which we do not know in advance the Monty will certainly show us a goat. He shows that the resulting non-standard version of MHP has a game-theoretic (minimax) solution involving randomization. Neat but totally irrelevant to presently discussed issues here.

As mentioned before Georgii presents the simple solution after merely remarking that the specific door numbers can be ignored. Barbeau only gives the simple solution and does not mention any debate. Selvin presented a simple solution in his first letter and endorsed another (contributed by Mr Hall himself) in his second. Bell commenting on Morgan said that since the gap between simple and conditional solution is so tiny under the usual full symmetry assumptions - given your initial choice X=1, whether host opens 2 or 3 is independent of whether C=1 or not - it was a matter of taste whether or not one pays any attention to the matter. As far as I can see, the only authors who think there is some debate have been firmly criticized by others, or are clearly unqualified to judge. And the editors here whose first language is German might try actually understanding the papers they cite before misinforming us as to their content. Richard Gill (talk) 11:55, 8 June 2011 (UTC)

It is not clear to me whether the purpose of this article is to describe a notable and interesting riddle, or to make a bunch of mathematicians look ridiculous for having no better way to get published than by quibbling about the underlying meaning of a parlor game. The only thing that does seem clear to me is that emphasizing this academic sideshow results in an article that is, for the general reader, clear as mud. ~ Ningauble (talk) 16:44, 8 June 2011 (UTC)

Hear, hear! Richard Gill (talk) 17:57, 8 June 2011 (UTC)

Rvertion of Gerhard's change to the lead.

Gerhard, I reverted your change to the lead because it seemed to be making too much about one of the much-argued-about points. My wording was intended to be strictly factual and based very closely on a tertiary source. Martin Hogbin (talk) 14:43, 28 May 2011 (UTC)

What source? -- Rick Block (talk) 16:24, 28 May 2011 (UTC)

Rosenthal - [The unconditional solution is] 'correct, but I consider it "shaky" because it fails for slight variants of the problem'. Note that I do not even say that the simple solutions are correct, just that they have been criticised. Martin Hogbin (talk) 16:27, 28 May 2011 (UTC)

Rosenthal is a primary source, not a tertiary (or even a secondary) source. "Some sources criticize" is wording that might conceivably come from a secondary source which I why I asked. -- Rick Block (talk) 17:17, 28 May 2011 (UTC)

Maybe tertiary is wrong but Rosenthal is not the originator of the criticism, that was Morgan. Rosenthal passes an opinion on the criticism. That is the kind of thing that we should base the article on. Martin Hogbin (talk) 19:18, 28 May 2011 (UTC)

Jeffrey Rosenthal's book and his paper are both splendid secondary sources. All editors here should read them carefully. They are written some time after Morgan, Vos Savant, Morgan, Selvin. The author knows the literature. He is writing as a popularizer and as a teacher, not as a mathemattics researcher (his research is on very advanced areas of probability). He has some personal points of view (he prefers Bayes rule to Bayes theorem). He introduces some catchy notions (Monty Fall and Monty Crawl). Does having a personality convert such work into a primary source? Richard Gill (talk) 13:17, 11 June 2011 (UTC)

I would revert all recent additions to the lede starting with this IP addition[6]. No matter what your position is, putting all the details into the lede is counterproductive. A high school student, when he reads the lead, is not going to understand the significance of the dispute. Trying to put two distinct problems in his head at the start invites confusion. The last paragraph of lead intended to introduce alternate interpretations and their (unexpected) consequences. Glrx (talk) 21:44, 28 May 2011 (UTC)

I strongly disagree. First of all the lead is not written for highschool students in particular and second the point is primarily not that he fully understands the dispute (the later chapters are for that anyhow), but that he is at least aware of the existence of a dispute or of ambiguities.--Kmhkmh (talk) 00:58, 29 May 2011 (UTC)

You are correct. The lead should not be written for high school students. It should be written for elementary school students. Please read WP:TECHNICAL for detailed instructions as to what level the lead must be written to. (please note that it has been shown by example that it is entirely possible to write a lead that isn't too technical following either of the POVs we are discussing. This has nothing to do with that debate.) Guy Macon (talk) 01:55, 29 May 2011 (UTC)

(e/c) I took out the paragraph because it lacked foundation. It referred to simple solutions before there was any mention of multiple solutions. Pointing out a dispute over unknown solutions is inappropriate at that place. If there is mention, it should follow the paragraph that introduces variants. At that point, it could state that some MHP solutions have trouble / do not work / do not apply / fail for the problem variations. Glrx (talk) 17:08, 29 May 2011 (UTC)

Glrx, I agree with your point about making the lead as simple and not confusing as possible. On the other hand, the lead should be a summary of the article. How do we balance these two requirements. Martin Hogbin (talk) 16:58, 29 May 2011 (UTC)

There still seems to a misunderstanding of "understanding" and the purpose of the lead. Reading that there are ambiguities, different problem versions or a dispute requires zero understanding. It is merely a matter of noticing. The purpose of the lead is to summarize/capture the most important aspects of an article/topic. If we are not doing that we might not bother with writing a lead in the first place.--Kmhkmh (talk) 17:30, 29 May 2011 (UTC)

State of the overhaul/changes

Just a question of where we right now.

My last understanding was that Martin was modifying the structure to something he considers more appropriate and that we take it from there. In particular waiting for a feedback of other editors (Rick in particular) to see how we'll proceed and what else might need to changed still.

Is that still the procedure we're pursuing now? And if so are Martin's modifications complet or where are we right now exactly?--Kmhkmh (talk) 14:59, 28 May 2011 (UTC)

I have completed my change in structure. Let me make clear yet again that this structure is not intended to promote my POV or even to be a compromise. It is structure that allows the simple solutions to be presented and discussed in a way that allows the vast majority of our readers to understand the problem and why the answer is 2/3 but which also allows the academic arguments to presented fully, based on what reliable sources say. Martin Hogbin (talk) 15:47, 28 May 2011 (UTC)

Alright as far as I'm concerned, the current structure is fine with me (at least at first glance). So let's what the others (and in particular Rick is saying), maybe we are finally getting somewhere.--Kmhkmh (talk) 15:57, 28 May 2011 (UTC)

My understanding is that Guy's intent is that we work on this version until it's as good as we can make it, and then I'll make a copy in a draft space that I'll work on along with anyone else who's interested. Are you both saying that you're happy with this version? I'll note that my objection that the current version is POV remains unaddressed (IMO Martin's recent change to the lead amplifies this POV issue). Does anyone want to try to address this objection before I make a copy in a draft space? Does everyone understand exactly what my objection is, or do I need to spell it out again? -- Rick Block (talk) 16:53, 28 May 2011 (UTC)

Rick I cannot see how my recent addition amplifies the POV issue. I have added to the lead the undeniable fact that there is some criticism of the simple solutions. I agree with Kmhkmh that the lead should reflect the article as a whole and should therefore mention that the simple solutions have been criticised. I chose to add a sentence based on a good source that neatly sums up the situation. We can amplify this in the body of the article.Martin Hogbin (talk) 19:27, 28 May 2011 (UTC)

Right. You are fairly clearly not understanding my objection. I guess I'll have to spell it out again (I will, but don't have time at the moment). -- Rick Block (talk) 20:04, 28 May 2011 (UTC)

That was and is my intent. They key is to (A) get agreement that the two versions properly represent the two views (otherwise the side which the consensus goes against will keep arguing for many more years) and (B) to bring the two versions as close to each other as possible (thus making the choice clear for uninvolved editors arriving at a consensus.) Guy Macon (talk) 18:07, 28 May 2011 (UTC)

But as I have said several times before, this version does not represent my views on the subject or indeed the views of the majority of editors of this page it is just a structure that conforms to WP:technical, is the way that most good text books are written and allows all POVs to be properly expressed without confusing the reader. No one but Rick objects to it.

If you want me to rewrite the article to reflect the dominant POV (expressed by Gerhard and which I reverted and the recent IP) I would be happy to do so. Martin Hogbin (talk) 19:27, 28 May 2011 (UTC)

"No one but Rick objects to it" is manifestly untrue. Glrx expresses concern above [7]. Glopk (who has been successfully chased away [8]) objects to it [9]). Handy2000 seems to object to it (based on comments like this [10]). Nijdam (who has been successfully made the subject of arbcom restrictions) objects to it. Coffee2theorems (who hasn't commented in a while) seems to object to it (based on comments like this [11]). -- Rick Block (talk) 20:04, 28 May 2011 (UTC)

I was talking about current editors. If you want to look over the life of the article, there is a strong majority on favour of the simple solutions.

I am not sure what you mean by 'Nijdam (who has been successfully made the subject of arbcom restrictions)'? I was as surprised as anyone over Arbcom's decision to ban Nijdam and I certainly did not propose or support the action, in fact I objected to it. Martin Hogbin (talk) 20:53, 28 May 2011 (UTC)

I think he just means that Nijdam has an opinion but cannot express it because of an arbcom decision. I don't think he was implying support for or opposition to that decision, just noting that Nijdam's opinion exists.

It is my considered opinion based upon extensive reading of the archives that neither side can lay claim to an overwhelming majority. Also, neither position is unreasonable, both positions have support from reliable sources, neither position is currently being pushed, both positions have good arguments, and neither side is misbehaving. This is simply a good-faith content dispute, no more and no less.

I would very much like one of you to rewrite the article to reflect your POV, and to work with the other to get the two versions as close as possible so that there is a clear choice going into content dispute resolution .

I would suggest that someone write a version on a separate talk subpage and the others edit it as follows:

Starting point:

Paragraph one

Paragraph two

Paragraph three

Paragraph four

Paragraph five

After first edit by other side:

Paragraph one

Paragraph two

Version A Paragraph three

Version B Paragraph three

Paragraph four

Paragraph five

And so on. At each disagreement, I would like you to all to try to agree if possible. Maybe one side could decide to abandon a paragraph and accept the other. Maybe a change in a paragraph would make it easier for the other side to accept it. Maybe we can get it down to one different sentence or maybe even just a few words. Minimal differences is the goal here. It would be hilarious if after all this time we were able to agree on everything, but I doubt that will happen. I am looking for compromise where possible and a clear difference in versions where there really is a disagreement about content.

Also, I would like to see everyone participate in this, not just Martin and Rick I don't think at this point we have a third incompatible POV, but if we need one we will make a version with a third POV.

At the end I want everyone to agree that "yeah, I like version X. I think I can make a strong argument for it during content dispute resolution. I expect the other side to accept it if the consensus is against them and I will accept it if the consensus is against me."

