Talk:Monty Hall problem/Archive 36

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Archive 34

Bayes and combining doors

Following discussion with Nijdam and suggestions from Richard I am going to suggest the following intuitive approach to using conditional probability and Bayes' rule to expand the 'Combining doors' solution later on in the article.

We could show the same two pictures but with the posterior probabilities being shown as 1/3? and 2/3? to show that we need to justify that these are the correct probabilities after the host has revealed the goat.

We currently say, '...the 2/3 chance of hiding the car has not been changed by the opening of one of these doors'. This is correct but we provide no justification for it.

If we start with the standard case, where the host must always reveal a goat behind an unchosen door but where we do not know which door the host has opened, the calculation of the posterior odds using Bayes rule becomes very simple. The probability that the host will reveal a goat is 1, regardless of the position of the car thus the posterior odds are equal to the prior odds.

We could then consider the case where the host chooses a door randomly, which just happens to hide a goat. To find the odds, after the host has revealed the goat we need to multiply the original odds by (the probability that the host will reveal a goat, given that the car is behind one of the two unchosen doors) divided by (the probability that the host will reveal a goat, given that the car is behind the originally chosen door), this ratio being 1:2 giving the posterior odds as 2:2.

We could then move on to the case where the host must always reveal a goat behind an unchosen door and reveals a goat behind one specific door (say door 3). We could start with the case where the host is defined to choose evenly when he has a choice (or we take a Bayesian perspective) and show that the odds remain 2:1 in favour of switching.

Finally we could consider the Morgan variants, where the host has a (known) door preference, and give the Morgan results.

This may seem quite long but it covers the standard problem and some variants in an intuitive way that is mathematically sound. Does anyone think this approach is worthwhile? Martin Hogbin (talk) 15:19, 17 February 2013 (UTC)

Let's forget about the idea of 'combining'. The 'combined' probability does not change, but the individual probabilities do. How does this contribute to the understanding? Why does the combined probability not change? The main argument for this is the unchanged probability of the chosen door. How is the combining of any help? Nijdam (talk) 11:51, 23 February 2013 (UTC)

Yes, I believe a simple Bayes is appropriate.[1] Glrx (talk) 01:29, 18 February 2013 (UTC)

Hear, hear. Richard Gill (talk) 17:48, 21 February 2013 (UTC)

Something along the lines I have described above might be considered OR by some. On the other hand it might be considered a routine calculation. What do you think? Martin Hogbin (talk) 19:13, 21 February 2013 (UTC)

Routine calculation. Glrx (talk) 00:23, 22 February 2013 (UTC)

For "professionals" one can give a routine calculation. But for laypersons this is less than useless: they can't follow the calculation, they gain no intuition from it. cf Erdos: the calculation does not explain why things are how they are. For laypersons one must give *ideas* not formula manipulations. The idea is: the actually observed host's action (opened door 3, not 2) is twice as likely as when the car is behind the second door as when it is behind the first door. Before we saw that, the first and second doors were equally likely to hide the car. Afterwards the second door is twice as likely as the first door to hide the car. Richard Gill (talk) 06:36, 22 February 2013 (UTC)

I am not sure you understood what I was asking. I was proposing adding something along the lines I have given above. This is not the argument that you give, which I think most people will find hard to follow, but a different one, based on the 'combining doors' solution but fixed using Bayes' rule. Do you not like that? We could have both, of course. Martin Hogbin (talk) 09:47, 22 February 2013 (UTC)

You are right Martin in saying "hard to follow", and I say: hard to follow without having clearly answered the "why?". Redundancy regarding "that important key" is most welcome. Richard: Yes, you are absolutely right in clearly saying that it is on this "why".

It should be "made obvious" that, as soon as the guest did select his first door, and still before any other action, a given asymmetry did already pop up within the pair the two unselected host's doors. Now "one" of his two doors became twice as likely to be opened by the host than his "other" door. This asymmetry is a fact, but we still do not (and need not) distinguish which one is which one. Later the host will show us which of his two doors did pop up to be twice as likely to be opened, and not his "other" door, caused by the guest's first decision.

Richard, in order to understand your true argument it could be helpful to say my ceterum censeo: In the standard version, the host is absolutely free to choose which door to open in only 1 out of 3, but he definitely is slave to the act in 2 out of 3: slave to the the guest's first decision. In that 2 out of 3 being absolutely bound to open the door that he opens, and never his other door. Consequently, in that 2 out of 3 the door opened (#2 and not #3 resp. #3 and not #2) clearly discloses that the car actually definitely IS behind his second door. – Overall: the door opened by the (part time slave) -host "always / in any case" was / is twice as likely to be opened than his second door, so in consequence the car is twice as likely to be behind the door offered, than to be behind the player's first choice: (2:1).

Yes, it is on the "why". And on clearly showing that "why". And on that occasion you can show the difference to some quite aberrant variant of a host who never is (part-time) slave to the act: (1:1). Gerhardvalentin (talk) 11:08, 22 February 2013 (UTC)

I don't think the calculation "is less than useless" for lay persons. I think the formula can be explained, and the significant feature is that dividing by P{Monty reveals a goat}=1 doesn't change the the probability the car is behind the contestant's door. It also plays into if Monty doesn't know where the car is, then P{}=2/3 and the contestant's door now has a 1/2 probability. Glrx (talk) 05:30, 23 February 2013 (UTC)

Bayes' formula is no use for people who can't relate to formulas. 99% of all people. Baye's rule "posterior odds equal prior odds times likelihood ratio", on the other hand, is meaningful, intuitive, easy to apply. Richard Gill (talk) 19:01, 27 February 2013 (UTC)

Which calculation are we talking about?

I am suggesting that we start with a trivially simple Bayes' rule calculation for the case where a (undefined) door is opened by the host to always reveal a goat. This is, I think, what Glrx is suggesting above.

We multiply the prior probability by the probability that host will reveal a goat given the car is behind one of the two unchosen doors divided by the probability the host will reveal a goat given the car is not behind one of the two unchosen doors. This quotient is obviously 1, since the host always reveals a goat, regardless of where the car is.

We can then move on, as Glrk suggests above, to the case where the host chooses an unchosen door randomly and it just happens to reveal a goat. Here we can apply Bayes' rule and explain what it is in one hit.

If the host chooses randomly and the car is behind the originally chosen door it is certain that the host will reveal a goat. On the other hand, if the car is behind one of the two unchosen doors, the host has only a 1/2 chance of revealing a goat. Since we only count cases where a goat is revealed we must multiply our original odds by 1/2, changing 2:1 in favour of switching to 1:1. In fact I wonder if it might be best to have this calculation first.

Finally we can consider the case where the host always reveals a goat and has opened door 3. Now we have covered all the angles. Martin Hogbin (talk) 11:24, 23 February 2013 (UTC)

I don't understand this. If no door numbers are specified we can't use Bayes, and don't need Bayes. There is a chance of 1/3 that the player chooses the door hiding the car. There is a 2/3 chance that he chooses a door hiding a goat. In the first case the host will open a goat door and a player who switches gets a goat too. In the second case the host will open a goat door and the player who switches gets the car.

Bayes is only interesting and useful when we take account of door numbers. Richard Gill (talk) 18:55, 27 February 2013 (UTC)

Why can we not use Bayes' rule when there are no door numbers? We use the rule to calculate the posterior odds that the car is behind the two unchosen doors combined given that the host has revealed a goat. We know the odds of the car being behind the two unchosen doors are initially 2:1 and we use Bayes rule to show that they remain the same after the host has revealed a goat. Martin Hogbin (talk) 23:48, 27 February 2013 (UTC)

The probability the host will open a door and reveal a goat is 1, whether the car is behind the player's door or behind another door. So the Bayes factor equals 1, and the posterior odds equals the prior odds, 1:2. This is not a very interesting application of Bayes' rule. But of course, we already know that everything becomes simpler if we discard door numbers. Some insight is required to know that one can discard door numbers (though some people do it blindly/intuitively). Richard Gill (talk) 16:06, 16 March 2013 (UTC)

That may not be a very interesting application of Bayes' rule but it shows that the odds do not change when the host opens a door. This can be compared with the case where the host chooses an unchosen door randomly that just happens to hide a goat, where the Bayes factor is 2 making the odds evens. This nicely distinguishes between the two cases, which is the thing that puzzles people the most.

Of course, if we are fussed about door numbers we can calculate the Bayes factor given that the host has opened door 3. This is (1/2) / (1/2) = 1 and shows that the posterior odds remain 1:2 after door 3 has been opened and the combining doors solution is fixed. ~~ — Preceding unsigned comment added by Martin Hogbin (talk • contribs)

Yes one can apply Bayes in two steps, thereby fixing the combined doors solution. I have written this solution up in a number of places. Obviously I can't be the one who adds a reference to the article. And probably other people did the same before me, anyway. It's completely elementary. Richard Gill (talk) 13:57, 18 March 2013 (UTC)

I agree that when introducing a new tool (e.g. Bayes' rule) it is useful to show how it works in different problems, different contexts. This helps make people feel at home with the new concepts, and also shows the power of the new tool. It provides a systematic way to attack all kinds of variants to MHP and thereby explains why different variants have different solutions, Richard Gill (talk) 10:04, 17 March 2013 (UTC)

Richard, are there any references in the literature to the 'combining doors' solution failing if the host chooses randomly? Martin Hogbin (talk) 12:51, 17 March 2013 (UTC)

Not as far as I know. It's a flash-of-insight solution unique to the special situation of standard MHP, it is not the result of following a general strategy which would allow you to solve every possible variant of MHP. Richard Gill (talk) 13:52, 18 March 2013 (UTC)

So what should we add to the article?

My suggestion is that we add:
1) Bayes' rule applied to the random-choice host (no door numbers), showing how the odds of the car being behind the original door are changed.
2) Bayes' rule applied to the standard case (no door numbers) showing how the odds do not now change.
3) Bayes' rule applied to the standard case in which the host reveals a goat behind door 3.
Martin Hogbin (talk) 17:43, 18 March 2013 (UTC)

Good plan! Richard Gill (talk) 06:49, 20 March 2013 (UTC)

What sourcing do we need for this? Can it be considered a routine calculation? Does anyone object to this? Martin Hogbin (talk) 09:32, 20 March 2013 (UTC)

Routine WP:CALC: give a sourced expression and then plug in the numbers. An explanation of what Bayes means should be a ref to an appropriate text. Glrx (talk) 19:03, 20 March 2013 (UTC)

Jeff Rosenthal (book chapter; journal article) does some of these calculations. Possibly Jason Rosenhouse (book) too. -Richard Gill (talk) 05:59, 21 March 2013 (UTC)

What I would really like to say is something like this:

Let us consider the variation in which the host chooses at random one of the two doors not chosen by the playerand it just happens to hide a goat. Before the host opens a door, the odds of the car being behind two unchosen doors combined, rather than the door originally chosen by the player are 2:1.

We need to revise these odds to reflect any information that we may obtain when the host reveals a goat behind one of the two unchosen doors. To to this we can multiply the original odds by the proportion of times that the host would reveal a goat if the car is behind one of the two unchosen doors (which is 1/2 as the host was equally likely to have revealed the car) divided by the proportion of times that the host would reveal a goat if the car is behind one of the originally chosen door (which is 1 as the host can only possible reveal a goat).

This changes the odds to 1:1, meaning that there is no advantage in switching.