BTW, if this works, I want a BarnStar (smile). Guy Macon (talk) 21:05, 28 May 2011 (UTC)

I'm perfectly willing to give this a try. However, since the compromise approach I favor is to treat this as a POV dispute (i.e. source X says something, source Y says something, source Z says something) the outline cannot be "the Truth is X; some sources criticize this". This definitely implies X is the majority (and correct) opinion and marginalizes both sources that say anything else (even those that don't criticize X) as well as the (relatively few) sources that have the temerity to explicitly criticize those sources that say X. An outline that consists of "Simple solutions" and "Other solutions" puts "Simple solutions" in a favored position relative to any solution presented in the "Other solutions" section and is nowhere near neutral. In any other domain the fact that this approach conveys a POV would be entirely obvious:

Religions

Christianity

Other religions

Paganism

Islam

Judaism

An outline consisting of "X" and "Other than X" promotes "X" to a position of primacy. Many reliable sources present "simple solutions". However, many other reliable sources present solutions based on conditional probability. Some, but nowhere near all, of the sources presenting solutions based on conditional probability criticize "simple solutions". Structuring the article around "simple solutions" therefore gives these solutions undue weight. Creating two versions of the article, identical except for individual paragraphs, doesn't seem possible.. Again, I'm perfectly willing to give this a try - but I think the two versions will differ by more than a few paragraphs. -- Rick Block (talk) 05:11, 29 May 2011 (UTC)

@Guy, there is a historical majority supporting the simple solutions as my evidence[12] to Arbcom showed, please tell me if I have missed anyone. I believe that there is a consensus for the current structure, but why do we not ask? What needs to be done is general editing to make the first section clearer and of more value to the general reader. Martin Hogbin (talk) 09:17, 29 May 2011 (UTC)

@Rick, the last thing in the world we want is a POV dispute. The current structure is designed to avoid such a dispute since all POVs can be presented properly within the current structure. Nothing in the current article supports any particular POV. In fact, in the outside world there is no POV argument, just an interesting topic that can be studied with to various levels of complexity. We must not base the article on any perceived POV dispute here. Martin Hogbin (talk) 09:17, 29 May 2011 (UTC)

Rick, concerning structure vs. creating two versions of the article identical except for individual paragraphs, that's a goal, but if the minimum difference requires different structures, then it has to have different structures. It's worth trying for one structure, of course. I am not convinced that Martin will reject any structure other than "X and Other than X", so why not try to make a structure that he will accept? If the content dispute really does require two structures, perhaps we we can figure out a way to make the difference clear to anyone evaluating the content dispute. I don't want to make any assumptions as to what the document that succinctly conveys the essence of the content dispute will look like. I just want it to be easy to understand and not so lengthy that nobody will wade through it. There may be a few bumps along the way, but I think we can do it.

Martin, the question of which position had more support before the arbcom is interesting (and I have seen two counts at http://en.wikipedia.org/wiki/User:Martin_Hogbin/Monty_Hall_History and http://en.wikipedia.org/wiki/User_talk:Nijdam#Situation ) but that past consensus was formed in the middle of some serious misbehavior that is no longer happening - and some folks tend to oppose things just because someone who supports them pisses them off or posts lousy arguments. Also, some folks may have changed their mind, and the arguments at that time were very lengthy, so we don't know how many of the editors listed actually read and understood all the arguments on both sides. If indeed there was a clear consensus then, we can get a clear consensus again. Once we have a succinct case to show other editors, I will post a link to it on the talk pages of all the names you or anyone else mentions and give them another chance to weigh in and help to form the current consensus. Also, I will be asking for a short argument from each side, and it may be that one of the arguments will be so overwhelmingly good that it convinces nearly everybody. Obviously each of you thinks they have a really good argument and the others argument is flawed, but can you sell that to uninvolved or formerly involved editors? Finally, the simple fact is that this sort of "I assert that the consensus is X" argument has been made many times before and has never resulted in an end to the arguments. My plan includes everybody agreeing on what constitutes a consensus and agreeing to abide by it if it goes against them. Guy Macon (talk) 10:03, 29 May 2011 (UTC)

Guy we had a quick poll on the structure [13] which (after your change to neutral) shows a majority of 5 to 2 in favour of this structure. There is no sign that anyone else has changed their mind and after over 2 years of argument I think it is very unlikely that anyone will do so, thus there is never going to be unanimity however long and hard we work at it. We have to proceed on the basis of a clear majority decision. If there no longer is a consensus to keep the article structure as it is then I am happy to discuss the subject while a new consensus is reached but first it needs to be shown that the current consensus is no longer extant. Martin Hogbin (talk) 10:42, 29 May 2011 (UTC)

Two different questions. We aren't discussing the structure of the current article. We are discussing the structure of the two versions described in my plan. Rick did not propose changing the structure of the article at this time. I am agnostic as to what version stays up while we settle the longstanding content dispute. Guy Macon (talk) 12:21, 29 May 2011 (UTC)

There is no dispute to settle. We have a consensus structure that covers both content POVs. Martin Hogbin (talk) 13:14, 29 May 2011 (UTC)

Get back to me when you get a single person from the other POV to agree that with the above. Guy Macon (talk)

I have no need to do that. We have a consensus for the current structure, that is the basis on which decisions are made on WP. Martin Hogbin (talk) 14:32, 29 May 2011 (UTC)

Again, not talking about the current structure. You do not have a shred of evidence to support your assertion that there is no content dispute to settle. Nor do you have a shred of evidence to support your assertion that the current structure "covers both POVs." You cannot just wish away the POV of those who disagree with you. If Rick were to edit the page to reflect his POV and unilaterally declare that it covers your POV whether you agree that it does or not, you wouldn't be happy with that.

A while back I proposed that everyone let you attempt to write a compromise in the hope that everyone would be happy with it. I though that you should be given a chance rather than having other editors fight you tooth and nail on every detail while you were attempting to craft a compromise I also suggested that Rick should get a turn trying to reach the same goal. You voiced zero objections to either proposal, only to sandbag me later with objections to both. I believe that most of the participants here approved your structure and let you do your work unimpeded based upon my recommendation. There was never any consensus that your compromise succeeded in making both sides happy, and there certainly was never any consensus that the longstanding content dispute does not exist or has been resolved. As I said, I am agnostic as to what the page should look like while we resolve the content dispute, but I strongly oppose any suggestion that we not make any attempt to resolve the content dispute through content dispute resolution Guy Macon (talk) 22:36, 29 May 2011 (UTC)

I had no objection to your proposal but I did not specifically agree to it and I am not sure who else did either. The reason that I changed the structure of the article was that there was a consensus [14] to do so. My stated reason for this structure was that it allowed all POVs to be expressed properly. I genuinely believe that to be true but if Rick chooses to believe otherwise that is his prerogative, however if he does not like the structure he is arguing against a consensus which is disruptive.

My recent addition to the lead (now removed) was an attempt to have an opposing view to mine represented in the lead, based on what reliable sources say on the subject.

As I have said before, I have no objection to participating in any dispute resolution process although I do not hold out much hope. I should also point out that I, and other editors, will not be starting out from the compromise position that we have now but I will be promoting what I believe to be the dominant POV here, that the simple solutions are perfectly adequate and anything else is an academic extension answering a different problem. I have so far specifically avoided this POV in the interests of compromise. Martin Hogbin (talk) 23:27, 29 May 2011 (UTC)

Again the only thing about the current version that is relevant to my plan is whether you have completed your overhaul, which you have indicated that you have. For the purpose of moving forward to content dispute resolution, It doesn't matter what is in the current version or who agrees with it. I am trying very hard to get everyone to either talk about moving forward to content dispute resolution or to talk about the current version of the article without mixing the two. Guy Macon (talk) 13:11, 30 May 2011 (UTC)

I'm hearing that the current version of the article is OK with most folks here. Before making a copy (per Guy's plan) I'll make some changes in this version that I imagine won't be terribly controversial. If anyone objects, please feel free to revert (no need to discuss). I'll limit the rate of these changes to no more than one per day. -- Rick Block (talk) 16:41, 30 May 2011 (UTC)

I am not happy with the current version. As I said earlier, it serves neither side well.

I am also opposed to Guy's plan for a separate page. When there are two pages, then what happens? We take one or the other? Debate different merges? What about intervening changes to the main article?

I would rather see slow edits to the main page (by Rick and others). Let's have the debate as the edits come up rather than postponing it to a future date.

Glrx (talk) 18:11, 30 May 2011 (UTC)

I agree that a separate page is not necessary.

Could you explain more about what you do not like about the current version. Martin Hogbin (talk) 20:27, 30 May 2011 (UTC)

Glrx, What would you suggest that we do the first time there is a dispute about an edit relating to the longstanding content dispute? Seeing as you oppose my plan to resolve that dispute, what is your alternative plan? Or are you advocating more years of argument without any resolution in sight?

Glrx, Your questions about my plan make it appear that you have not read and understood what my plan is. You appear to think that the page I describe is for some purpose other than for the specific purpose of presenting it to other editors as a part of content dispute resolution as I have explained several times. Could you take a moment and read the plan? I will be glad to address any questions you still have after that. Guy Macon (talk) 22:15, 30 May 2011 (UTC)

Martin - Glrx made his position clear above [15]. Do you not find this clear?

Guy - Glrx is suggesting a WP:BRD approach. I don't think there's any harm in getting as far as we can with such an approach. If we get stuck at a point where we can't reach consensus (i.e. where there's no version we can ALL live with) creating the two versions you're talking about might make sense. My understanding is that Martin is insistent that the so-called simple solutions be presented in their own section, and that the article NOT include any mention (even in simple terms) of conditional probability or the difference between a solution that addresses the chances of predetermined switch vs. stay strategies as opposed to a solution that addresses the probability in a specific case (such as player picks door 1 and host opens door 3) until well AFTER this section is presented, and he's claiming that this presentation represents "consensus". We could try a couple of versions that present a more integrated approach and see if anyone reverts - although I think we might know the answer already. -- Rick Block (talk) 00:04, 31 May 2011 (UTC)

I am game for anything that might work, if for no other reason than being able to say we tried X, Y and Z. I am going to take a break and unwatch this page for a few weeks. If anyone thinks they need me, please leave a note on my talk page. Thanks! Guy Macon (talk) 00:44, 31 May 2011 (UTC)

I may do the same. I had hoped that I had found a way that all POVs on this subject could be fairly represented according to WP policy and normal academic practice but it seems that not everyone accepts this and some still see it as a dastardly plot to get my own POV into the article. I will continue to watch the page and may make the occasional comment but it is time for someone else to try to sort this article out. No doubt there will continue to be a steady stream of people coming to the article and asking why a simple and incredibly unintuitive puzzle has been turned into an exercise in academic arrogance and pedantry. Martin Hogbin (talk) 08:33, 31 May 2011 (UTC)

I just came here and that's exactly what I was wondering. It looks like this poor article has endured a long history of disputes. How can an article about a math problem be POV? I haven't looked into this in depth, but I'm hoping everyone is familiar with WP:FRINGE and understand that the quickest way to avoid claims of POV is to simply state what reliable sources say. Night Ranger (talk) 14:49, 31 May 2011 (UTC)

Something that is in virtually every introductory probability textbook can hardly be called "fringe". The issue here is some editors seem to think the absolutely standard approach to this problem, presented in virtually every relevant textbook, is an "exercise in academic arrogance and pedantry". -- Rick Block (talk) 15:04, 31 May 2011 (UTC)

No. The first issue here is that most editors do not find it necessary to add health warnings in the initial part of the article which tell the average reader that some academic writers in mathematics think there is something wrong with the simple arguments. A second issue is that a steady stream of people come to the article and ask exactly what Martin just wrote: "why has a simple and incredibly unintuitive puzzle been turned into an exercise in academic arrogance and pedantry?". Interestingly, this is also the opinion of probability professionals themselves: editors Richard Gill and Boris Tsirelson, to mention just two. Who have moreover patiently explained time and time again that if you do insist on bringing in conditional probability, there are a host of ways to do it more easily than copying the "standard treatment" from the elementary probability textbooks, where MHP is used as a fun example to liven up a class on Bayes theorem, as just mentioned professionals have repeatedly mentioned. Richard Gill (talk) 05:37, 10 June 2011 (UTC)

Criticism of the simple solutions is Fringe.