I suspect that his may be considered OR but I wonder if there is any way that we can apply Bayes' rule and explain what it means at the same time. A bare calculation, in proper notation will be of little help and interest to most people. Martin Hogbin (talk) 10:32, 21 March 2013 (UTC)

I'd mention Bayes and state what it does.

The Monty Hall problem can use some techniques from the theory of probability. Bayes' theorem provides a formula for how the probability of event A (the contestant's door hides the car) is affected by the occurrence of an event B (the host revealing a goat). Bayes' formula is

P(A|B)={\frac {P(B|A)\,P(A)}{P(B)}}

where P(α) is the probability of event α and P(α|β) denotes the conditional probability of event α given that β happened. (Give an explanation of the formula...)

For the MHP, the probability the car is behind the contestant's door, P(A), is 1/3 because each door is equally likely to hide the car. The probability that the host reveals a goat, P(B), takes a little thought. There are two goats, so at least one of the two remaining doors will hide a goat. Monty knows which door hides the car, so Monty will not choose to open that door. Consequently, the probability that Monty reveals a goat behind one of the other two doors, P(B), is 1. Similarly, P(B | A), the probability of event B (Monty reveals a goat) given A (the car is behind the initially selected door) is also 1. Substituting the numbers in Bayes' formula gives

P(A|B)={\frac {P(B|A)\,P(A)}{P(B)}}={\frac {1\,(1/3)}{1}}=1/3.

The formula tells us that the conditional probability that the car is behind the initially selected door is unchanged at 1/3 (i.e., the same probability before Monte revealed the goat). That also means that switching is advantageous because the probability that the remaining door has the car is 2/3.

Bayes' formula can also be used to calculate the probability for a different problem where Monty does not know which door hides the car. In that case, Monty randomly chooses one of the two remaining doors. Monty might reveal a goat or he might reveal the car. We are interested in the situation where Monty reveals a goat (if he reveals a car, the issues of switching is pointless. If the contestant's door hides the car (that is, A), then Monty can only select a goat. Consequently, P(B|A) is a certainty (1.0). Monty's chance of revealing a goat is 2/3. (A footnote could explain two ways of calculating this number.) Therefore

P(A|B)={\frac {P(B|A)\,P(A)}{P(B)}}={\frac {1\,(1/3)}{2/3}}=1/2.

This result, 1/2, corresponds to many people's intuition. If Monty is ignorant and behaves like a second contestant, then the contestant's door and the other unopened door are equally likely to hide the car and switching doors does not improve the chance of winning the car. However, in the original problem Monty knows where the car is and avoids choosing it, so it is beneficial to switch.

Glrx (talk) 16:59, 21 March 2013 (UTC)

That looks good to me but I would prefer to use Bayes' rule which talks in terms of odds and is bit simpler. Martin Hogbin (talk) 20:34, 21 March 2013 (UTC)

And moreover, it has all been done in the literature: Rosenthal's book "Struck by Lightning" (chapter 14 is on MHP), and his published article on MHP http://probability.ca/jeff/writing/montyfall.pdf. Note: in the common situation when prior odds are equal, Bayes' rule is even simpler still: the posterior probabilities of various hypotheses given some particular evidence are proportional to the so-called "likelihood" of these hypotheses given the evidence - by definition the likelihood of each hypothesis is the probability of the evidence under the hypothesis. Rosenthal calls this simplified version of Bayes "the proportionality principle". Richard Gill (talk) 12:30, 22 March 2013 (UTC)

second controversy

Why do we have all the citation needed stuff there? The information can essentially be found in Morgan's original publication and later replies, the letter by Martin & nijdam and Rosenhouse's book.

Also a paragraph based footnotes might be a good choice here.--Kmhkmh (talk) 11:42, 17 March 2013 (UTC)

Kmhkmh, what is this 'second controversy'? Martin Hogbin (talk) 12:11, 17 March 2013 (UTC)

Ok I'm slightly stunned now, of course i was referring to Monty_Hall_problem#A_second_controversy. I would add the sources myself, but I don't have access to them right now with the exception of Rosenhouse's book.--Kmhkmh (talk) 14:05, 17 March 2013 (UTC)

I have copies in a dropbox folder, anyone who is interested send me an email and I'll share them. Richard Gill (talk) 13:42, 18 March 2013 (UTC)

Sorry, Kmhkmh, I understand now. I was thinking of discussions here, where the Morgan issue was the first (and only) controversy. Martin Hogbin (talk) 17:36, 18 March 2013 (UTC)

The fundamental justification, if one is really needed, for my placing [citation needed] tags in the "controversy" section is simply this: any he said / she said narrative about a contentious exchange simply must cite sources.

However, although I would personally find it interesting to peruse the sources, I must say that I think the whole Morgan et al. contretemps is being given undue weight in the article. I agree with Richard Gill's earlier remarks that the "controversy continues" subsection is completely out of place,[2] and that the Morgan affair needs to be downplayed.[3] ~ Ningauble (talk) 18:37, 21 March 2013 (UTC)

What do you suggest? Martin Hogbin (talk) 20:31, 21 March 2013 (UTC)

Ok, let me clarify with three specific suggestions. Some are expressed provisionally, and I indicate my preference.

If this material is to be included in the article then I suggest that Wikipedia:Verifiability is not optional. It says "Please remove unsourced contentious material about living people immediately." (It also says "Editors might object if you remove material without giving them time to provide references; consider adding a citation needed tag as an interim step" – which is what I did.) The current article reports that vS and M accuse each other of mendacity and evasion: this is contentious material that needs specific citations. This is not really just a suggestion: as policies go, this is a relatively firm one.
If this material is to be included in the article then I suggest, as mentioned above, that it is out of place. The initial "media furor" was about widespread naïve 50/50 responses to the riddle, and it may be debatable whether it is best to include this before or after presenting any solutions. A "second controversy" over vos Savant's method of solution and the putative necessity for using conditional probability should not be presented before any solutions are presented. Discussing contention over vos Savant's method of solution before even giving her solution is not just a confusing way to organize the article, it is contrary to both proposals in last year's RfC. If the article is to cover debates about or between vS and M then I suggest placing it in a later section about the merits of different methods of solution, presented after the solutions themselves.
The previous two suggestions were contingent on "If this material is to be included in the article". My third suggestion, as mentioned above, which is my personal preference, as should have been clear there, is to downplay the Morgan affair. As much as I like the idea of calling Morgan, et al. on the carpet for their mendacity, the article should focus on the Monty Hall problem and not on the personalities of its discussants. It is one thing to describe the relative merits of different approaches, briefly, to the extent that mathematical pedagogy is relevant for the average Wikipedia reader. I suggest doing so, briefly. It is another thing altogether to astonish and confuse our readers with a catfight.

I hope these suggestions clarify what I mean by {{citation needed}} and, perhaps to some extent, by "undue emphasis on contention." ~ Ningauble (talk) 17:21, 22 March 2013 (UTC)

I think this is getting a bit surreal now. Obviously it is appropriate to complain about the lack of citation/sources in that section. However as I wrote above the content is (to my recollection at least) more or less correct and sources are available. So why isn't simply the person who added that content adding the according sources now or alternatively somebody more or less agreeing with that section and having access to all involved sources? Instead we seem to be at the verge of potentially producing much ado about nothing.

@Richard Gill: Since apparently have access to all the related sources couldn't you simply add the citation. I know it is isn't necessarily your responsibility or problem however since you didn't mind to work extensively on the article in the past, I don't quite see what the problem is in doing such a small edit yourself now and why you're offering the sources to others instead. Or is their any content you object toh and hence don't want to source?--Kmhkmh (talk) 21:24, 22 March 2013 (UTC)

P.S.: I see no real reason for "downplaying" the "second controversy" as such, because that controversy was somewhat influential in the academic discussion of the subject and is as such described in (summarizing) secondary sources like Rosenhouse's book. There the subject is treated in 2 separate sections as well (L'Affaire Parade (pp. 22-26), The American Statistician Exchange (pp. 26-31)).--Kmhkmh (talk) 21:33, 22 March 2013 (UTC)

It's not one citation. It's a whole heap of citations (at least, there is a whole list of notices to add references).

Let's Make a Deal: The Player's Dilemma: Comment

Author(s): Richard G. Seymann

Source: The American Statistician, Vol. 45, No. 4 (Nov., 1991), pp. 287-288 Published by: American Statistical Association

Stable URL: http://www.jstor.org/stable/2684454

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

Author(s): J. P. Morgan, N. R. Chaganty, R. C. Dahiya, M. J. Doviak Source: The American Statistician, Vol. 45, No. 4 (Nov., 1991), p. 289 Published by: American Statistical Association

Stable URL: http://www.jstor.org/stable/2684455

Letters to the Editor

Author(s): William Bell, M. Bhaskara Rao

Source: The American Statistician, Vol. 46, No. 3 (Aug., 1992), pp. 241-242

Published by: American Statistical Association

Stable URL: http://www.jstor.org/stable/2685225

Title: MORGAN, J. P., CHAGANTY, N. R., DAHIJA, R. C., AND DOVIAK, M. J. (1991), "LET'S MAKE A DEAL: THE PLAYER'S DILEMMA," THE AMERICAN STATISTICIAN, 45, 284-287: COMMENTS BY BELL AND RAO

Author(s): Marilyn vos Savant, John P. Morgan, Narasina R. Chaganty, Ram C. Dahiga, Michael J. Doviak, Nicholas R. Farnum, Duncan K. H. Fong

Source: The American Statistician, Vol. 45, No. 4 (Nov., 1991), pp. 347-348 Published by: American Statistical Association

Stable URL: http://www.jstor.org/stable/2684475

Morgan, J. P., Chaganty, N. R., Dahiya, R. C., and Doviak, M. J. (1991), “Let’s Make a Deal: The Player’s Dilemma,” The American Statistician, 45 (4), 284–287: Comment by Hogbin and Nijdam and ResponseRichard Gill (talk) 07:54, 23 March 2013 (UTC)

I have made a couple of deletions to reduce the personal aspect of the text. WE can add things back if we have good sourcing. Martin Hogbin (talk)

(edit conflict, I have not reviewed Martin's latest changes) Thanks Richard. I know you have researched the literature quite extensively, but I am still confused about the correspondence between these sources and statements in the article:

The article refers to "subsequent letters to the editor", but the first two citations here are discussion that ran together with the original Morgan, et al. paper (as sometimes happens when the editors of a journal feel a paper ought not stand on its own), not subsequent letters. Where the article says "In particular, vos Savant defended..." it looks like it refers to a letter by vos Savant. Are we still missing a citation for this, or is one of these a secondary source describing her defense? I assume (?) that where the article says "Later (2011), they did agree...", it refers to the "Comment by Hogbin and Nijdam and Response", but we need a more specific citation for where/when this appears.

I would also like to note that when the article says Morgan, et al. agree that, given random choice of goat, conditional and unconditional probabilities give "the same value", this is hardly news: their original paper makes much ado about the right answer for the wrong reason. They make a stronger statement in their rejoinder to Seymann about legitimate basis for the solution, not just the value. ~ Ningauble (talk) 13:46, 23 March 2013 (UTC)

Yes, sorry, I forgot about the vos Savant contribution. I´ve added it to the list.

PS The fact that Morgan et al´s contribution even generated a really angry response by vos Savant adds, in my opinion, to the noteworthyness of this second controversy.