Presentation of the formal conditional solutions is not fringe. These are 2 distinct issues, a fact which both 'sides' fail to address. 184.81.169.186 (talk) 16:53, 31 May 2011 (UTC)

Dr. Steven Selvin wrote this 15 years before vos Savant or Morgan. vos Savant argued these same points with Morgan. A problem with a biased host is different than the Monty Hall problem of either Selvin or vos Savant, both making this very clear in their writings.

"Excerpted from The American Statistician, August 1975, Vol. 29, No. 3 On the Mony Hall Problem I have received a number of letters commenting on my "Letters to the Editor" n The American Statistician of February, 1975, entitled "A Problem in Probability." Several correspondents claim my answer is incorrect. The basis to my solution is that Monty Hall knows which box contains the keys and when when he can open either of two boxes without exposing the keys, he chooses between them at random."

166.216.194.83 (talk) 06:00, 2 June 2011 (UTC)

Move of section describing the problem

I assume this is a work in progress. As it stands the article progresses from the wp:lead (meant to be an introduction and a brief "standalone" summary of the entire article) to "simple solutions" without making it clear what problem the simple solutions address. The point of the previous placement of the "Problem description" section was to explain what the "usual" problem is (this is not a variant, but the "main" interpretation), specifically that the host must open a "goat door" (has not opened a random door and fortuitously revealed a goat rather than the car), the host must always make the offer to switch (doesn't get to make the offer more or less often depending on the player's initial pick), and if the host has a choice of two goat doors to open must choose between them "evenly" (uniformly randomly). Without this section the exact problem that the solutions are addressing is entirely unclear. -- Rick Block (talk) 18:19, 31 May 2011 (UTC)

The introduction already has this paragraph about the usual problem:

The Monty Hall problem has some implied assumptions about how the game is played. For the most part, the assumptions are about fairness (for example, the car is equally likely behind any of three doors). The contestant can initially choose any door. Monty will always open another door and give the contestant an opportunity to switch (the game is played the same way each time). A crucial point, however, is that Monty knows what is behind the doors and will always open a door that reveals a goat. Always revealing a goat makes sense; if Monty revealed a car, then the game would be uninteresting. But Monty's revelation is expected and unsurprising: the contestant knew at least one of the two unselected doors hides a goat; Monty identified something that was expected. Consequently, Monty revealing an expected goat does not turn the game into a 50-50 coin flip. That Monty can always reveal a goat is a key to switching being a better strategy. A more precise statement of the Monty Hall problem is given below.

I moved the problem description down because it has more detail than needed for the flow of the article. The issues surrounding the problem interpretation issues should come out as the solutions are discussed. The value of q, for example, isn't important if the doors are indistinguishable; it can have an excellent introduction when the decision tree is explained because a value must be assigned to each arc.

The two subsections that split the two solution groups should also move, but I'll wait for the current move to sit for a time.

The solution section lacks an intro. Additions welcome.

Glrx (talk) 18:36, 31 May 2011 (UTC)

Stylistically, anything in the lead should be a summary of something elsewhere in the article (this is why leads typically don't have references - the references are provided in the "full text" in sections following the lead). Read a selection of any of the WP:Featured article to get a sense for what I mean. The lead is NOT an introduction (in the normal sense), but an "above the fold" intro/summary of the entire article, which logically starts after the lead. -- Rick Block (talk) 18:46, 31 May 2011 (UTC)

So, a stylistically better solution is to abbreviate the existing bit in the lead about the problem description and restore the moved section :) -- Rick Block (talk) 18:47, 31 May 2011 (UTC)

Most of the problem description has now been moved back. With the current text, variations are now present in the problem description. That's bad for a novice audience and has little impact on the solutions (which currently do not address solving the variations anyway).

A better plan is to move the paragraph problem description that was in the lead to where its commented out version is now; just uncomment it. Put the problem description back down about where I had it; if you like, call it an Extended problem description.

Keep the article simple up front; let the complexity come in later when it has context and importance.

Glrx (talk) 01:29, 1 June 2011 (UTC)

I don't see any variations, just the "usual" additional assumptions - motivated by the observation that the Whitaker/vos Savant wording leaves various aspects of the host's behavior unclear. Are you specifically talking about the second sentence, the one starting with "For example ..."? The point of this sentence is to substantiate (with a reference) the claim in the previous sentence that "certain aspects of the hosts's behavior are not specified". Calling these "implied assumptions" rather than "usual" begs the question of who says these are implied. Barbeau says they are part of the standard analysis. Krauss and Wang say they are assumed by most people whether they're explicitly mentioned or not. A paragraph like this was in the article at the time of the last wp:featured article review, at which point the source for essentially each and every independent thought (basically sentence by sentence) was questioned since the referencing standard for FAs is nearly literally that each sentence needs a citation. If we're going to go with the wording currently commented out (that was in the lead), we're going to have find a citation for essentially each sentence. The commented out text and what's actually there now don't read very differently to me. Can we work on the wording rather than replace the entire paragraph? -- Rick Block (talk) 06:30, 1 June 2011 (UTC)

What "most people" see as "the natural assumptions" is irrelevant. After all, most people initially give the wrong answer because they "see" the problem in an unhelpful way. Moreover, how most writers of chapters on Bayes theorem in introductory probability texts formalize the problem tells us something about the mechanics of conventional probability calculus, but it is also not necessarily the way the problem has to be solved. It is the way the problem is conventionally solved within a certain pedagogical context.

Authorities from game theory and decision theory see the problem as a decision problem and have short powerful arguments for switching which do not require making any probability assumption at all.

Ordinary people can rapidly understand that switching gives the car with probability 2/3 assuming only that you initially hit a goat with probability 2/3. The good way to play is "aim at a goat and switch". Since it's inconceivable that one could improve on 2/3 overall success rate, it is indeed an entirely academic exercise to try to figure out conditional probabilities (which is one way of proving rigorously that 2/3 can't be beaten). There are also direct proofs. E.g. dominance. Editors of this page should study the modern secondary literature on MHP rather than remaining mired in the academic debates of 20 years ago. In particular, it is *not* true that the simple versus the conditional approach corresponds to whether a player must decide in advance or only after seeing an open door with a goat. Marilyn/Craig tells us to imagine a game show. We are told the rules: the host will open another door and reveal a goat,because he knows where the car is. To solve the problem correctly we have to imagine the whole process (car is hidden, player's choice, host's choice, player's decision).

Richard Gill (talk) 12:58, 5 June 2011 (UTC)

Per Krauss and Wang (a fairly recent source, not a 20 year old source), most people imagine the situation suggested in the problem statement - i.e. they're standing in front of a closed Door 1 which they initially picked, they can see a closed Door 2, and the host has opened Door 3 showing a goat, and it is at this point (in this situation) that they are asked whether to switch. Virtually all probability textbooks and many papers in the literature (current as well as "20 years old") explicitly address this specific situation with the machinery of conditional probability.

You're suggesting the question should be interpreted to be asking whether, as a preselected strategy (knowing the rules), switching is better than staying. Although they don't exactly say it, this is effectively how most popular sources presenting "simple solutions" approach the problem, and is how at least some game theorists approach it as well (although as far as I can tell, game theorists often relax other constraints on the host, such as whether the host must make the offer to switch, in which case one of the minimax solutions is the host makes the offer only if the player initially selected the car and in general staying with your original choice is never worse than switching!).

My contention is that as Wikipedia editors our opinions (including Professor Richard Gill's opinion) about which of these approaches is "right" is completely irrelevant. The question we should be talking about is what is the relative prevalence of these views among reliable sources. Most authors of probability textbooks seem to think the conditional question is what should be addressed (for whatever their reasons). A not insignificant selection of sources, starting with unnamed respondents to Selvin's original letter to American Statistician, continuing with Morgan et al.'s paper written in response to the nationwide furor vos Savant's columns instigated, further continuing with occasional columns, papers and books all explicitly saying the conditional probability is what should be addressed provide ample evidence that the notion that the question is about the conditional probability as opposed to a preselected strategy is at least a (if not the) dominant view in the relevant academic field. -- Rick Block (talk) 16:55, 5 June 2011 (UTC)

Kraus and Wang are psychologists, who study cognitive processes. As a mathematical puzzle (brainteaser) MHP is a trick question. It is deliberately posed to put the listener on the wrong track, so as to make the "punch line" - a simple and powerful argument why you should switch, not stay - as cathartic as possible. It's a joke. A surprise.

It is a rather biased POV that MHP "belongs" to one specific academic community. It's prevalence as an entertaining example in probability texts is due to its didactic use in getting students into Bayes' theorem. It belongs equally well to the academic community of decision theory. It can be argued that it belongs better there, if anywhere.

Finally, since developments in academia are leading to ever shorter and more incisive and at the same time comprehensive analyses of MHP, editors of the Wikipedia article could better spend their time assimilating the latest insights so as to be able to write the most accessible and at the same time complete overview of the field rather than wasting time and energy on side issues.

Why should the problem be solved by tying your hands behind your backs? Does it make any sense that you are only told Monty was *certainly* going to open his door and offer you a chance to switch, *after* you have chosen a door (thinking it was your final choice) and *after* Monty has opened his? Craig Whitaker asked Marilyn about switching and staying in general. Marilyn added the parenthetical ", say, Door 1, " and ", say, Door 2, " for dramatic effect, deliberately in order to encourage the "wrong" answer. Richard Gill (talk) 15:18, 6 June 2011 (UTC)

You say above "As a mathematical puzzle (brainteaser) MHP is a trick question". Says who? Is this the prevalent opinion among reliable sources or is this the personal opinion of Richard Gill? Is the MHP really a trick question, or it it perhaps instead a straightforward conditional probability question about the prior probabilities (clearly equal, both 1/3) and posterior probabilities (which intuitively seem must also be equal) of whether the car is behind Door 1 and Door 2 given the event "host opens Door 3" (after the player picks Door 1)? The straightforward conditional probability is perhaps tricky in the sense that most people don't have the tools to figure it out, but it is certainly not a trick. Lacking the proper tools, most people rely instead on the strongly held (but incorrect in this instance) intuition that the (conditional) probability must be split evenly between the two remaining doors (per Falk). As you well know, saying the (conditional) probability the car is behind Door 2 is 2/3 after the host opens Door 3 is saying something entirely different than that a player who follows a preselected strategy of switching wins with probability 2/3. I'm really not at all interested in arguing who's right here - the relevant question remains the relative prominence of these views among reliable sources. I think it would really help if you would focus on this question and stop trying to convince everyone that your view is the right one. -- Rick Block (talk) 04:10, 8 June 2011 (UTC)

So the article "has to be" just a monotonous lesson in conditional probability calculation, at all costs "conditioning on door numbers 1 and 3"?

But as "door numbers" are of no relevance whatsoever for finding the correct answer: "switch here and now, in this very special game" (see the relevant sources), there is no benefit at all to just "condition on door numbers". This cumbersome detour, untruly promising "maybe closer results" for this very special game, is not appropriate for the WP article. We can show the results in odds-form, yes, and we can refer to the Bayes article, yes. But time for sophisms and failings is over, it is not appropriate to decay the famous paradox to a lesson for pupils of conditional probability theory, as it tenaciously is / was for the last years. Gerhardvalentin (talk) 08:00, 8 June 2011 (UTC)

I'm not saying the article has to be "just a monotonous lesson in probability calculation". I'm only saying that this approach to the problem must not be marginalized, because to do so means the article is not (from WP:NPOV) "representing fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources". The approach based on conditional probability is at least as prominent as the "simple solutions", so should get equal (not favored) billing. What I'm hearing in response is repeated claims that the conditional approach should be presented effectively as a footnote. -- Rick Block (talk) 00:59, 9 June 2011 (UTC)

You are right, Rick, the conditional approach should be presented not only as a footnote. You can present it in odds form of the Bayes theorem, easily to grasp for anyone. And you can show that any "host's bias" is completely irrelevant to answer the question, as long as you do not "know" about such bias, its direction and its prominence. And even if you "knew" of the most extreme cases, that the "chance" :-) of 2/3 that your initial choice was wrong can rise to certainty or fall to 1/2, but never below. So you should show that any host's bias is completely irrelevant to answer the famous question, as it is never unfavorable to switch.

It should be shown that the chance to win by switching will always remain within the fixed range of at least 1/2 (but never less) to 1. And in the situation given, in this one very special game the question is about, as you never can "know" any closer rate for the actual chance to win by switching than the stone hard 2/3 (equal to the overall success rate of "always switching" what never can be beaten), that the candidate should switch. Period. And for further irrelevant "conditioning on door numbers" you can provide for a link to Bayes theorem. We should try to make the article readable and understandable for grandma and grandson, also. And we can succeed, if all of us try. Gerhardvalentin (talk) 08:51, 9 June 2011 (UTC)

MHP without Probability

I made a reference a while ago here, by now archived, to a completely new approach to MHP recently discovered by A. Gnedin (and posted by him on arXiv.org, mathematics section). It uses the game theoretic notion of Dominance (game theory)

Here are now two references. The author Alexander V. Gnedin (Sasha) is a highly reliable authority in probability and optimization from Utrecht University, Netherlands.

"The Doors"

http://arxiv.org/abs/1105.5809

"The Monty Hall Problem: Switching is Forced by the Strategic Thinking"

http://arxiv.org/abs/1103.3890

Please read this carefully.

Let C be the number of the door hiding the car, let H be the number of the door opened by the host.

Consider players Alice and Bob who have the following "pure" strategies:

Alice: choose door 1, stay (whatever H)

Bob: choose door 2, switch (whatever H)

If C=1, Alice wins, Bob wins

If C=2, Alice loses, Bob loses

If C=3, Alice loses, Bob wins

Bob wins whenever Alice wins, and also wins in some situations where Alice loses. More generally, any "choose (however) and stay" strategy is beaten by appropriate choose and switch strategies. Stayers are Losers! They can be ignored.

Optional extra (for completeness): Consider also player Caspar who has the following "mixed" strategy:

Caspar: choose door 1, switch if H=2, stay if H=3

Note that Caspar can never end up with door 2. Compare Caspar with Bob, as before:

If C=1, Caspar wins or loses depending on H, but Bob always wins

If C=2, Caspar loses, Bob wins

If C=3, Caspar wins, Bob wins

Bob wins whenever Caspar wins, and also wins in some situations where Caspar loses.

Any strategy of choosing and then switching or staying depending (perhaps) on what happens next is beaten by a strategy of choosing (a different door) and then switching, whatever. Not only are stayers losers, those who can't make up their mind in advance are losers too.

If the rules of the game are known in advance, it is only a fool who considers the possibility of staying. The only question the player needs ask himself is which door to choose initially. Stayers are losers! This can be decided in advance!

Choosing it completely at random is a good way not to worry if one is doing the right thing.

You'll win the car with probability 2/3. The conditional probability of winning given your own choice and given the door opened by the host is completely irrelevant.

This is an example of the game-theoretic notion of dominance at work. It is a total solution, and it is completely elementary also (you can explain it to anyone).

Short version: "choose anything, stay" is beaten by "choose anything else, switch". Which anyone can understand.

(Complete version: show that mixed strategies are useless too, as I did above in small print. Not surprising once you have shown that all "stay" strategies may be discarded in favour of "switch" strategies).

Note that we do not have to say anything at all about the quizmasters's choice or about our knowledge or ignorance thereof.

We do have to take a step back and see our two choices (initial door, then switch or stay) as part of one total strategy. People intuitively make the wrong conclusion (stay because it's 50-50) not only because our instincts of conditional probability are poor, but also because of our visual orientation: we "see" two closed doors, one open with a goat. We forget the process behind this. We even forget our own role in that progress.

One can use MHP inside a probability class to illustrate proper (careful) conditioning. To do this one needs to add extraneous frequentistic probability assumptions, or to work with formal subjective probability (ignorance in, ignorance out).

But when we consider the whole process and think strategically the problem (to switch or to stay? - remember, that is what Marilyn/Craig asks us!) has a trivial and obvious solution.

"Choose x and stay is beaten by choose y and switch".

Since we apparently know the rules of the game in advance, we'ld be fools ever to think of staying. No point in postponing that decision. We'll initially simply choose the *least* likely door to hide the car and switch. If in doubt (or paranoid) choose at random (for the paranoid: with a secret random generator).

Consideration of conditional probabilities is quite a waste of time. At best it's a useful exercise in the Bayes theorem class.

Richard Gill (talk) 14:14, 7 June 2011 (UTC)

Putting probability back in

Of course, if all doors are initially equally likely to hide the car, and also the host is equally likely to open either door when he has a choice, the chance that the other door hides the car cannot depend on which door was chosen, which door was opened. Switching gives the car with probability 2/3, whether or not you take account of specific door numbers involved in any particular instance.

Whether or not this fact is prominently mentioned in the beginning of the article is the entire substance of a more than two years running content dispute. Though already in the discussion of Morgan et al., William Bell pointed out that it was so obvious that it could be considered a matter of taste whether or not one pays explicit attention to it. Indeed Selvin, who introduced the whole thing, apparently would have agreed with Bell whole heartedly, since he gave both simple arguments for "unconditional probability is 2/3" as well as more elaborate for "conditional probability is 2/3" side by side, with no comment whatsoever on the difference!

Some academic writers in probability theory like to consider the case that the host is not equally likely to open either door when he has a choice. Such writers do invariably assume that initially the three doors are equally likely to hide the car. No-one comes up with a sensible reason why one would know that car-hiding is done with equal probabilities but door-opening is done with unequal probabilities, but still, it gives a nice opportunity to teachers and popularizers of probability to illustrate Bayes formula - preferably in the odds form, see Jeffrey Rosenthal's excellent paper "Monty Hall, Monty Fall, Monty Crawl" or his highly succesful book "Struck by Lightning: The Curious World of Probabilities". Even the infamous Morgan et al., who publicly bashed Vos Savant over the head for not appearing to care about conditional probability, admitted in their reaction to the 2010 note by Hogbin and Nijdam (correcting a hitherto unnoticed mathematical error in Morgan et al's original paper) that the scenario of their paper was a purely academic diversion.

Anyway, for those interested in academic diversions and conditional probability, the Morgan et al. scenario is easily dealt with using the domination (game theory) of switching.

Suppose we only know that the three doors are equally likely to hide the car, but we don't know the host is equally likely to open either door when he has a choice.

We know that any strategy involving staying is beaten "case by case" (dominated, in game theory parlance) by another strategy of choosing a different door and switching. Choosing any door and switching has overall success rate of 2/3. By domination, no strategy can have a better success rate. Therefore the conditional probability of winning by switching, given the door chosen by the player (whatever it is) and given the door opened by the host (whatever it is), must be at least 1/2.

Proof of last statement: according to the law of total probability, the overall success probability of any strategy is equal to the sum over all situations which can be distinguished by the player (number of door chosen by him, number of door opened by host) of the conditional probability of winning by this strategy in that situation times the marginal probability of the situation. Therefore, the overall success rate of any strategy can only be improved if the conditional probability of winning in some particular situation can be improved by changing the strategy in that situation. Since 2/3 by always switching can not be improved overall, the conditional success probability of this strategy in any situation cannot be improved either. Hence the conditional chance of winning by switching can never be smaller than 1/2.

Richard Gill (talk) 05:07, 10 June 2011 (UTC)

And this relates to the content of the article how? Or did you intend to post this on the /Arguments subpage? -- Rick Block (talk) 04:24, 11 June 2011 (UTC)

Good questions, Rick. Did you read Gnedin's papers yet? Or my own, for that matter?

1) I am not going to edit the article myself so if I think there is potential content which I think could be interesting for editors because it can be important for the page, I will draw attention to it on the Talk page.