PPS In Morgan et al´s response to Hogbin and Nijdam´s letter is written "We take this opportunity to address another issue related to our article, one that arose in vos Savant’s (1991) reply and in Bell’s (1992) letter,and has come up many times since. To wit, had we adopted conditions implicit in the problem, the answer is 2/3, period." Richard Gill (talk) 15:16, 23 March 2013 (UTC)

Couldn't you be so kind and simply add those sources and we'd be done with it? Btw. I agree with you regarding the noteworthiness (as outlined above). --Kmhkmh (talk) 15:05, 23 March 2013 (UTC)

Very busy (and I would prefer other editors also studied the sources carefully - one of the problems with MHP is that editors tend to think the problem can be solved by their own common sense and there´s no need to study the literature). But I at least added author (year) citations to make clear who is saying what when. Still to be done: links to proper bibliographic references. All are already in the reference list except for Rao, whose letter is published together with Bell´s. Richard Gill (talk) 15:16, 23 March 2013 (UTC)

Well I agree editors not (carefully) reading the sources has been longstanding problem of this article (in the German version as well). But that's the thing with most evangelists, they tend to stick to reading their own scripture rather than that of others. Be that as it may, thanks for adding the sources.--Kmhkmh (talk) 16:34, 23 March 2013 (UTC)

false conclusions due to incomplete mathematics

The conclusion in the article that switching is advantageous is due to the incomplete and innacurate mathematical formulations in the calculations leading to that conclusion.

Simply stated the "probability collapse" referred to in the erroneous conclusion that there is a 2/3 advantage to switching doors fails to recognize that the 1/3 value of the first opened door with a goat has to be distributed between the remaing two doors not only on the originally unchosen switch door. The MHP is actually two seperate and mutualy irrelevant sets of conditions, values and outcomes. The first set of probabilities concludes with the outcome where a zero value door is opened. The second set of conditions, values and outcomes must distribute the original 3/3 values with mathematical equality amongst the remaining two doors. There is no mathematical justification for distributing the 1/3 value of the first opened door upon the unchosen of the remaining doors. Both doors remaining doors are equal.

The choice between the final remaining 2 doors is the essence of the MHP. The information derived from outcomes of the original conditions, whether there were three doors, ten doors, or a million doors is has no probabalistic value to the second set of conditions where there are two doors and one car. Since information has no probabalistic value the only values relevant to the essence of the MHP (the choice between two doors) are the value of the prize and the number of doors. 98.164.120.241 (talk) 16:03, 21 March 2013 (UTC)

The Monty Hall problem is probably the most unintuitive simple probability puzzle in the world with most of the population initially believing that there is no advantage in switching. Read the section about the response to vos Savant's correct answer. The fact is, proven and agreed by every reliable source, that switching doubles your chances of winning the car.

Read the article and see if any of the arguments convince you. If not, try one of the simple simulations. You will find, to your surprise, that it really is true. When you have done that and convinced yourself that there is an advantage to switching, come back and help us improve the article in explaining the correct result more clearly. Martin Hogbin (talk) 16:47, 21 March 2013 (UTC)

You know, anon is right: we do have false conclusions due to incomplete mathematics. Who says the contestant doesn't want the goat? Goats are great. Can you milk a car? Can you eat a car?

As for having to distribute the 1⁄3 value of the door Monty opens between the two remaining, anon, you've not shown this to be true. These are not two separate and mutually irrelevant sets of conditions, values or outcomes. Monty has two restrictions placed on him: he has to open a door with a goat and he cannot open the door chosen by the contestant. So, imagine you're Monty; if the contestant initially chooses the door with the car, you can open either of the two remaining doors since they both have goats, but if the contestant initially chooses a door with a goat, you have to choose the other door with a goat; on average about one in three of your contestants will initially choose the door with the car and would loose if they swap but two in three will choose a door with a goat and would win if they swap.

What would happen if we lift either or both of these restrictions? We then have three possibilities.

Monty he has to open a door with a goat but he can open the door chosen by the contestant.
Perhaps the contestant writes his choice down but doesn't tell Monty. In this case there is a fifty-fifty chance of winning but should you switch? If Monty chooses a different door to yours, it doesn't matter whether you switch, but there's a one-in-three chance Monty chooses your door, in which case you have half a chance of winning if you switch and no chance of winning unless you switch.
Monty can open either door not chosen by the contestant.
The contestant chooses which door to keep closed and Monty flips a coin to decide which of the remaining doors to open. There's a 1⁄3 chance that the contestant chooses the right door at first, in which case the door Monty opens will definitely have a goat. There's a 2⁄3 chance that the contestant chooses one of the wrong doors at first, in which case the door Monty opens has a 50% chance of having the other goat. Do you switch? If the door Monty opens is the one with the car, of course you switch. If the door Monty opens has a goat, you've neither got anything to gain or loose by switching.
Monty can open any door.
You choose a door but Monty either doesn't care or doesn't know which and then he opens any of the three doors at random. What do you do? There'll be a one-in-three chance Monty opens the door with the car making the choice easy. If he opens the door with a goat, you have a fifty-fifty chance of winning. Should you switch? It depends on whether Monty opens your door. There'll be a one-in-three chance Monty opens the door you chose; if it's a goat, switch, and if it's the car stick with it. If he opens a door you didn't choose, switch if it's the car but if it's a goat your chances are fifty-fifty whether you switch or not.

Changing how Monty chooses which door to open changes the game entirely. In this problem Monty's choice is restricted. Of the two remaining doors one of them was not available for Monty to open since the contestant initially chose this door. Therefore the two remaining doors are not equal. One of the two remaining doors is the one the contestant chose to have kept closed, the other is the one Monty, who knows where the car is, left closed. Did Monty leave it closed on a whim because the contestant had already picked the winning door (with a one-to-three chance) and it didn't matter which of the other two he opened or was he forced to leave it closed because the contestant had picked the door with the other goat (with a two-to-three chance) and he wasn't about to open the door with the car? JIMp_talk·cont 06:04, 24 March 2013 (UTC)

I think every variation of the MHP has already be thought of, including the farmer who would rather win the goat. Come to think of it, I am not sure we have ever given the goats the choice of which door to open. Martin Hogbin (talk) 23:19, 24 March 2013 (UTC)

view from the outside

Administrative goings on elsewhere caught my eye. I'm a long-time fan of Monty Hall's show and old enough to have watched it in the original as well as a student many moons ago (Cooper Union) of statistics, probability, and game theory. The comparison offered from 2005 is straightforward and approachable. Today's version doesn't know if it's focused on the controversies or the mathematics or if it is merely a compendium of everything ever said pro or con (regarding the "2/3rds...") and of all mathematical models which apply in whole or in part.

As the article and the quandary are the "Monty Hall problem", I'm also puzzled as to how and why the article lead metastasized into its current form. There's a lot of stuff in there that should simply be in the body. Also, if we're going to box quote anyone, it should be Monty Hall or no one. Featuring a quote calling the "better odds if you switch" derivation crap is rather giving that position prominence.

Too many cooks stirring the pot?

Some retrospection on how the article got to its current state might assist in improving it. Just saying. VєсrumЬа ►TALK 23:34, 23 March 2013 (UTC)

Vecrumba: the Monty Hall problem was made famous by Marilyn vos Savant and her statement of the problem is repeated at the head of just about every journal article, book, or newspaper article on the problem. That is what we have in the lead. The box quote which you complain about is an illustration of the media controversy. By the way, do I understand that your remark that a derivation of "better odds if you switch" is crap, means that your personal opinion is that the player should not switch doors? Unfortunately, whether you like it or not, the consensus in the literature is that you do have better odds if you switch.

But anyway, if you don't like the lead, why not draft an alternative? Richard Gill (talk) 07:37, 24 March 2013 (UTC)

The following is a copy of something I posted at Wikipedia:Arbitration/Requests#Amendment request: Monty Hall problem. I think you will find the series of links in the middle to be of interest (especially the newer ones near the end):

In my considered opinion, we should rethink this issue and consider new solutions.

This is the longest-running content dispute on Wikipedia, and is featured at WP:HALLOFLAME.

After well over a million words, we have not reached a consensus on article content. To this day Talk:Monty Hall problem is full of spirited debates about what the content of the Monty Hall problem page should be. Another million words are unlikely to change that.

This has reduced the quality of the page, as evidenced by the fact that it is a former featured article. A comparison of the present page with the with the (featured 2005 version) is instructive.

Every avenue of dispute resolution has been tried, some repeatedly. Unlike many articles with unresolved content disputes, this does not appear to be the result of any behavioral problems. Instead, it is an unfortunate interaction between editors, each of whom is doing the right thing when viewed in isolation.

In my opinion, it is time to ignore all rules and start considering new ways to solve this, the longest-running content dispute on Wikipedia.

I propose applying a 6-month topic ban -- no editing of the MHP page or MHP talk page -- on every editor who was working on the page two years ago, one year ago, and is still working on the page today (this of course includes me). I predict that within a few months the remaining editors (and perhaps those who have gone away discouraged) will create an article that is far superior to the one we have now, and they will do it without any major conflicts. Giving the boot to a handful of editors who, collectively, have completely failed to figure out what should be in the article will have a positive effect. Of course it should be made clear that this does not imply any wrongdoing on anyone's part, but rather is an attempt to solve the problem with a reboot.

Two years is enough. It is time to step aside and let someone else try. --Guy Macon (talk) 01:39, 24 March 2013 (UTC)

I like this idea, both for this article, and in general for long-running-dispute (or talk-heavy) articles that are effectively owned by a small group of editors. Old revisions aren't deleted; they will still be here in 6 months too :-) – SJ + 07:26, 25 March 2013 (UTC)

The is no long running dispute here, just the normal WP cooperative editing which you are most welcome to join. Martin Hogbin (talk) 10:09, 25 March 2013 (UTC)

I think there is no content dispute going on here, there is no problem to be fixed. Guy: you did resolve the content dispute! Congratulations! People who think the article could be a whole lot better can come in and make proposals, they can draft alternatives and we can discuss them. They can be daring and rewrite whole parts of the article! They can cut out stuff which they think is unnotable and too specialistic. They might however also need to spend some time studying the sources since what makes things tricky with MHP is that everyone has an instinctive feeling that they know the answer and the sources are irrelevant. Yes: the old timers should hold back.

It is indeed interesting to look at the history of the article. Why the article was once "featured", short, coherent. Because it took sides in a dispute which is out there in the literature as to what is a right solution and what is a wrong solution. It is much easier to write an article if you are allowed to pick a particular POV than when you are not allowed to do that.

Guy: why don't you become an involved editor of the page rather than a conflict resolver? Richard Gill (talk) 07:43, 24 March 2013 (UTC)

If you are correct about there not being any disputes to resolve (and I am certainly not arrogant enough to assume that I can't possibly be wrong), that would make a lot of sense. If, however, I am correct, the last thing this page needs is one more voice. --Guy Macon (talk) 08:10, 24 March 2013 (UTC)

Guy if you think there are any real content disputes here please tell us what they are. As Richard says, you played a major part in resolving the long-standing dispute, which is now history.

We do have a couple of constructive discussions in progress. One is on adding a Bayes' rule explanation somewhere. No one has argued against this in principle but we are all looking for a way to produce a mathematically sound and well sourced explanation that is accessible to as wide a range of readers as possible. This is standard WP cooperative editing and anyone is welcome to join in.

There have also been suggestions that we have one or two of the simplest and most convincing solutions before the section on media response. I have personally shied away from pushing this because it might inflame old passions if done in the wrong way. Again, by working together I actually think we can add at least one simple explanation to the start of the article without causing any unnecessary friction.