2) The old quarrels still dominate the talk pages. New editors are drawn into the old fight. The same arguments are being trotted out again and again, everything is being endlessly repeated. I felt that drawing attention to new developments on MHP which actually make formal Bayesian calculations redundant might bring some fresh air into this stifling atmosphere: Choose x and stay is beaten by choose y (different from x) and switch. Stayers are losers! You can decide in advance that you will switch, without any considerations of probability at all. You know in advance what the host is going to do, don't you?? Or did you arrive just now from the neighbourhood of Alpha Centauri, find yourself on the stage, and was only told *after* the host opened a door revealing a goat and offering you the chance to switch that he was going to do this anyway????

3) While wikipedia editors replay an academic fight in the pages of the American Statistician of more than ten years ago, the academic community moves on and new approaches are discovered to MHP which hugely simplify any formal mathematical analysis. See for instance the peer-reviewed articles on Monty Hall Problem in statprob.com or citizendium.org. Read my paper [16] published early this year in Statistica Neerlandica, and read the papers of A.V. Gnedin (go to arXiv.org and do an author search "Gnedin, A" in the mathematics section). Those editors who want to focus on a formal mathematical approach to MHP should be spending more time on their own "permanent education". Otherwise they will become dinosaurs.

4) The issue which you still find so important that all discussion here continues to be focussed on it, opinions become polarized, new editors get frustrated and move away again, and the article continues to stagnate, can be dealt with in the context of the conventional (fully specified, probabilistic, symmetric MHP) by the words: "If all doors are initially equally likely to hide the car and the host is equally likely to open either door when he has a choice, the chance that switching will give the car cannot depend on which door was chosen by the player and which door was opened by the host".

5) If you really want to pay attention to the biased host case, it can be taken care of by the words: "We know that any strategy involving staying is beaten "case by case" (dominated, in game theory parlance) by another strategy of choosing a different door and switching. Choosing any door and switching has overall success rate of 2/3. By domination, no strategy can have a better success rate. Therefore the conditional probability of winning by switching, given the door chosen by the player (whatever it is) and given the door opened by the host (whatever it is), must be at least 1/2."

6) Please let's give the article a life again! Forget the old issues. They have become NON-ISSUES. Perpetuating the old quarrel is about egos and ownership, not about content, it's not about creating a better article for our readers. Let the article start simple and move later to complex. Let it not be dominated by the idea that there are simple solutions for simpletons and Bayesian probability calculations for mathematicians, and nothing else. The gap between simple and complex has narrowed so much (if you understand MHP well enough that it makes sense that you are an editor of the MHP page) that there is no longer an issue.

Richard Gill (talk) 08:59, 11 June 2011 (UTC)

The middle way: synthesis and synergy, not opposition

The most often discussed version of MHP is the one with the natural probability assumptions - if probability represents knowledge (the so-called subjectivist interpretation) - that initially all doors are equally likely to hide the car and that if the host has a choice, he's equally likely, as far as we are concerned, to make either choice. (The man in the street is a subjectivist in his understanding of "probability". As Kraus and Wang's finding confirms!) Everyone is agreed that the answer to Whitaker's question is "switch". Everyone is agreed that the chance that you'll get the car by switching is 2/3, and this is not only true overall but also given the specific numbers written on the door chosen by the player and the door opened by the host. Admittedly, some people don't or won't see or make the distinction, but they'll agree when it's pointed out to them: it's 2/3 and the specific numbers are irrelevant. Both intuitively and in the technical sense, the specific number on the door which gets opened is independent of whether or not the initial door hides a goat.

Switching gives the car if and only if you initially pick a goat. The chance you initially pick a goat is 2/3. So the chance of winning the car by switching is 2/3. The chance can't depend on the specific numbers of the doors concerned in a specific case (e.g., Door 1; Door 3) by symmetry. So the chance of winning the car by switching in the specific case that player chose 1 and host opened 3 is also 2/3.

I fail to understand why the article can't contain such a complete and at the same time elementary and non-technical solution - integrating a correct "simple" solution and a correct "conditional" solution - as early as possible, right in the lead. Synthesis! Avoid the technical concept "conditional probability"; use the intuitive concept "independent". Do it early. Richard Gill (talk) 15:11, 11 June 2011 (UTC)

As far as I can tell, the issue seems to be that some people do not want it pointed out that there's a difference between the overall chance of winning by switching being 2/3 and the chance being 2/3 given the specific numbers written on the door chosen by the player and the door opened by the host - and these people insist the early parts of the article must not mention this difference and also must only say the former and not the latter. I.e. they're not interested in synergy (original wp:synthesis is a no-no), but only promoting what they perceive to be the "one true way" the problem should be approached. I have long since given up trying to convince anyone here of anything other than that this approach violates fundamental Wikipedia content policies, specifically WP:NPOV. Both approaches are prevalent in the literature. Both must be given roughly equal billing. Showing the synergy between these approaches would be fabulous. Doing this is the point of the "unified solution" sections I've been suggesting since December 2009. Here's yet another try. Note that rather than completely avoid conditional probability the intent here is to provide an approachable conditional solution - while remaining strictly neutral about which approach might be considered "better". -- Rick Block (talk) 17:50, 11 June 2011 (UTC)

According to the problem statement above, a car and two goats are arranged behind three doors and then the player initially picks a door. If the player has picked, say, Door 1, there are three equally likely cases.

Door 1	Door 2	Door 3	result if switching
Car	Goat	Goat	Goat
Goat	Car	Goat	Car
Goat	Goat	Car	Car

A player who switches ends up with a goat in only one of these cases but ends up with the car in two, so the probability of winning the car by switching is 2/3. In other words, there is a 2/3 chance of being wrong initially, and thus a 2/3 chance of being right when changing to the other door.(vos Savant 1990).

Another way to analyze the problem is to determine the probability in a specific case such as that of a player who has picked Door 1 and has then seen the host open Door 3 (Morgan et al. 1991). The probability in a specific case can be determined by referring to the figure below or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138). Referring to either the figure or the tree, in the cases the host opens Door 3, the car is behind Door 2 with probability 1/3 but is behind Door 1 with probability 1/6 (the sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time). Switching wins twice as often as staying, so the probability of winning by switching for players who pick Door 1 and see the host open Door 3 is 2/3. The intuition behind this solution is that the host must always open Door 3 if the car is behind Door 2, but opens Door 3 only half the time the car is behind Door 1 - so a player who switches wins twice as often as a player who doesn't.

A formal proof that the conditional probability of winning by switching is 2/3 using Bayes' theorem is presented below, see Bayesian analysis.

Car hidden behind Door 3	Car hidden behind Door 1		Car hidden behind Door 2
Player initially picks Door 1

Host must open Door 2	Host randomly opens Door 2	Host randomly opens Door 3	Host must open Door 3

Probability 1/3	Probability 1/6	Probability 1/6	Probability 1/3
Switching wins	Switching loses	Switching loses	Switching wins
If the host has opened Door 3, this case has not happened		If the host has opened Door 3, switching wins twice as often as staying

Please note that the words I wrote

Switching gives the car if and only if you initially pick a goat. The chance you initially pick a goat is 2/3. So the chance of winning the car by switching is 2/3. The chance can't depend on the specific numbers of the doors concerned in a specific case (e.g., Door 1; Door 3) by symmetry. So the chance of winning the car by switching in the specific case that player chose 1 and host opened 3 is also 2/3

do not support one particular point of view on how MHP "should" be approached. They gently introduce the reader to the fact that MHP is approached from many different points of view in the literature. They represent a synthesis of two mathematical proofs (brief, informal version) but the synthesis is not original - it was given by discussants of the Morgan et al. paper. Why can't something like that be in the lead? Richard Gill (talk) 05:12, 12 June 2011 (UTC)

I have some suggestions for small improvements of Rick's text, which in general seems to me to be fine.

(1) He writes: "in the cases the host opens Door 3, the car is behind Door 2 with probability 1/3 but is behind Door 1 with probability 1/6 (the sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time)."

The words "in the cases .." suggests what is coming next is already going to be a conditional probability. I would prefer "the car is behind Door 2 and the host opens Door 3 with probability (1/3)x1=1/3, while the car is behind Door 1 and the host opens Door 3 with probability (1/3)x(1/2)=1/6. These are the only possibilities in which Door 3 is opened. Altogether, the host opens Door 3 with probability 1/3+1/6=1/2. Given that he opens Door 3, the car is behind Door 2 (1/3)/(1/2) = 2/3 of the time".

(2) He writes: "Switching wins twice as often as staying, so the probability of winning by switching for players who pick Door 1 and see the host open Door 3 is 2/3".

I would prefer "Not only overall (as we already saw), but also for players who pick Door 1 and see the host open Door 3, switching wins the car 2/3 of the time." Richard Gill (talk) 15:22, 17 June 2011 (UTC)

Another Big Problem with this Article

This article, as it is written, isn't about The Monty Hall problem, it is about Marilyn vos Savant's column about it. That is a slight exaggeration but only slight. --Ericjs (talk) 04:34, 21 June 2011 (UTC)

There are several good reasons for this.

Marilyn Vos Savant's column made the problem world famous; put it into the public domain, so to speak. It triggered a huge literature, both popular and academic.
Almost every source quotes literally Marilyn Vos Savant's (ambiguous, incomplete) wording of the problem.
Different sources have formalized Vos Savant's question in different ways. One of the reasons the article is so long is because there are so many different Monty Hall Problems. What they have in common is that they are all attempts to extract from Vos Savant's words a collection of logical or mathematical assumptions which can be analysed logically or mathematically. Richard Gill (talk) 10:22, 23 June 2011 (UTC)

POV?

How did this article ever pass FA with a neutrality tag? Is it still necessary? Joefromrandb (talk) 08:11, 1 June 2011 (UTC)

It didn't. The article is considerably different that it was as of its last FA review (see the history box at the top of this page). The neutrality tag was recently added in response to "consensus" changes to restructure it in a way that in the opinion of several editors here gives undue WP:WEIGHT to the "simple solutions" (in preference to the standard approach using conditional probability used in virtually every introductory probability textbook). Those promoting this structure claim it does not create a bias, but other than arguing "no it doesn't" have made no attempt to address this concern. So, yes, the tag is still necessary. -- Rick Block (talk) 14:04, 1 June 2011 (UTC)

Virtually - but not every - textbook in elementary probability solves MHP in this way in order to have a fun example for their mystified / bored students. None, as far as I know, give a good reason why you "must" solve it this way. The reason is quite simple: because there are plenty of other ways to solve MHP under their own restricted formulation, and even more ways to solve equally sensible mathematical formulations. The general reader has little reason to be exposed to Bayes theorem. Those who can stomach Bayes will already know the law of total probability, which shows that the only way any conditional probability would actually staying would be if there was a way to do better, overall, than the success rate of 2/3 obtained by always switching. *If* the general Wikipedia reader would imagine for a moment that 2/3 could be beaten, there are even plenty of elementary ways to dispell any doubt.