Your suggestion that all the regular editors walk away from this article is not one of your better ones, it is totally against the ethos of WP and has received no support from anyone else. Far better would be for you to join in the ongoing and civil discussions on how to improve the article. Martin Hogbin (talk) 10:27, 24 March 2013 (UTC)

@Gil, I think you misunderstood me re: "By the way, do I understand that your remark that a derivation of "better odds if you switch" is crap, means that your personal opinion is that the player should not switch doors?", which is that if the article was to feature any spot-quote, a quote from Monty Hall on why the reveal/switch offer works for the show, not a spot quote highlighting the opinion on one side of the equation (no pun). Well, I've rewritten content elsewhere for some contentious topics,... might try the lead.

This does remind me of another threesome, on a three lane highway in traffic the other lane really is moving faster most of the time, statistically speaking. VєсrumЬа ►TALK 21:07, 24 March 2013 (UTC)

Monty has made a comment about this problem and it said two things. Firstly he agreed with vos Savant (and everybody else) that it is better to switch in the circumstances described in the problem and secondly he pointed out that, in the real show, he only offered money and never the opportunity to swap. Martin Hogbin (talk) 21:48, 24 March 2013 (UTC)

Vecrumba, I'm glad I had misunderstood you! Richard Gill (talk) 07:56, 25 March 2013 (UTC)

Question: is the above a disagreement concerning what the content of the Monty Hall problem page should be or is it something else? This talk page gets a lot of words posted to it each week. Surely those words must be about something. I'm just saying. --Guy Macon (talk) 22:10, 24 March 2013 (UTC)

You will have to ask VєсrumЬа that, I am not sure exactly what his/her point is. Martin Hogbin (talk) 23:22, 24 March 2013 (UTC)

Well, for one, I think the article could strike a better balance between being scholarly and being more approachable for the general reader: "... is a veridical paradox because the result appears impossible but is demonstrably true", for example, makes my eyes water because it reads like someone showing off vocabulary followed with a tautological definition for the ignorant masses. For me, the lead beats "counter-intuitive" into dead horse-dom. (I overstate, but on WP that's empirically the only way to make a point.) All that said, I don't mean to merely snipe. Text, sans templates except for quote, of a simpler lead follows. I'm not necessarily advocating for it as a substitute, only to hopefully make my point regarding approachability. (outdenting...)

"The Monty Hall problem or paradox is a probability puzzle named ~~for~~ after the host of the American television game show Let's Make a Deal. In the classic case, Monty shows a contestant three doors and asks them to select which one hides the prize. Once they choose, Monty opens one of the two remaining doors to show the prize is not there. (Alternately, a competing contestant also chose a door and Monty opens their door first.) The show's moment of drama comes when Monty offers the contestant the opportunity to change their choice to the other remaining door.

"The intuitively obvious answer is that the prize is behind one of two doors, a 50:50 chance; there should be no advantage to picking one door over the other, that is, switching. Probabilistically, however, mathematics, computer simulations, and less traditional methods such as quantum strategical analysis, have demonstrated that the contestant has a 2/3rds chance to win if they switch, only 1/3rds if they stay with their initial choice.

"Steve Selvin first posed the problem within the mathematics community in a letter to the American Statistician in 1975 (Harv, Selvin, 1975a) (Harv, Selvin, 1975b). Marilyn vos Savant published one of the most widely known statements, and solution, of the problem in her "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990a):

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. 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?

"Her answer was that the contestant should pick the other door because switching gave them a 2/3rds chance of winning the car.

"Vos Savant's column ignited a firestorm of controversy, prompting 10,000 letters to the editor, a majority claiming she was wrong (Harv, Tierney, 1991) including PhD's disputing her solution. In actuality, the Monty Hall problem is a restatement of identical (Three Prisoners problem) or related earlier (Bertrand's box paradox) probabilistic puzzles each with similarly counter-intuitive solutions. However, this is the best known variant of the genre because its origin is rooted in an icon of popular culture. The original problem and variations, including the effect of differing assumptions, continue to be examined in the mathematical literature. These are ~~examined~~ discussed in further detail below."

My two cents. VєсrumЬа ►TALK 19:02, 25 March 2013 (UTC)

'named for'??? Martin Hogbin (talk) 19:33, 25 March 2013 (UTC)

"Named after" is probably better, changed. I was going to ask if there was some different Monty Hall, but that's how small offenses grow into big ones. IMHO there's been some of that here. Just saying. VєсrumЬа ►TALK 23:20, 25 March 2013 (UTC)

a minor flaw

Currently the article writes : "In Tierney (1991), the mathemagician and Stanford professor Persi Diaconis stands up for vos Savant, see Diaconis (1988) for his work on symmetry in statistics."

Sadly enough Tierney (1991) doesn't even mention Diaconis nevermind describing him as standing up for vos Savant.--Kmhkmh (talk) 17:52, 25 March 2013 (UTC)

Have you read all of Tierney's article in New York Times? It's three pages long. It does mention Diaconis and Diaconis does support vos Savant (incidentally I also corresponded with Diaconis about this ...).

It's true I had an ulterior motice in having contributed those lines. My personal point of view is that if we are solving MHP with objective Bayesian probabilites (all probability is derived from ignorance, and symmetry of ignorance implies equal probabilities) then the irrelevance of the specific door numbers to the question whether you should switch or stay follows from the symmetry. Having discarded door numbers we have a simpler problem and it is solved by one of the simple solutions. For instance, the combined doors solution! Now what I just said only will only have any meaning to a mathematician but my point is that that mathematical approach, I believe, is the mathematical formulation of many ordinary person's intuitive approach to MHP. If we are solving MHP intuitively, then that is the kind of probability we use and the symmetry is immediate and intuitive.

I refer to William Bell's (1992) discussion of Morgan et al. (1991): the difference between the simple and the conditional solutions is a simple observation which is so obvious that it is hardly worth mentioning; the Morgan et al affair was a storm in a tea-cup.

I refer also to Kraus and Wang's psychological analysis of how ordinary people come to understand that switching is the right thing to do. One has to discard the initial visual but static image of three doors, one open with a goat, and replace it with something else. One of the possible replacements follows the "less is more" principle. Drop the door numbers. Your initial door versus the other two. Remember Erdos? Even after seeing a simulation, and seeing a formal calculation with Bayes' theorem (unfortunately, we wasn't shown Bayes' rule), he still didn't "see" why the answer was 2/3. To "see" the right answer you have to "re-see" the problem. Richard Gill (talk) 19:00, 25 March 2013 (UTC)

My bad I did indeed overlook pages 2 and 3. However Diaconis still doesn't do what you claim in that NYT article. The 2 parts mentioning him are:

Actually, many of Dr. Sachs's professional colleagues are sympathetic. Persi Diaconis, a former professional magician who is now a Harvard University professor specializing in probability and statistics, said there was no disgrace in getting this one wrong. "I can't remember what my first reaction to it was," he said, "because I've known about it for so many years. I'm one of the many people who have written papers about it. But I do know that my first reaction has been wrong time after time on similar problems. Our brains are just not wired to do probability problems very well, so I'm not surprised there were mistakes."

Still, because of the ambiguity in the wording, it is impossible to solve the problem as stated through mathematical reasoning. "The strict argument," Dr. Diaconis said, "would be that the question cannot be answered without knowing the motivation of the host."

I can't see how that translates to "standing up for vos Savant [in the debate with Morgan et al]".

As far as simplicity is concerned that's in the eye of the beholder and I have no interest in arguing anything in this neverending debate. However as far as the article is concerned, that line & citations should be fixed by either modifying the line or using a different citation to support it.--Kmhkmh (talk) 19:42, 25 March 2013 (UTC)

Just go ahead and change the text to what you think is appropriate. Diaconis is (and was then already) a famous probabilist. He didn't tell Tierney that vos Savant's method of solution was wrong and she only coincidentally got the right answer, though knowing him, he would have jumped at the opportunity if he thought it reasonable. Not only did she get the right answer, he has absolutely no criticism on how she derives it. And I happen to know that he is very aware of the role of symmetry in this. But I don't have a wikipedia reliable source to give you to support my claim. Notice that Diaconis' only note of criticism bears on the other debate, namely that her problem statement is open to misunderstanding, though her own solution makes perfectly clear how she intends the reader to understand the problem - the way in which, as she emphasized, most of her critics understood it too. Richard Gill (talk) 07:53, 26 March 2013 (UTC)

A reminder that the dramatic device as employed on the game show did not rely on door number for effect. And I would suggest that the naming of the problem answers for the motivation of the host, that is, to generate drama. As for "standing up" perhaps slightly reworded reflecting first reaction wrong? Personally, I don't see that there is any ambiguity in the wording or the situation. I do see a lot of effort expended on statistical speculation regarding the effect of motivations and initial conditions. Just saying, perhaps a re-balancing of forest and trees? Getting back to other stuff, no, really. VєсrumЬа ►TALK 19:51, 25 March 2013 (UTC)

The "there is no content dispute" hypothesis

Let us consider the hypothesis that "There is no long running dispute here, just the normal WP cooperative editing" and "there is no content dispute going on here".

An article that has been around for years and which has no content disputes settles into a fairly stable form, with occasional additions of citations or minor rewording. The article talk page has very little activity because there are no content decisions to be made.

Let us consider the amount of activity in a 60 day period for the following pages, chosen because they have "problem" in the title:

Birthday problem:
13 revisions by 10 users
Talk:Birthday problem:
0 revisions by 0 users

Three Prisoners problem:
4 revisions by 3 users
Talk:Three Prisoners problem:
1 revisions by 1 user

Two envelopes problem:
12 revisions by 5 users
Talk:Two envelopes problem:
9 revisions by 6 users

Sleeping Beauty problem:
2 revision by 2 users
Talk:Sleeping Beauty problem:
0 revisions by 0 users

Monty Hall problem:
266 revisions by 29 users
Talk:Monty Hall problem:
498 revisions by 22 users

The Two child problem could also be added to this list! Richard Gill (talk) 13:05, 27 March 2013 (UTC)

Is this the activity level of an article with stable content, or is this the activity level of an article which has an ongoing discussion about what the content of the article should be?

Now let us consider selected quotes from this talk page (I have purposely deleted the names and the actual arguments so that this does not rekindle an old argument. The point here is whether there is evidence of a content dispute):

"[part of article] is highly misleading"

"It is not misleading but there is an argument that it requires additional logical steps to be correct."

"That's your opinion and the opinion of some sources. And of some editors. But not all sources, not all editors. There's no point in discussing this again. The discussion has gone on for five years or so and as far as I know, nobody involved ever changed their minds."

"You are speculating that [source] thinks like you. A way of thinking which is not even discussed in published literature. Pure speculation and highly biased. I am arguing from knowledge of the context in which the article was written, and knowledge of the preceding literature."

"I am not suggesting that we add my interpretation of this solution to the article. Just that we leave our own personal interpretations out of it."

"Do you have a copy of the article that I can read. If it clearly supports your view that [description of content of article] then we can, of course, say that in the article."

"I have been bold and made changes to [article] to explain exactly what I mean."

"I've been bold also, and reverted your changes."

"Some people (somewhat perversely) consider there to be [argument]. To start half way through the problem with a hand-waving argument about starting where convenient is not acceptable if we want to be thorough."

"This is not perverse, it is eminently sensible. If you want to decide rationally on a course of action you need to know [argument]"

"The perverseness is in assuming that [argument]"

"It's also not perverse to [argument]. [argument] is an eminently sensible way to solve the problem in a principled way. That's why all the probability texts do it that way."

"There was no handwaving in my description of the solution. I told you explicitly that [argument]. Absolutely sensible, absolutely normal."

"Not exactly clear or simple. Why not start with [argument], just as you do with [argument]?"

"Clear and simple if you've learnt some formal probability theory. Why not? Because it would be a complete waste of time."

Are these the comments one would expect for an article with stable content, or for an article which has an ongoing discussion about what the content of the article should be? Comparing them with the talk pages of the other "problem" articles listed above is quite instructive.

I think that it is pretty clear that there is indeed a long-running dispute about the content of this article. --Guy Macon (talk) 17:07, 25 March 2013 (UTC)

It is interesting to note that the discussion now apparently moves from disputes over content to disputes over disputes (over content). It seems editors here can't exist without disputing something.--Kmhkmh (talk) 17:40, 25 March 2013 (UTC)

"Perhaps when a man has special knowledge and special powers like my own, it rather encourages him to seek a complex explanation when a simpler one is at hand." --Sherlock Holmes

--Guy Macon (talk) 18:22, 25 March 2013 (UTC)

That is an interesting (cherry picked!) collection of quotes. Guy's theory is that there is an ongoing content dispute. There certainly *was* a content dispute, I thought that it was resolved through Guy's patient and hard work, which led to a working "consensus" as to how the article should be structured. Since the new consensus structure differed from the structure the article had before, a lot of work was done by a lot of editors, old and new, to restructure the article according to the new consensus. Naturally this led to a lot of discussion. I don't think it is quite finished yet. My first conclusion is that Guy should wait for at least two years for the dust to settle, before concluding that there is some kind of terrible dispute still going on which needs some draconian intervention to fix. (By the way I think there is much less activity on the talk page than in the hey-days of the dispute, and much more serious editing of the article: so I disputes Guy's interpretation of his statistics)

Now another thing is, do we expect this situation ever to change? Guy proposes that all editors who were fairly active over the last two years should stop editing the article for the next six months. There certainly is some merit to this as the old hands do from time to time still repeat to one another what they have been saying for years. On the other hand, let's suppose all the old hands not only quit editing but also quit talking on the talk page, and also quit talking on the Arguments page. Many of the old hands had, at least, taken the trouble to read a lot of the literature on MHP. You know, the reliable sources on which wikipedia articles are supposed to be based? Preferably ternary sources?

My prediction is that in no time there would be new quarrels going on here among new editors. Possibly the article will get drastically rewritten by adopting just one of the numerous Points of View which exist, out there, on how MHP ought to be solved. If you want a powerful concise clear article on MHP it should be written by one person with a clear and consistent point of view. The old encyclopedias had a single authoritative person write each article. Many on-line encyclopedia's today are still organized in this way. An authoritative editorial board invites authoritative scientists and scholars to each write an article about their own pet subjects.

My prediction is that if Guy's proposal is adopted, we will see no superficial change in the volume and nature of exchanges on this talk page, but the Wikipedia-Quality of the article will decline markedly. (Sure, some readers will love the result; but others will hate it. Probably better that everyone hates the article a little. I'm reminded of Democracy - a very bad political system but unfortunately there isn't a better one). Richard Gill (talk) 18:44, 25 March 2013 (UTC)

Any subject which requires a specific level of skill or knowledge will only attract a small community of editors. My experience with "WP:HIATUS" is that it at best delays the inevitable where controversy is concerned; worse, fresh crops of editors in contentious areas tend to introduce more opinionated and/or less informed content. (For example, I haven't written a probability equation in > 35 years.) My experience has been that all too many editors equate ignorance with neutrality, the ultimate "WP:HUBRIS" rendering any opinion, no matter how belief-system based and fact free, as encyclopedic. VєсrumЬа ►TALK 19:42, 25 March 2013 (UTC)

Richard Gill makes some very good points, which anyone who is considering implementing my suggestion (which would pretty much have to be arbcom -- nobody else has the authority) should carefully consider. --Guy Macon (talk) 20:22, 25 March 2013 (UTC)

I haven't looked for folk going to Arbcom for enforcement requests in the past. My experience is that such actions only add to recriminations and don't solve anything, although with luck truly obnoxious editors will eventually get topic banned. (Personally, if someone's not going to be nice, I'd rather not deal with them--that doesn't mean we can't disagree vociferously, but off topic.) Arbcom won't rule on content. On your suggestion, I expect they will suggest that if someone wants to recuse themselves from the article, they can certainly do so.

If differing positions are possible as genuine differences of interpretation based on the same agreed-upon baseline, then it's quite possible some won't be persuaded no matter what, but then that same lack of persuasion should be evident in secondary and tertiary source and be reflected in the article. VєсrumЬа ►TALK 23:35, 25 March 2013 (UTC)

Discretionary sanctions

Notice: There is an application to remove the authorization for discretionary sanctions from this article at Wikipedia:Arbitration/Requests#Amendment request: Monty Hall problem. ~ Ningauble (talk) 15:36, 13 March 2013 (UTC)

Ningauble, thanks for pointing this out. Would you care to clarify your present criticisms on the article, posted there: "Distortions in the current article, such as the inadequately sourced and apparently incorrect narrative under "A second controversy", and the (mis-) interpretation of the context sources refer to under "Criticism of the simple solutions" "? I agree that sources should be added to "A second controversy". That's not difficult to do. I disagree with your qualifications "incrrect" and "misinterpretation". But I'm trying to keep away from this page, so it would be good if new editors would discuss these issues with you here. Richard Gill (talk) 10:10, 17 March 2013 (UTC)

Ninguable, I too am puzzled by your comments. Where is the 'environment of antagonistic browbeating' who are, 'the most pugnacious or masochistic contributors'?

I see no sign of newcomers being driven away, In fact we have had some very useful contributions from them. Martin Hogbin (talk) 12:09, 17 March 2013 (UTC)

@Richard: I elaborated a little on "A second controversy" in a separate section below. I am generally dubious of things that are not cited, but confess that presuming they are incorrect is a matter of pushing the burden of proof rather than asserting they are wrong. The present section is apparently questionable, or at least ambiguous, in using parenthetical remarks cited as "Later (2011). It would be good to verify and clarify whether the putative source explicitly contrasts "given the situation the player is in" with "averaged over all possible situations" in the manner indicated, or whether the parentheses indicate these are editorial interpretations/clarifications of the source.

@Martin: I am not going to name names and cite cases because I was not requesting arbitration enforcement, I was only recommending that discretion remain available. I apologize for using some loaded language in commenting on the amendment request. Please feel free to interpret my remark about pugnacity and masochism as a reflection of my own lowly timorousness. If I have repeatedly walked away from discussions here for extended periods when I could not stand it anymore, it was unfair of me to assume anyone else is as pathetically weak-willed as I.

I am not going to delve into the "Criticism" section at this time because I would rather go after some low-hanging fruit before restarting something that has already been filibustered and stonewalled to death, or, if you prefer, because I can't walk and chew gum at the same time. ~ Ningauble (talk) 17:49, 22 March 2013 (UTC)

The remarks "given the situation the player is in" and "averaged over all possible situations" are editorial clarifications - uncontroversial unchallenged interpretations, attempts to convey the meaning of the technical terms unconditional and conditional to the general reader. Richard Gill (talk) 20:06, 23 March 2013 (UTC)

I think the attempt at editorial clarification is misplaced (if the article needs to define what conditional probability means, it belongs in the section on using conditional probability), unclear (averaged over what possibilities?), and confusing (averaging is really not a defining principle here, the defining principle is dependence on the observed evidence, or in this case, an aspect thereof). I recommend removing the parenthetical comments from this paragraph.

If my recommendation is not followed then, at the very least, the text should make it very clear that this is not what the cited source says, or a paraphrase of what the source says, but is an editorial comment about something to which the source refers. ~ Ningauble (talk) 14:28, 29 March 2013 (UTC)

Leonard Mlodinow

What is the functiion of the section on Leonard Mlodinow in the introduction??? Nijdam (talk) 21:44, 9 April 2013 (UTC)

Personally, one long quote already there (not Mlodinow) should be good enough, no? Extensive quoting is usually a sign that editors have arrived at an impasse in creating content and have fallen back to the he said/she said model. At least that's been my experience with multiple viewpoints (so that is different approaches to same agreed to situation or problem). VєсrumЬа ►TALK 02:04, 10 April 2013 (UTC)

I agree. I think most editors here find it excessive. There was a discussion of how we could concisely add the fact that it matters that the host knows where the car is and therefore can always reveal a goat. This is an essential part of the problem, it was discussed right at the start by vos Savant and it continues to confound people.

The general discussion stalled and Gerhard was prompted to write something as a starting point. I think what we have now is too long and should be made very clear and concise, without quoting any specific source.

Could somebody suggest something. Martin Hogbin (talk) 09:22, 10 April 2013 (UTC)

Discussion: The simplest explanation to begin it with:

I think the simplest explanation is the following and it should start the article since it has the highest ..probability to achieve success in understanding the solution: "Your first choice will be bad two thirds of the time by definition, hence since the host those two thirds of the time will reveal the only bad choice left, switching when you are in those bad choices two thirds of the time, you will win. i.e. Only one third of the time that you were initially winning, switching loses. Note: The central theme is 'switching' and not initial choosing. Note: The reason human intuition is easily fooled is that rarely in nature an invisible hand of a 'host' helps us in such an obscure way." --fs 23:21, 25 March 2013 (UTC)

I think Monty Hall himself had a simple explanation. I've probably said enough for one day, don't want to overstay my topic newbie welcome. VєсrumЬа ►TALK 23:46, 25 March 2013 (UTC)

Vecrumba, I am not sure where you get the impression that you might outstay your welcome from. It is a principle of WP that new editors are always welcome. You may be challenged on your arguments, logic, or sourcing but that does not mean that you are not welcome. Martin Hogbin (talk) 09:42, 26 March 2013 (UTC)

You can read Monty Hall's view of the MHP problem in the Tierney New York Times article of 1991, and in a letter to Steve Selvin of May 12, 1975. In the letter he writes "Oh incidentally, after one has seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him." This seems to me to be a not clear way to give a simple explanation. The simple explanation being: focus on the door you chose in the first place. The chance that there's a car behind it was 1 out of 3. This has not changed. The reason it hasn't changed has two parts (a) because the host was always going to open another door and reveal a goat, whether or not the car is behind the first door, and (b) the particular door he opened, 2 or 3, doesn't make any difference. Most ordinary people don't bother to add (b) to the explanation and see no point in adding it, but the specialist can remark that this is the point where we make use of our lack of knowledge not only of where the car was hidden - that was used in part (a) - but also of how the host chooses to open a door. Because we know nothing about that, the particular choice he made is completely noninformative to us. He is 50-50 likely to open door 2 or door 3, whether or not the car is behind door 1. The symmetry of our total ignorance of how the car is hidden and of how the door is opened means that the particular door numbers are irrelevant. Most people "know" that intuitively, don't put it into words, and wouldn't even know the words to use if they did. Several editors have said here that "symmetry" is a much too difficult word for ordinary folk; but the fact remains that symmetry is a key tool in problem-solving (it enables one to make a difficult problem more simple) and that people use it intuitively all the time.