The article on Bayes Theorem already contains MHP as a worked example, so putting enormous weight on it here amounts, it seems to me, to be an extremely biased point of view, probably inspired by some half-understood extremist points of view which were roundly criticized in the academic literature 20 years ago. Richard Gill (talk) 14:34, 7 June 2011 (UTC)

You're talking about your opinion of "The Truth" here, not what references say. If textbook authors don't say why they are solving it using conditional probability, you're simply speculating about their reasons. Whether they think you "must" solve it as a conditional probability problem is quite another question - equally lacking in any evidence whatsoever (I'll offer up Grinstead and Snell, who explicitly say that not solving it this way is not addressing the question Craig/vos Savant asked). The point is that from a survey of sources you'd find

hundreds of more or less reliable popular and non-math academic sources presenting "simple solutions" including both sources (like vos Savant) that show the chance of winning of a strategy of switching (preselected in advance of seeing which door the host opens) is 2/3 as well as sources (often the same sources) that claim the posterior probability (after the player picks Door 1 and the host opens Door 3) of Door 2 must be 2/3 because (?!) the prior probability (before the host opens Door 3, before or after the player picks Door 1) of Door 1 is (obviously) 1/3 and the posterior probability of Door 3 is (obviously) 0.
many dozens if not hundreds of textbooks and other academic, nearly exclusively math, sources presenting solutions based on conditional probability using Bayes' theorem or decision trees.
a few (perhaps more than a dozen), nearly exclusively economics, sources presenting solutions using game theory
a few (I've named at least half a dozen, there are quite possibly more and they are not all "20 years old") sources criticizing "simple solutions" for not addressing the conditional probability
very few (2?) sources criticizing the criticizers

Now, go read WP:NPOV and then look at the outline of the current article. Does the current article "fairly represent all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint"? Or does it give enormous weight to "simple solutions"?

Again, as Wikipedia editors our opinions (including Professor Richard Gill's opinion) about which of these approaches is "right" is completely irrelevant. The question we should be talking about is what is the relative prominence of these views among reliable sources. You apparently agree virtually all (but not every) textbook in elementary probability solves MHP using conditional probability (i.e. you apparently agree with #2 above). Do you disagree with any of the other points? If you respond, please respond about sources, not about your opinion of The Truth. -- Rick Block (talk) 16:15, 7 June 2011 (UTC)

The living person Richard D. Gill (mathematician) is a reliable source. You didn't take any account of his publications in your literature survey. I speak on this talk page as a real person, not as an anonymous editor (of the article on MHP). I've explained why lots of textbooks in elementary probability include MHP as an illustration of Bayes Theorem: because if you add extra probability assumptions, it can be solved that way, so makes a fun example for a probability class. Gill (2011) shows how by making successively stronger assumptions about the game, one can get successively stronger reasons for switching. That paper also argues that the assumptions need to be justified and this require thinking about (and choosing) an interpretation of probability. Gill argues that there is no one "right" solution but that the task of the mathematician is to offer the consumer a menu of options. Does the user need a Volkswagen or a Mercedes? If he wants the latter he'll have to put down more cash. Grinstead and Snell are wrong in stating that the simple solution doesn't answer Vos Savant's (Whitaker's) question. In mathematics we do not argue by Authority but by logical, mathematical proof, so on this point anyone can see that Grinstead and Snell are bad teachers. Morgan et al. were thoroughly taken to task for similar arrogance. Of course one might refer to these arrogant opinions of Morgan c.s. but then one should also refer to the criticism of the criticism. I recall that poor Nijdam was opposed to the simple solutions because he didn't want his students to come to him saying that the conditional solution is unnecessary because Wikipedia gives the simple solution! Seems he was not succeeding it getting across to his students *why* it should be done that way, just as he didn't succeed to us either. So he wanted Wikipedia rewritten in order to buttress his Authority in the classroom!

Agreed: it is hard to write an article on a topic like MHP when much of the published material is evidently poor quality.

The reason we are still here quarelling is that the conditionalists never ever succeeded in giving sound reasons which the ordinary reader can appreciate, why the ordinary reader *has* to learn how to do it their way. So you should be interested to learn that Gill's own publications do contain material on this point! That could be useful in bringing your agenda forward! Understand the sound reasons for giving a conditional answer, appreciate the limitations involved, and appreciate the fact that the sound reason for wanting a conditional probability depends very much on what you mean by probability anyway. A good article on MHP simply can't be written by cutting and pasting without understanding from published literature.

Personally I'd simply recommend focussing on mathematical truths (correct logic) and let the reader judge for themselves what solution best serves their purposes. Lay bare the relationships between the different approaches. MHP is fun, rich, full of unsuspected new facets as one studies it deeper. If you want to promote the conditional approach (I'd like every reader to come out understanding it at the intuitive level, and appreciating it!) you must be very careful not to alienate your average reader. The average reader is just not interested that some academics said that other folk had done it wrong. Don't write that up front. Focus on writing a good section on the conditional approach, and come up with a constructive, not a destructive, motivation for it. Give the readers who are new to MHP a chance to get used to the idea that switching is smart and that if probabilities are involved, they're not 50-50, before moving to the next level. And build on the simple solutions (e.g. by using the symmetry proof) so that the reader actually benefits from their experience on level 1.

P.S. Anyone who understands Bayes' theorem obviously knows the law of total probability and hence realizes that the only possbile reason one could want to look at the conditional probability is if one wants to be sure that no strategy whatsoever can beat the overall success-rate of 2/3 achieved by the "always switch" strategy. We already know from the simple solution that "always switch" (2/3) beats "always stay" (1/3). Is it conceivable that staying or switching depending on X =(1 say) and H (=3 say) could do even better than 2/3, overall? In my experience of talking to amateurs and professionals, no one for a moment imagines that some complex mixed strategy could do better than either of the two extreme pure strategies. And of course, there exist a multitude of argments for this, some simple, some complex. But anyway it is clear that rigorously ruling out this ludicrous possibility is an optional extra for specialists. Actually the whole simple versus conditional issue is about making a mountain of a molehill. Richard D. Gill (mathematician), Richard Gill (talk) 15:07, 27 June 2011 (UTC)

Are you deliberately refusing to answer the question or do you simply not know the relative prominence among sources of various approaches to the problem? -- Rick Block (talk) 01:08, 9 June 2011 (UTC)

Yes I'm deliberately not answering the question. Because "ask a stupid question, get a stupid answer". The concept "relative prominence" is not well defined. I've patiently explained for the nth time why counting the number of elementary probability textbooks and comparing it to the number of elementary game theory textbooks is stupid. I'm not saying that the conditional approach should not be in the article at all. Start simple and move to complex later. Keep it simple. Present the facts of the matter in an attractive way and stimulate readers to think for themselves. There is no academic debate. There never was one. There is an ambiguously posed brain teaser which has many fascinating facets. It can be used as a vehicle for illustrating basic ideas in probability, game theory, psychology, ... MHP is fun. It is amusing! It is not a religion. Richard Gill (talk) 21:45, 9 June 2011 (UTC)

Just weird. WP editors should not forget about the topic of this article. It's about the question whether the candidate should better switch or better stay. And obviously it's about the paradox that most people – due to incorrect assessment of the given situation – tend to come to an incorrect result and consequently give the wrong answer. And the article should help to get the correct result and the correct answer. That staying has a chance to win of "1 out of 3", whereas switching has the double chance to win of "2 out of 3". The correct total solution has to be presented.

There are sources and sources. Even sources that talk past the issue and never did grasp the total solution. But that say that it doesn't matter that you never will know better than that the chance to win by switching will be "2 out of 3". Sources that assert that you "eventually could know even a tiny bit better, though", if you just "assume" to know better. If you just assume to have additional information on the actual location of the car in this one special game that the famous question is about. Then you effectively really "could" come to a closer result. Actual sources confirm that such misinterpretation is farcical, indeed.

Because all of that does not address the question posed and is completely irrelevant and completely useless to give the correct answer "switch". No matter what door number the contestant has first selected, and no matter which door number the host has opened. Sticking on door numbers shows that one did not grasp the "problem" and did not grasp the total solution. As this is a question of mathematical correctness, it is important to cite the correct sources that say that irrelevant and completely useless "solutions" are what they are: a farcical misinterpretation of the topic.

Yes, you should present conditional probability in odds form also, to show the correct solution. But without the ridiculous claim of "sine qua non". Sources are of importance, yes. Especially actual sources, capable in evaluating dated obsolete misconceptions and misinterpretations. Inevitable. There is no way to avoid. Gerhardvalentin (talk) 20:33, 7 June 2011 (UTC)

Gerhard - I can't tell if you're disagreeing with my characterization of sources above or not. Do you agree, yes or no, and if not with what in particular do you disagree? -- Rick Block (talk) 23:44, 7 June 2011 (UTC)

Rick, it is still the matter of understanding the sources, or not. Of understanding what all the sources say, or forgetting about the essence, for insignificant partial aspects. Gerhardvalentin (talk) 08:29, 8 June 2011 (UTC)

If the census of reliable sources offered by Rick Block above is accurate, it would seem the section of Criticisms is both overly prominant, and overly lengthy. I could see these issues as not properly reflecting NPOV. 166.216.194.171 (talk) 02:46, 8 June 2011 (UTC)

Yes 166.216.194.171, you are right. Overcharge of irrelevant two pennies (you could know more if you just knew more), not addressing the answer to the famous question, but belonging to textbooks for teaching and learning conditional probability theory and Bayes, examples in droves. Gerhardvalentin (talk) 08:29, 8 June 2011 (UTC)

Unified Solution vs. Argument Without Bounds

I have been watching the recent discussion with great interest.

Clearly the best solution would be to come to a solution that the consensus supports. So let me ask; given the recent discussions is such a solution likely? Or does it look like any such effort would be shipwrecked by some editors (including those not currently participating) who in good faith insist on content not acceptable to other editors who are also acting in good faith? In other words, does a basic and intractable content dispute still exist?

The optimist in me sees a half-full glass and believes that editors working together in good faith can come up with content that the consensus supports. In that case, the problem can be solved for good.

The pessimist in me sees a half-empty glass and believes that someone will then jump in and claim that no consensus exists, someone else will claim it does, and the argument will continue without resolution indefinitely. In that case I think I would need to restart my plan to get a ruling on the underlying content dispute so that we can all tell one side or the other that consensus is against them. I don't have any idea which side would prevail - it all depend on who has the most compelling argument.

(The Engineer in me, of course, does not see the glass as being half-empty or half-full, but rather sees the glass as being twice as big as required to hold the liquid.)