Notice that I'm also using Bayes' rule in a completely intuitive way here. It seems to me that the best solution to MHP is one which is at the same time verbal and intuitive, and mathematically rigorous. Use Bayes' rule and symmetry and you have a mathematically rigorous version of the combined doors argument, which is intuitively appealing to anyone ... no maths training required.

I agree that a short solution should be part of the lead. Other editors disagreed. Richard Gill (talk) 08:17, 26 March 2013 (UTC)

The simplest solution: if you chose a goat first and switch, you will win the car, if you chose the car, you will win a goat. You are twice as likely to pick a goat. Robo37 (talk) 08:48, 26 March 2013 (UTC)

I agree that a simple solution( or two )should be given at the start of the article, with a simple solution in the lead. The question is, which are the simplest, and best, solutions?

I agree with Robo and Vecrumba that some wording of the 'opposite of your original choice if you swap' solution is probably the best. I do not think that there is much opposition to having a solution/explanation in the lead so I propose that we discuss how we should add some simple solutions. I am going to propose in a section below that we discuss how to word the 'opposite of your original choice if you swap' solution for inclusion in the lead. Martin Hogbin (talk) 09:36, 26 March 2013 (UTC)

Proposed simple solutions for the lead

Martin's suggestion

Players initially have a 2/3 chance picking a goat and those who swap always get the opposite of their original choice; the car.

I say 'players' rather than 'a player' to silently cover the two issues mentioned by Richard above. I think it is also important to say ' always get the opposite' to exclude the 'ignorant Monty' case. Martin Hogbin (talk) 09:36, 26 March 2013 (UTC)

Isn't the end of the sentence a bit too short? How about instead of "the car" write "so players who always swap have a 2/3 chance of getting the car". Richard Gill (talk) 13:02, 27 March 2013 (UTC)

Yes, the English is a bit mangled now I read it again. I was trying to make it as simple and short as possible. I see you slipped in another 'always', I can see why but I do not think it is really necessary if we say 'players', If we leave it, in we get:

Players initially have a 2/3 chance picking a goat and those who swap always get the opposite of their original choice so players who always swap have a 2/3 chance of getting the car.

I am still not sure how convincing and understandable this is. In the end nothing seems to convince many people except doing a simulation themselves. Martin Hogbin (talk) 13:43, 27 March 2013 (UTC)

This seems to me very understandable, and I know it convinces many people. Richard Gill (talk) 18:05, 27 March 2013 (UTC)

How about breaking it up into three sentences?

Players initially have a 2/3 chance picking a goat. Players who swap always get the opposite of their original choice. So by swapping players have a 2/3 chance of getting the car.

But to me it seems there are two things missing: an explanation of why players who swap always get the opposite of their original choice (don't assume anything to be obvious) and a spoiler warning for those who want to figure it out themselves (maybe everything's obvious).

Vos Savant suggested a strategy to maximise the contestant's chances of winning the car. His response was that the contestant should switch to the other door. After one of the goats has been revealed only the car and the other goat remain. So contestants who switch always get the opposite of their original choice. Initially contestants have a 2/3 chance picking a goat. By applying the strategy of always switching this becomes a 2/3 chance of getting the car.

Perhaps something like this we could put in place of the third ("Vos Savant's response ...") paragraph. The first sentence introduces the topic of the paragraph (the winning strategy) and thus alerts the reader to the fact that the answer is coming. The second gives the winning strategy. The third sentence explains why by switching you get the opposite. The rest is just the above argument reworked.

Perhaps it's getting a little lengthy. JIMp_talk·cont 00:02, 28 March 2013 (UTC)

Caution: No ambiguity, it must be very clear what you are talking about. Players get the opposite only if it is given that no hidden unspoken conditions may arise. Leonard Mlodinow (The Drunkard's Walk, page 55) says that it is on the host’s role: "The host is fixing the game". So you should say

The host, after the guest first did select his door, in any case will open one of his two unselected doors in order to show a goat but never the car. Result: Players initially have a 2/3 chance picking a goat and those who swap always get the opposite of their original choice. So players who always swap have a 2/3 chance of getting the car. Gerhardvalentin (talk) 00:29, 28 March 2013 (UTC)

I like that, I think it is better in two or three parts. It discreetly covers all the angles (some editor believe that we should cover cases where the host fixes the game) without over-complicating things.

Leonard Mlodinov (author of a popular book about randomness) is referring to standard Monty Hall problem when he says "the host is fixing the game". I think he means by this (though he doesn't say) that the host's action *is* informative of where the car is. After all, opening door 3 is twice as likely when the car is behind door 2 as when it is behind door 1 (the player's initial choice). However, I don't think his analysis is particularly notable. It's very shallow compared e.g. to that of Rosenthal Richard Gill (talk) 08:01, 29 March 2013 (UTC)

Yes, I see what you mean. You are saying that Mlodinov said that the host was 'fixing the game' by always revealing a goat and thus helping the (smart) player. I will have a look at the video. Martin Hogbin (talk) 23:49, 29 March 2013 (UTC)

Nijdam and Rick, would you be happy with this wording in the lead? Martin Hogbin (talk) 09:18, 28 March 2013 (UTC)

I'd like to remind folk that in the two-contestant scenario, the appropriate situation presents itself a majority of the time and the host doesn't have to "fix" anything. Once contestant A has picked a door and it is revealed to be a goat, contestant B is given the option to change their choice.

Obviously the host can know what's where and orchestrate the situation for dramatic advantage, but that's merely dramatic variation.

In the one-contestant situation, I would venture with all that adrenaline pumping, there's no candidate that did the logic in their head at the moment that it was best to switch. It's a bit speculative that the host, by revealing the goat, is intentionally and successfully assisting the informed contestant. VєсrumЬа ►TALK 23:14, 30 March 2013 (UTC)

What is the two-contestant scenario? Except where stated otherwise, the article concerns itself with the standard version of the problem where we have one contestant and a host who always open a door to reveal a goat (he can do this because he knows where the car and goats are) and always offers the swap.

The game was never actually played as described in the problem so we all have plenty of time (many might say too much) to decide on the best action to take. Martin Hogbin (talk) 00:32, 31 March 2013 (UTC)

Ningauble's suggestion

I would recommend against including a solution or explanation in the lead. Giving the numerical answer and noting that it met with incredulity is enough.

I say this not to avoid a "spoiler", though I am not aware of any other articles on mathematical problems that attempt to explain the answer in the lead, but because there are too many different demonstrations in the body of the article, and too much contention about thir relative merits, for any one explanation to be an adequate or neutral summary introduction to the article. ~ Ningauble (talk) 19:01, 28 March 2013 (UTC)

Given the size of the article, problem and variations, I would keep the lead as simple as possible. Remember, it is not necessary for the host to even know where the "car" is (although knowing assists in dramatic effect). In a two-contestant situation, as soon as you open the first contestant's door and it's revealed the "car" is not there (2 out of 3 times), the other contestant can be offered the opportunity to switch.

Explanations should be left to the body of the article. One can intellectually understand that the 1:3 chance of having picked the car never changes if you stay with your initial choice, meaning the "left over" door is now the 2:3 favorite. The counter-intuitive part is that the contestant's reaction is "Oooooh, now I have a 1:2 chance instead of 1:3", so what we have to explain is why revealing the third door isn't the one doesn't reset the odds. IMHO, it's going to take more than the lead to make the point properly (and cover the effect of initial assumptions, those who still choose to disbelieve, and so on). 21:36, 28 March 2013 (UTC)— Preceding unsigned comment added by Vecrumba (talk • contribs)

Please distinguish chances from odds. 1 out of 3, 1/3, is a chance. 1 against 2, 1:2 is an odds, ie 1 chance for, 2 chances against.

Of course it takes more than the lead to make the point properly. That's why there's an article as well as a lead. The lead is a brief overview/summary. Seems to me that since all reliable sources are agreed that the contestant should switch and that the reason has something to do with a chance of 2/3, this belongs in the lead. If there's a one sentence reason for 2/3, give it too. I see a thousand good reasons to come straight out with a (the?) punchline, and no reasons to delay. The literature is not in doubt that the player should switch. The whole point is that this initially appears counter-intuitive to most, but on deeper thought is indisputable. We are not writing the screenplay of a movie, but an encyclopedia article. And our task is not to amuse or titillate but to inform. Richard Gill (talk) 07:54, 29 March 2013 (UTC)

Ninguable, for much of the life of this article, including the time it was a FA, there has been a simple explanation in the lead. It is the most frequently suggested change by visitors. Martin Hogbin (talk) 11:09, 29 March 2013 (UTC)

Martin, I don't believe that is a good description of the article history. In the article milestones at the top of this discussion page, you can review links to the version that was promoted to FA in 2005 and the versions that passed FA review in 2007 and 2008. None of them include explanations in the lead section. The version that failed FA review in 2011 included an explanation in the lead. If there is any correlation between explanations in the lead and featured article status, it is a negative correlation. ~ Ningauble (talk) 13:04, 29 March 2013 (UTC)

Several readers of the article have suggested having a simple explanation early on in the article. Putting it in the leads seems to me to be the best way of achieving this but we could perhaps put this explanation early on the body but I am not sure how it would fit in with what we have. Martin Hogbin (talk) 13:14, 29 March 2013 (UTC)

I completely agree that some explanation should be provided earlier than in the present article.

As indicated in my suggestion #2 at "Second controversy" above, I think it is absurd to have an entire section about technical arguments over methods of solution before offering the reader any explanation of the answer at all.

I also think that the section on "The problem" is too long-winded. A brief clarification of ambiguities is needed for a solution to make sense, which is good reason not to put a solution in the lead, but this is a lot to wade through. The Whitaker quotation doesn't add anything to vos Savant's version in the lead, and the clarifications are buried in the third paragraph.

The article really needs to say "here is the problem" and "here is an explanation" up front, before reader's eyes glaze over, but it is too much for the lead, which needs to give an overview of the whole article. ~ Ningauble (talk) 15:12, 29 March 2013 (UTC)

How about a section right at the start called 'Introduction' with a short problem description and a simple solution? Martin Hogbin (talk) 16:38, 29 March 2013 (UTC)

Good idea! Richard Gill (talk) 18:27, 29 March 2013 (UTC)

I have now put an introduction at the start. Maybe it could be expanded a little. Martin Hogbin (talk) 23:50, 29 March 2013 (UTC)

A clear and simple explanaion in the lead

This discussion is highlighting the cause why the article has been misty and opaque for years. Marilyn vos Savant tried to present the scenario of an obvious paradox, some say a paradox that has already been known centuries ago. They say that back then one red and two black cards allegedly were put on a table, face down. But be as it may, I'm sorry to see above a complete misinterpretation of Mlodinow's words. Nobel prize winning psychologist Daniel Kahneman & colleagues (1982) & Leonard Mlodinow (“The Drunkard’s Walk” 2008) very clearly disclose the cause for the unintuitive famous paradox. It's what I said for years, eg in the RFC. Mlodinow says it very clear and impressive, and anyone can see Mlodinow on youtube and hear how he clearly shows up the real key. It's not about maths, it's about apperception. Erdös probably would have been thrilled. Not even ignoring it might not be enough. Gerhardvalentin (talk) 23:09, 29 March 2013 (UTC)

So far I only know Mlodinow's book and I didn't find that specially good on MHP. But it's certainly not bad. But if you disagree then clearly it is useful for some people. I did not know he also talks about MHP on Youtube. Please give us a link!