Opinions and advice sought as to the prospects of a resolution without going to content dispute resolution. Guy Macon (talk) 23:43, 23 June 2011 (UTC)

As I suspect everyone here knows, IMO the current version of the article is not NPOV as it presents the "simple solutions" as the primary way to address the problem - which (also IMO) does not represent (per WP:NPOV) "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources". If those in favor of the current version remain unwilling to address this issue in some fashion, then I'd say we have an intractable content dispute. I've suggested both introducing the "simple solutions" as one of several approaches with forward references to sections presenting other solutions (specifically including a solution based on conditional probability) and presenting a simple solution as well as an approachable conditional probability based solution in a combined "solution" section. If anyone has any other suggestions for how to address this issue I'm open to other ideas as well. -- Rick Block (talk) 21:48, 26 June 2011 (UTC)

That was my feeling as well. What I am seeing is yet another good-faith effort to arrive at "compromise" or "unified" content that doesn't satisfy both sides of the intractable content dispute. The good news? Still no hint of user misbehavior of any kind. I really think we need to walk this through the steps of content dispute resolution and end up with one side or the other being told that the consensus of the larger community is against them. BTW, I reject the assertion that we already have a consensus as being factually untrue. Consensus is not just a matter of votes, and both sides clearly have sufficient adherents and good enough arguments that it is clear that we have no consensus.

To that end, I need help in one specific area. I need some sort of concise document that clearly shows the differences between the two views. It could be a couple example pages created as talk or user sub-pages, but I am agnostic as to the format or location. The key here is that (1) it is short and easy for someone from the outside to evaluate and (2) everyone agrees that it represents the two positions. Then I need short arguments making the case for each version. Once I have that, I can start walking it through content dispute resolution. The alternative is not just arguing without end, but arguing until the year-long blocks expire and we are joined by at least one person who is known to be disruptive, then back to arbcom. I cannot do this alone - I don't have the background and I want to remain neutral (easy enough - I really do think that both positions have good arguments and have no idea which way the consensus will go). If someone has a better plan, I am all ears. Otherwise, who will help me? Guy Macon (talk) 00:34, 27 June 2011 (UTC)

To make a rational decision about the weight and NPOV issue I'm raising I think we also need an agreed upon summary of the sources. The summary I suggested above ([17]) has not been disputed by anyone (although Richard Gill apparently thinks determining relative weight among sources is a stupid question [18]). Moreover, there seem to be a variety of other issues folks have with the article in addition to the POV issue I'm raising (e.g. Glrx's comments above [19] and Tijfo098's [and other's] FAR comments [20] which have resulted in the article being demoted). To get this article back to featured status will require someone or some group of editors to spend a considerable amount of time and energy on it - resolving the POV issue is simply one of many things that will need to be done. -- Rick Block (talk) 02:41, 27 June 2011 (UTC)

Rick: I have refined my answer to you about the sources in the section you referred to (POV?), immediately after your passage with the summary of sources, which strikingly omits any mention of the latest academic literature, which deals explicitly with the conflict we are talking about here. Richard Gill (talk) 15:11, 27 June 2011 (UTC)

Here are some thoughts I just came to on a related paradox and related problematic wikipedia article: two envelopes problem.

There are at least three completely different two envelope problems. With and without opening the first envelope, with and without probability. They all have numerous very easy solutions, at least, easy in their own terms. And they have different easy solutions for readers with different backgrounds. Probabilists, economists, logicians all see different issues. A layperson can "defuse" the paradox (es) by common sense, but is not equipped to even see what the different academics are on about. Should the layman care? Probably not. Should the academic professional care that the layman doesn't care? Probably not. If the academics failed in the past to communicate their concerns to the laypersons, we can hardly expect wikipedia editors to be able to do it now. The literature is vast and complex. A big challenge to an encyopediist who has to write both for laypersons and for *all* the academic communities. Solution: collaborative editing. It requires that *all *participants recogniise the "relativity" and hence incompleteness of their own point of view

Richard Gill (talk) 09:58, 27 June 2011 (UTC)

Rick, tell me how to determine relative weight of popular, elementary statistical pedagogical, elementary game theoretic pedagogical literatures. I don't think it's stupid. I think it is immensely difficult and ultimately highly subjective. What about relative weight of readership? ("popular" or "academic"?). That is more easy to determine, and I think is a prior issue. Richard Gill (talk) 10:06, 27 June 2011 (UTC)

+1 to Richard Gill and +1 to Guy Macon, and I appreciate Guy Macon's contributions of 23th June and today's 00:34h, especially his words that we need some sort of concise document that clearly shows the differences between the two (or even more?) different views, and imho also picturing the sticking cruces and the nagging of "we could know better indeed, if we just knew better, but while we don't know better, let's just assume we knew better, then we are free to suppose that we really know better than 2/3". Gerhardvalentin (talk) 13:42, 27 June 2011 (UTC)

I am strugging to figure out what a Guy's rightly desired summary of the content dispute should look like. And don't get anywhere. The problem is that we have a popular literature on a popular brain teaser called MHP; we have an academic literature where MHP is used as a pedagogical example to illustrate Bayes' theorem in the elementary probability class; we have a less visible and more inhomogenous literature from game theory, decision theory, and mathematical economics where MHP is a pedagogical example of a decision problem; and we have a big literature in psychology and pedagogics and maths education where the focus is not on "the right answer" (a "right answer" is taken for granted) but on our mental processes in coming to it. The writers of sources in all those fields appear to be writing for the whole world, and they are writing for *their* whole world as they see it, but in fact they are writing primarily for their colleagues in their own field. We'll have readers from all these areas and more, we have editors from al these areas and more. Editors have simply got to forget their personal preferences and personal background and work together so that something attractive can be put together using all this material, which does justice to its diversity. Thus fighting about relative weight of different solutions and how to avoid a biased point of view etc etc is simply working at the wrong level! If individual editors don't have the overview and background in order to write an article in which the reader hardly notices that material from very different communities has been merged, we had better split the article into separate main sections "MHP in popular literature", "MHP in teaching statistics", "MHP in game theory", "MHP and psychology", etc. The lead will simply say that MHP is approached in different ways in these different literatures and each separate section will have it's own lead and be more or less self-contained. Richard Gill (talk) 15:34, 27 June 2011 (UTC)

Synthetic Solution a la Richard Gill

Switching gives the car if and only if you initially pick a goat. The chance you initially pick a goat is 2/3. So the chance of winning the car by switching is 2/3. The chance can't depend on the specific numbers of the doors concerned in a specific case (e.g., Door 1; Door 3) by symmetry. So the chance of winning the car by switching in the specific case that player chose 1 and host opened 3 is also 2/3.

"The chance can't depend on the specific numbers of the doors concerned in a specific case (e.g., Door 1; Door 3) by symmetry" is only correct if Monty tosses unbiased coin. But Monty's behavior, and the implied Bayesian computations (under the assumption of randomization of his choices) are irrelevant for the optimality of always-switching, in the sense that "choose 1 then switch whichever door revealed" gives 2/3 for every Monty's behavior, and it is impossible to do better. May I suggest therefore a simple and hopefully inambiguous improvement

"Switching gives the car if and only if you initially pick a goat. The chance you initially pick a goat is 2/3. So the chance of winning the car by switching is 2/3. It is impossible to have higher winning chance with any other strategy of contestant (choosing a door and deciding on switch), because, no matter how Monty opens the doors when he has a choice, for at least one out three locations of the prize the contestant will lose. Thus the strategy of choosing door 1 and switching all the time is optimal among all possible strategies. Because it is optimal overall, it is optimal in the situation when door 3 is revealed. "

A combinatorial fact underlying the statement "for at least one location of the prize the contestant will lose" is the observation that for every strategy profile (deterministic way of playing for Monty and contestant, presuming hiding action of the quiz-team) there at least one particular door (maybe two doors) which will be either revealed or mistakenly unchosen on the final stage. — Preceding unsigned comment added by Machtindex (talk • contribs) 14:00, 28 June 2011 (UTC)

There is so much published material about the MHP that IMO there's little justification for including anything other than well established solutions that have been independently published dozens of times. Richard has recently published his solution, but it is not one of the ultra-common approaches of which there are at least two: vos Savant's (i.e. a strategy of switching wins in 2 out of 3 equally likely car locations), and the direct conditional probability approach (i.e. given the player has selected door 1 and the host has opened door 3 [choosing uniformly randomly if the car is behind door 1], the conditional probability of winning by switching is [ 1/3 ] / [ 1/3 + 1/3*1/2 ]). Since these are the two predominant approaches, each appearing in dozens and dozens of reliable sources, they should be the approaches that are "featured" in this article. Other approaches can of course be mentioned, but only in proportion to their WP:WEIGHT among sources. -- Rick Block (talk) 18:27, 28 June 2011 (UTC)

@machtindex:

1) I defend the mention of symmetry as follows. Please realise that I am using probability in the subjectivistic sense: "my" probabilities reflect my knowledge, they do not (necessarily) reflect the physics of real world random generators (coins or brains). Very many people use probability in the Bayesian/subjectivist sense. I suspect almost all lay-persons. It is not a question of knowing that Monty uses an unbiased coin. It is a question of not knowing anything, it is a question about your knowledge being symmetric in renumbering of the doors. If you happened to choose the door hiding the car, and it was Door 1, you would be equally surprised if Monty opened Door 2 as if he opened Door 3. Using subjectivist probability we can actually solve the MHP. But not with frequentist probability, unless we are given further information. All we know is what Vos Savant told us, so from a subjectivist point of view, the car is, for us, equally likely to be behind any of the three doors, and if we choose the same door as the door hiding the car, Monty is equally likely, for us, to open either of the other doors. Vos Savant didn't tell us that Monty made his choice by tossing a coin, unbiased or not! Richard Gill (talk) 18:55, 28 June 2011 (UTC)

2) Sure, if we take a frequentist point of view, and allow a biased host, then it is also the case that we can't do better overall than 2/3 win-chance (by always switching). Therefore no conditional probability can *favour* not switching. Therefore it is a waste of time to try to figure out what the conditional probability is. All this, provided that the car is initially hidden completely at random. But how do you know that you can't do better overall than 2/3 win-chance? I know it is true and I know a lot of elementary proofs, proofs which don't require Thomas Bayes' machinery or unilluminating calculations. They just require a little careful thought. I have published all the proofs I know. However Rick Block will tell you that we must stick at simply mirroring the existing published literature, strictly proportional to the number of publications, irrespective of quality, originality, date of appearance, etc etc. Previous editors forbade me even to mention the fact that you can't do better overall than 2/3 and hence no conditional probability can indicate not switching, because apparently this elementary observation (elementary for anyone who knows probability) had not been published in the literature. Fortunately now it is published, by me. According to Rick, wikipedia readers must now wait another 20 years till hundreds of writers of elementary statistics texts have done MHP this way too. I don't think this is a good way to write a good article. But if the rules say that's how it is to be done, then that's how it is. We'ld better write good articles in other online encylopedias where editors are allowed to use some intelligence. Richard Gill (talk) 18:55, 28 June 2011 (UTC)