I agree that MHP - as a popular brain teaser - is more about perception than about mathematics. But logical deduction is also important. Kraus and Wang is all about this interaction between perception and deduction.

I like the intro! Richard Gill (talk) 09:01, 30 March 2013 (UTC)

Thanks Richard. I think we should add a little more but it should be very clear and concise and I am not sure what is needed.

Gerhard I have looked at a couple of Mlodinow's videos and I cannot find where he says that the host is fixing the game, perhaps you could give us a link to this version. My impression from his videos that I have watched though is that Richard is correct and that by these words he would be referring to the host using his knowledge to help the (smart) player by always revealing a goat, rather than the Morgan scenario in which he signals to the player by his choice of door. Of course, I agree with you that this last case is a variation on the obviously intended problem. Martin Hogbin (talk) 10:24, 30 March 2013 (UTC)

He says this in his book (I put the relevant section of the book in the dropbox folder). Richard Gill (talk) 12:37, 30 March 2013 (UTC)

Links: link to The Drunkard's Walk (complete version, cover page contains links to all chapters 1 to 10 (relevant is chapter 3 page 41 to 59): (http), or simply [4]

Link to Uni Hawaii: (ppt)

Link to The Drunkard's Walk on youtube: relevant 3 minutes from min. 25:00 of total 51:35

Mlodinow on page 53 of The Drunkar's Walk: "AS I SAID EARLIER, understanding the Monty Hall problem requires no mathematical training. But it does require some careful logical thought ..."

And page 54: "the host, who knows what's behind all the doors, opens one you did not choose, revealing (a goat). In opening this door, the host has used what he knows, to avoid revealing (the car), so this is not a completely random process." Very illustratively he is spreading this item in saying that only in 1 in 3 the initial choice of the contestant was the car, only in this "Lucky Guess scenario" the host is free to open either one of his two doors as both hide goats. But the probability of landing in the Lucky Guess scenario is only 1 in 3.

"The other case we must consider is that in which your initial choice was wrong. We'll call that the Wrong Guess scenario. The chances you guessed wrong are 2 out of 3, so the Wrong Guess scenario is twice as likely to occur as the Lucky guess scenario. [...] Unlike the Lucky Guess scenario, in this scenario the host does not randomly open an unchosen door. Since he does not want to reveal the (car) he chooses to open precisely the door that does not have the (car) behind it. [...] So the process is no longer random: the host uses his knowledge to bias the result, violating randomness by guaranteeing that if you switch your choice, you will get the (car). Because of this intervention, if you find yourself in the Wrong Guess scenario, you will win if you switch and lose if you don't. [...] owing to the actions of the host, you will win if you switch your choice."

"The Monty Hall problem is hard to grasp because unless you think about it carefully, the role of the host goes unappreciated. But the host is fixing the game. The host's role can be made obvious if we suppose that instead of 3 doors, there 100." [...] And he says after the host opened 98 doors, the chances are still 1 in 100 that the car was behind the door you chose, and still 99 in 100 that it was behind one of the other doors. But now, thanks to the intervention of the host, there is only one door left representing all 99 of those other doors, and so the probability that the car is behind that remaining door is 99 out of 100. And Mlodinow asks: "Had the Monty Hall problem been around in Cardano's days (centuries ago), would he have been a Marilyn vos Savant or a Paul Erdös?"

So "the key" is that the host in 2 out of 3 is slave to the act, as he never "always" can open one of the two unselected doors "at random", because in that 2 out of 3 he is forced not to show the car, but only the second goat. Richard's conclusion "the actually observed host's action (opened door 3, not 2) is twice as likely as when the car is behind the second door as when it is behind the first door. Before we saw that, the first and second doors were equally likely to hide the car. Afterwards the second door is twice as likely as the first door to hide the car" is not "valid per se", it is a consequence that is true only under the constitutive key. That consequence has to be shown in the spotlight of this constitutive prerequisite, otherwise invalid.

Yes it is a consequence. Depending on where the car actually is, the host has different numbers of choices, hence the chance of a particular choice depends on where the car is, hence the particular choice is informative about the location of the car. Bayes' rule = logical deduction = intuitively obvious, once you have had your eyes opened. Richard Gill (talk) 05:16, 31 March 2013 (UTC)

Of course, it's just a consequence that anyone is able to see as soon as he is aware of "the key" to decode the paradox (spotlight on), the role of the host who never will show the car. Just from the beginning, this "key" should govern the article as clear as possible. Redundancy indispensable. Spotlight on, from the beginning to the end. We have to stop to misapply this constituent key as an "also-ran" only. Every editor should be ready to agree, eyes opened. To decode the paradox requires no mathematical training, it just requires to see the core of the paradox, to see *why* the role of the host is fixing the game. Without stressing that constitutive key the article will remain opaque. Please help the article to be as clear as possible: spotlight on. Gerhardvalentin (talk) 11:13, 31 March 2013 (UTC)

That prerequisite is the basic key, the eye-opener to understanding the paradox. It has to be distinctively emphasised, just from the beginning of the article, before any other considerations, and not misapplied as an also-ran. Without that explicit "key" the paradox remains opaque, resp. will not exist at all. Gerhardvalentin (talk) 17:04, 30 March 2013 (UTC)

I cannot work out exactly what this discussion is about. Gerhard can you explain what you are getting at. Do you like the introduction that I have added? Martin Hogbin (talk) 13:11, 31 March 2013 (UTC)

Martin, your proposal of such introduction is splendid. I like the introduction that you proposed to be added, together with my (any) clear underlining the precise behaviour of the host (he is absolutely avoiding to ever show the car, and if the contestant first did select a goat, then the host definitely will "choose" the second goat, thereby he circumvents randomness). This principle has to govern the whole article, the valid scenario for the famous paradox may never be neglected as an also-ran, mixed up with series of deviating scenarios. The host's normative behaviour and its effects is the cause for the paradoxical chances stay:switch of 1:2. And early enough the article should show the consequence of complete randomness in another host's behaviour (in case of complete randomness chances of 1:1).

Once more: Bayes is beautiful and can be shown, but is not necessary to decode the paradox. And any pondering on host's bias as per Morgan can never be helpful for the decision Marilyn asked for, such dreams of any such unknown bias should be reserved for students of probability theory. Gerhardvalentin (talk) 15:44, 31 March 2013 (UTC)

Gerhard, I have already added the introduction, the wording of which excludes the 'ignorant Monty' and 'Morgan' scenarios without bringing them unnecessarily to our readers' attention. What else do you think we should say there? Martin Hogbin (talk) 16:28, 31 March 2013 (UTC)

Regrettably the sources mentioned in the intro of the article do not attach appropriate importance to the essential key to decode the paradox, the vital consequence of the host's absolutely avoiding to ever show the car: randomness in opening a door in only 1 out of 3. But using his knowledge in 2 out of 3, being absolutely biased in opening his "selected" door, reducing randomness to 1 in 3 and so excluding full randomness. We should use appropriate sources that mention this crucial "key". I propose to use Mlondinow who, following Nobel prize winning psychologist Daniel Kahneman & colleagues (1982), expressively highlights the consequence of this deciding parameter of the MHP. The complete bias of the host in opening a door in 2 out of 3, if the contestant should have selected a goat, effectively guarantees the car by switching. That cannot be spotlighted enough. Should we use Mlodinow? Or psychologist Kahneman & colleagues, as not every excellent mathematician is also psychologist enough to realise that – with one's eyes open – understanding the famous paradox requires no mathematical training, but requires some careful logical thought. Gerhardvalentin (talk) 22:41, 31 March 2013 (UTC)

Mathematical training is the development of careful logical thought. And the best mathematical solutions to MHP, IMHO, are those which correspond "line by line", in just a couple of lines, to both intuitive and logically correct verbal deductions. For instance: a mathematical proof using symmetry and Bayes' rule is such a proof, a proof using intuitive and powerful ideas, not calculations or algebra. Richard Gill (talk) 05:43, 2 April 2013 (UTC)

The Monty Hall problem and the non-existent paradox

I can see what you are getting at now. I have always said that there are two things that puzzle most people about the MHP. The first, which we cover in the intro, is that the answer is 2/3. The second is that it matters that the host always reveals a goat, which requires that he knows where the car and goats are.

What is the wording that you propose and where should we put it? Martin Hogbin (talk) 08:45, 1 April 2013 (UTC)

Thank you for your comment. You are fully correct in saying what matters. Regrettably the article for years never clearly responds to the famous paradox and its preconditions, never tries to clearly show the valid scenario producing that famous paradox, but in the first line was and still is trying to report about sources that refer to quite invalid scenarios where the paradox never will arise. Marilyn vos Savant tried to present a valid scenario. Admittedly her first description was not fully complete, but her subsequent comments completed what matters, clearly showing the essential seminal role of the host. Basing on distinct modern sources, the article first should present her completed valid scenario where the paradox clearly arises, explaining that paradox, that – with ease – can be decoded and understood. And only after the work is done, the history of misapprehension and countless misunderstandings and aberrant scenarios where the paradox never can arise may be shown. Including Morgan et al. Historical negligibility, compared to the famous striking paradox.

The article should be structured in a way to help the reader to grasp what all of that is about, no more opaque wish-wash of mixing various different scenarios without clearly saying what scenario the actual segment is talking about.

The article should be structured following the outcome of the recent RFC, providing a chance for the reader to follow what matters. What about such a clear structure of the article. Gerhardvalentin (talk) 17:22, 1 April 2013 (UTC)

So why not add to the intro: "It is crucial that the host knows where the car is. Because of this, he always can open a different door to the door chosen by the player and reveal a goat; the solution 2/3 assumes that he always does both (i.e., open a different door, and reveal thereby a goat)". And if you want to give a literature reference to support this, you could cite Mlodinow, or Rosenhouse, or Rosenthal (excellent tertiary reliable sources). Richard Gill (talk) 07:53, 2 April 2013 (UTC)

Regarding Kahneman: in a blog about Kahneman I read "In his brilliant and fascinating book "Thinking, Fast and Slow", Kahneman coins a term called WYSIATI, which expands as "What You See Is All There Is". WYSIATI is about coherence and intuition, and explains how we are able to think fast and make sense of partial information in a complex world. However, it also explains many of our biases and judgements, and our tendency to use heuristics in situations when more thoughtful consideration of all probable scenarios will yield completely different outcomes." http://42ing.wordpress.com/category/daniel-kahneman/ I did not yet manage to read what Kahneman writes explicitly about Monty Hall Problem. Precise references, someone, please? But WYSIATI is a good slogan for the psychology which makes everyone initially give the wrong answer to MHP. Richard Gill (talk) 08:06, 2 April 2013 (UTC)

That sounds like like a good idea to me. I do not think though that there is any need to so prominently mention that he opens a different door from the one originally chosen by the host. Apart from one editor here I do not think there has ever been a suggestion that the host might open the originally chosen door. It would make a bizarre game show and, to most people, self-evidently change the problem to a different one. I suggest:

"It is crucial that the host knows where the car is. Because of this, he can always open one of the two the doors not chosen by the player to reveal a goat; the solution 2/3 assumes that he always does this."

Could we also say:

"If the host reveals a goat by chance the problem is changed and the solution is different." The could be referenced to vos Savant."