Richard - the (obvious) result that no conditional probability can indicate not switching (given the car is randomly located and the host must show a goat and must make the offer to switch) was shown 20 years ago, by both Morgan et al. as well as Gillman. They both showed that if you leave the host's probability of opening door 3 in the case the car is behind door 1 as a variable q, the chance of winning by switching is 1/(1+q), i.e. you're never worse off switching. Showing this same result without using the mechanics of conditional probability is simply another path to the same (already known) result. You apparently have a distaste for the mechanics (which I don't truly understand since you're a practicing academic in this field), but the point of any encyclopedia is not to pick some "best" way to approach things - it is rather to convey the totality of what is known about a topic. "Known" in this sense means published - and the more frequently something has been independently published, the more likely it is to be something that's important for a reader to know about the topic. This is what wp:weight is all about. A good article gives each view weight in proportion to the prominence of that view among reliable sources. If you disagree that this is a good way to write an article that's fine - but what you're saying is you don't agree with WP:NPOV which is one of the fundamental principles of Wikipedia. -- Rick Block (talk) 23:45, 28 June 2011 (UTC)

Obvious??? Not obvious to most readers, I think. And anyway, it is a superfluous result. It is much easier just to prove that P(win by switching | door chosen=x, door opened=y)

\geq

1/2 whatever the number x of door chosen and y of door opened, than to prove that this probability is precisely equal to 1/(1+q) and hence is

\geq

1/2; where q = P(host opens door y | car is behind x and the the player chose x).

In fact, if you find it intuitively obvious that no mixed strategy could improve on the 2/3 overall win-probability of "always switch" (compare with the 1/3 of "always stay") then you will not even require a proof. If you do want a proof, the domination trick of Gnedin's recent papers will give you a super-elementary proof. And all this is for the more tricky situation with a possibly biased host! Which I thought we had agreed is not the main MHP!

For the symmetric case, it's enough to note that by symmmetry the chance of winning can't depend on specific door numbers of any particular case so therefore must equal the unconditional chance. I just cannot imagine why it helps to tell readers that elementary textbooks on probability theory find the result 2/3 by a formal calculation by first principles from the definition of conditional probability, especially since the wikipedia article on Bayes theorem contains precisely this calculation as an illustration.

I am reporting mathematical truths here, this is nothing to do with a POV. I am not trying to push one side or another of some argument. I am not agreeing or disagreeing with Morgan's opinion of Vos Savant's intelligence. This particular little controversy was a storm in a tea cup, as the discussion contributions to Morgan et al. show.

There are a collection of mathematical facts. I hope all editors can agree on them. The important question is how to convey these to the layman reader in as palatable a way as possible, while at the same time also catering for specialists (students, professionals) from various fields (probability, game theory, mathematics education). Harking on about some bad-mannered and ill-informed remarks by various obscure writers of 10 years ago does not aid writing an article for the general public. Richard Gill (talk) 12:03, 29 June 2011 (UTC)

You are most certainly pushing a POV, specifically that the oft-published solutions that specifically address the conditional probability are "superflous" or "a storm in a tea cup" and, hence, both

1) that including such solutions in the article as an approach equally valid to the approach of addressing the probability of a strategy of always switching is unimportant

I do not say this, Rick! Such solutions should be in the article. The question is one of pedagogy: how to do it? They should be made simple and appetizing. Richard Gill (talk) 06:23, 30 June 2011 (UTC)

2) we should ignore the numerous authors (because they're "ill-informed", "bad-mannered" and "obscure" - ad-hominem much?) who explicitly say the MHP is asking about a conditional probability

I do not say this either! But don't alienate readers by copying dogmatic (unmotivated) statements. Give the good reason why it could be of interest to look at the conditional probability. MHP asks for a decision: switch or stay. It doesn't ask for a probability. Richard Gill (talk) 06:23, 30 June 2011 (UTC)

By "obvious" above, I meant obvious to someone such as yourself who understands the very basics of conditional probability (once again, this is not rocket science but rather high school level math approachable by any even moderately bright 15 year old). The probability the car is behind door 2 after the host opens door 3 is (by the elementary definition of conditional probability - no Bayes theorem required) the probability the car is behind door 2 divided by the sum of this plus the probability the car is behind door 1 (in the case the host opens door 3). If at the start the door 1 and door 2 probabilities are the same, then the resulting probability can obviously never be less than 1/2 - and if we assume the host picks evenly in the case the car is behind door 1, with very little thought, can be shown to be exactly 2/3 (1/3 / (1/3 + 1/6)). -- Rick Block (talk) 15:37, 29 June 2011 (UTC)

I'm all in favour of material such as what you have just written. What I object to is converting that in-line direct calculation into long formulas and suggesting that that is the only academically respectable solution. And I object to having the structure of the article dictated and dominated by the issue of the quarrel between Morgan and Vos Savant. The quarrel was resolved in the discussion contributions to Morgan et al's paper. The reader of the wikipedia article on MHP is interested in gaining insight into MHP. Our task is to find the pedagogical structure which maximizes the communication of insight. And today we have even more insight than twenty years ago. The staple content of introductory probability texts in the intervening twenty years is not of great interest to the present-day reader. Richard Gill (talk) 06:32, 30 June 2011 (UTC)

Yes Rick, it's really a matter of WP:weight and not of POV. The remarkable and really striking fact concerning the MHP is not that most people usually are able to give the correct answer at once, but the really remarkable and most striking fact (WP:weight) is that almost no-one is able to give the correct answer − nor to apprehend, not even if it had amply been explained to him that two or more doors have better chance than one single door. That's the "paradox".

The underlying psychological reasons have extensively been analyzed and explored, and this kind of literature is of real WP:weight regarding this paradox.
Once more: the MHP is about a solution that most people get "wrong". And that's of WP:weight.

But the phantasm and sophism to make believe that using "additional information about the actual location of the car", in absence of such "additional information" regarding "q" for the actual situation the famous question is about, has to be denounced. Because such hair-rising considerations never have addressed the paradox, nor ever helped to give a better answer than to switch, although such considerations, including an unknown "variable" q (i.e. without knowing its value) may very well be suited to teach pupils conditional probability theory. Not to answer the famous question, never ever, but just to teach and to learn conditional probability theory. Without any impact or WP:weight to correctly answer the famous question: "YES or NO?"

So you should claim no longer that such unhelpful hair-raising considerations have any WP:weight to give the correct answer to the famous question of that famous paradox, except for contents of textbooks for teaching and learning conditional probability theory. And that's where it belongs, without WP:weight for the correct answer to that famous question. Gerhardvalentin (talk) 21:28, 29 June 2011 (UTC)

Let's start with something simpler. Can we agree on what to name each position? Hopefully in plain English? Guy Macon (talk) 02:56, 29 June 2011 (UTC)

Big-Endian position=simplistic, Requires no knowledge of Bayes. Little-Endian position=conditional resolution: exercise from the 3 page of textbooks Middle-Endian position= Big-Endian solution implies Little-Endian, strong argument to re-write textbooks — Preceding unsigned comment added by Machtindex (talk • contribs) 06:39, 29 June 2011 (UTC)

I have carefully re-read the article in its main part. I think it is a great job. One only remark is that what is called "decision tree" is not a decision tree for the reason that choosing door logically follows placing the car. The best name would be "tree diagram" as in Grinsted eta all. Another remark to the sources of confusion: mathematicians (not laypeople) do not realise immediately that the basic-MHP (may I suggest this inplace of canonical/standard) is distinct from Monty Fall.

To the essense of the dispute: what Richard Gill says is the greatest discovery ever about the MHP, after the main fact that switching is much better. In depth, this is Bellman's dynamic programming: overall optimality is equivalent to the conditional optimality in every step of the decision process. If the page is willing to stay in line with time and the up to date knowledge, (which is understandable to schoolkids) then a synthetic section is a *must*. I strongly recommend the synthetic section.

To RickBlock: What you call "obvious" about MonteCrawling is incorrect for certain 4-door versions, as signaling by the sequence of 2 revealed doors Monty can inform you about the right location 100%-ly. It is a special and amazing feature of the basic version, which you cannot appreciate without going in higher dimensions.

I also do not see in the article connections with sociology, philosophy, economy etc. These need to be added by experts. I think it would be a good policy that Administrator simply invites good people to contribute. — Preceding unsigned comment added by Machtindex (talk • contribs) 13:42, 29 June 2011 (UTC)

@Machtindex: Our long-suffering "administrator" Guymacon is only active here because of past conflicts and interest in the topic. If you would like to encourage editors from relevant fields to join in the editing of the page on Monty Hall problem, you should try to recruit them yourself e.g. by placing announcements on suitable wikipedia project pages. Wikipedia is a collective effort! I think it has already been done for statistics, mathematics, and game theory. Richard Gill (talk) 13:06, 1 July 2011 (UTC)

Just to make sure nobody gets the wrong impression, I am not an actual Wikipedia administrator (Richard, of course, knows this, thus the quotation marks above), nor do I have any particular desire to become one. It is an interesting aspect of Wikipedia that if you are an administrator you are expected to not use your administrator powers in areas where you are involved but rather to act as an ordinary user. (Can you tell whether one of the participants in this discussion is an admin? No? Then he is doing it right.) Guy Macon (talk) 14:19, 1 July 2011 (UTC)

@Guymacon As far as statistical inference concerned, the Administrator is likely to be Rick Block. Whoever he might be I admire his work. Some three years he stands

(1) answering tonns of profanations, (2) resisting ambitions of highly educated profs to push forth their insights. The task of the site is to serve the general public by presenting things in understandable way, but also to make it attractive for students and everybody looking for new insights. I checked some other Wiki articles, and observed that some of the editors-mathematicians are busy with self-promotion and ignore important contributions. Administrator of mathematics-related didactical site, above understanding the things well should have ability to stay neutral and have a good taste. It is difficult to balance. Perhaps, there should be a group of administrators to compromise. I personally find the 3 year-long talk-pages discussions worth of publishing in a book. Are there copyright constraints?Machtindex (talk) 22:32, 1 July 2011 (UTC)

@Machtindex: Rick is indeed a so-called Admin which gives him certain powers in general, but specific wikipedia articles do not have Admins assigned to them. I understand that it was through his efforts that the page became a Featured Article at some stage. About republishing material on wikipedia: read [21]. Richard Gill (talk) 07:15, 2 July 2011 (UTC)