What about adding:

"This is because, if the host chooses randomly between the two doors not chosen by the player, and just happens to reveal a goat, we obtain information which changes the probability that the car is behind the door originally chosen by the player"

Do we have a source for this? Martin Hogbin (talk) 08:58, 2 April 2013 (UTC)

All sounds good to me. For the very last item, the source could be Rosenthal (the article, with Monty Fall and Monty Crawl alongside of normal Month Hall). Richard Gill (talk) 19:16, 2 April 2013 (UTC)

Make the language simpler and more direct. "It is crucial that the host knows where the car is..." states a key point in a round about way. If you know it is crucial, then the sentence makes sense.

The "This is because...." statement is confusing. Explain why the host's knowledge gets us the 2/3 solution. Explaining that with the converse host's ignorance gives a different problem adds confusion.

Glrx (talk) 05:07, 3 April 2013 (UTC)

Glrx, if you think the language can be better and if you think further explanation should be added (I don't) just go ahead and make those improvements - be bold. Let us know on the talk page what you did. Richard Gill (talk) 11:43, 4 April 2013 (UTC)

Glrx, it is not so easy, in my opinion, to explain why the host's knowledge gets us the 2/3 solution. What would be your explanation?

On the other hand I think it is easier to see that if the host chooses randomly and happens to reveal a goat he does tell us something about what is behind the unchosen door.

Having explained that, it then makes more sense to say that if the host knows where the goats are and we know that he can and will always reveal a goat we get no information on what is behind the original door when he does so. Martin Hogbin (talk) 18:18, 4 April 2013 (UTC)

Regarding a proper wording, what about listening to Leonard Mlodinow: The Drunkard's Walk — How Randomness Rules. Explaining "the Monty Hall problem" according to his book "The drunkard's walk" chapter 3 relevant pages 53–. The Monty Hall problem on youtube, min 25:00 – 28.00 of 51,35 in few short sentences (please excuse my bold face typing of relevant words) :

So now let's talk a little bit about conditional probability. [...] Briefly, that game was a game of the three doors. And behind one of them was a great prize like a car, and behind the other two was a booby prize like a goat.

And Monty would ask you to choose a door, and then he would open one of the two you didn't choose, okay? He opens that one to show a goat, and there's two doors left, and you had chosen one, but now he gives you the chance to switch to the other one. So if you switch or not it doesn't not matter? And the intuitive answer is that it doesn't matter. Most people think that it doesn't matter. And that's wrong.

So that's a good way to illustrate conditional probability. [...] You think "how can it not matter, it's random, it's 50:50, there's two doors. But it's not random. Because when he opens one of the other doors, he chose not to open a door that shows the car. So it's no longer random. Or as you might say in mathematics: You have been given more information. Things change, the probabilties change, and you have been given a certain condition, certain more information. And that's what we have to understand.

He says if Monty would show any of the two unselected doors, be it the car or a goat and did just happen to show a goat, then it's random, chances stay:switch = 50:50. But because Monty chose not to open a door that shows the car, "so it's no longer random." We have been given more information. Just in the beginning, we have to present the valid scenario.

For years I tried to say so by calling Monty's avoiding to show the car "no more symmetry, but asymmetry", but I was told that unsourced personal preferences are irrelevant. Now we have sources, let's find a proper wording to show "the key" just in the beginning. Gerhardvalentin (talk) 10:52, 3 April 2013 (UTC)

If you think you can make the article better, Gerhard, just go ahead and do it - be bold. But please remember "less is more". Less repetitions, say what you want to say just one time, as compactly as possible. Richard Gill (talk) 11:45, 4 April 2013 (UTC)

Thank you, but please consider: the MHP is about an intended famous "paradox" where – for answering the question stay:switch – the intuitive answer 1:1 is wrong but the "absurd" answer 1:2 is correct. The "key" to understand and to decode that intended paradox is the fixed behaviour of the host.

MvS said the only thing we know about the host is that he knows what's behind the doors, and always will avoid to open the door with the prize.

The article treats this *key* just as an "also-ran", mixed with various invalid scenarios that make he paradox impossible. And I hear editors saying that the article need not help the reader to understand and to decode the paradox, but is to show what sources say. Regrettably most sources refer to quite aberrant scenarios.

It's the tragedy of this article that the key to understand and to decode the intended "paradox" (1:2 and not 1:1) remains hidden. Your saying "Less repetitions, say it just one time" helps the article to remain covered by a smoke screen as it was for years now. No, the radical key to the paradox needs redundancy, it should govern the whole article. Once more: redundancy is inalienable. Gerhardvalentin (talk) 17:28, 4 April 2013 (UTC)

Un-hide it then. Say it once, clearly, right up front in the introduction, in the article. Don't keep saying it on the talk page. The more you say it on the talk page the less other editors will take any notice. Richard Gill (talk) 14:23, 5 April 2013 (UTC)

I try to insert Mlodinow as an aid to understanding the "paradox". – Imho the whole paragraph is important to finally grasp what matters (1:2 and not 1:1), and I hope later edits will improve lucidity, not again pulling wool over Mlodinow's assistance. Gerhardvalentin (talk) 19:10, 6 April 2013 (UTC)

The introduction

Following your suggestion I said it right in front of the introduction. I hope it is clear and short enough, and will remain short and clear enough.

As to the diagram in section The Economist, it would be fine to add at the right side of scenario "1." the words: "Lucky Guess scenario, P = 1/3". And at the right side of each of the two following Wrong Guess scenarios "2." and "3." the words: "Wrong Guess scenario, P = 1/3".

And below that diagram we should clearly say that the probability to be in each one of these three scenarios is 1/3 if no host's "bias" in showing one of his two goats is "known". But given that such host's bias in opening of one special door, its direction and extent would be "known" (Falk), then such known bias could increase our guess to actually be in the "1.) Lucky Guess scenario" from 1/3 to max. 1/2, reducing probability to be in one of both Wrong Guess scenarios to min. 1/4 each, or vice-versa could reduce the probability to actually be in the Lucky Guess scenario to zero, increasing probability to be in each one of both Wrong Guess scenarios to 1/2 each. I think that could open the eyes to understand clearer what infamous Morgan et al. meant (changing the 1/3 probability to actually be in one of those three scenarios). Am I correct? Can anyone help to do so? Gerhardvalentin (talk) 09:41, 7 April 2013 (UTC)

The Economist solution puts randomness in the player's choice. The location of the car is fixed, the choice of the host (when he has one) is irrelevant. It's a completely opposite approach to the usual Bayesian approach (including Morgan). Richard Gill (talk) 17:23, 7 April 2013 (UTC)

You seem to want to rewrite the article in order to teach. But: "Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter". See WP:NOT. Richard Gill (talk) 17:48, 7 April 2013 (UTC)

Well, there is randomness in the player's choice, and, for myself, I think that (Economit) recounting of the problem is far simpler and approachable for the reader than other approaches taken in the article. I also have to remind folk that naming scenarios with capitalized descriptions is the quickest way to make the average reader's eyes glaze over.

We might do well to lead with a simple explanation for the masses. That the Economist version is basically presented as a cartoon further down somewhere in the article with pretty much no further explanation or mapping to various formal mathematical views of the problem rather speaks to the "mathelitist" bent of the article as it currently stands. VєсrumЬа ►TALK 19:13, 7 April 2013 (UTC)

Richard, you got it wrong. Morgan and his infamous host's bias cannot change the actuality, but can only give additional info about what scenario the contestant "actually is in". Changing "1/3:1/3:1/3" for example to "0:0:1", if the host abnormally opened the door that he usually strictly avoids to open. But I leave Morgan now (I.P.R) and I abstain from such unnecessary cue to infamous M.

Wikipedia is no textbook, but announces what relevant sources say.

+1 to Vecrumba. Elucidation for the masses is overdue and necessary, and it was really not bad to add clear titles to the Economist diagram, saying afterwards that only in the 1) Lucky Guess scenario the host can randomly choose which goat he shows, but is bound to show his only goat in the two Wrong Guess scenarios. Anyone against it? Gerhardvalentin (talk) 20:45, 7 April 2013 (UTC)

Vecrumba: perhaps you should read some of the papers on MHP written by one R.D. Gill. It is even on my request that the solution from "The Economist" is back in the article. There are several main approaches to MHP, and each has both a "mathematical" and a "popular" version. The different approaches correspond to different understandings of "probability" - e.g. subjective vs, objective - and different emphases - e.g., is MHP about beliefs or about decsiions? If you like objective probability and decision making then, IMHO, the best solution to MHP is: choose your initial door completely at random and thereafter switch. You'll get the car with objective probability 2/3 and you cannot do better. I think it is no coincidence that "The Economist" prefers a popular solution whose mathematical incarnation is the game theoretic solution. MHP became well known in the game-theoretic and mathematical economic and decision theoretic literature through the paper of Nalebuff, post Selvin but pre Vos Savant. It leads an independent existence in that literature.

Gerhard: I don't understand at all what you are saying when you mix up Morgan and The Economist. Since these solutions have completely different philosophies there's simply no comparison. Seems like some OR and personal POV to me. The consumer simply has to study the small print and choose an apprach according to taste, budget, context... Wikipedia can, and should, expose the small print of each solution. You now seem to want to permeate the whole article with anti-Morgan propaganda. That's against the spirit of the conclusion of the RfC and it's against Wikipedia principles. You are a great fan of Mlodinow but his is not the only way to rhink about MHP. Sure, he'sa reliable source. But he pushes his favourite way of understanding MHP. He doesn't survey all ways. Richard Gill (talk) 08:37, 8 April 2013 (UTC)

I agree, the article (after much improvement) seems to deteriorate again, much along the old (pro Morgan, pro vos Savant) conflict line.--Kmhkmh (talk) 10:34, 8 April 2013 (UTC)

Besides that there's no OR at all, please read what I said above: I.P.R to Morgan, so R.I.P and no more Morgan. Richard, what you say is right, but only partly right. "You'll get the car with objective probability 2/3" is not true "per se", it is not true if the host always opens one of both unchosen doors at random. The article should prominently and clearly say that the intended paradox is based on a host who, in case the contestant selected a goat in 2 out of 3, will never show the car, but in this 2 in 3 will "choose" to show the second goat. Only then your saying "with objective probability 2/3" applies, but never otherwise. So we should not confuse any more but clearly say what we are talking about. Gerhardvalentin (talk) 11:47, 8 April 2013 (UTC)

We are talking about MHP. It is given that the host always opens a different door to the door chosen by the player and reveals a goat (which he can do because he knows where the car is hidden). If the player makes his initial choice completely at random and afterwards switches regardless of which door the host opens he'll win the car 2/3 of the time. No assumption needed or used about how the car is hidden, no assumption needed or used about how the host chooses a goat door to open when he has a choice. That is the whole point of the game theoretic (minimax) approach. A player who approaches the game in this spirit does not ask himself the question: "what is the chance I'll get the car given that I happened to choose door 1 and given that the host happened to open door 3?". Such a player is not interested in the answer to this question. The player knows that unconditionally, he'll get the car with probability 2/3, and he knows that he is using the only strategy which gives this cast-iron guarantee, completely independently of how the car is hidden or the how the host chooses a door to open! (Perhaps you should read not only Mlodinow but also Gill). Richard Gill (talk) 15:42, 8 April 2013 (UTC)

PS when I used the phrase "objective probability" I am using the word "objective" in its technical sense concerning different concepts of probability: frequentist/objective/ontological vs. Bayesian/subjective/epistemological. I don't know the abbreviation I.P.R. Richard Gill (talk) 15:45, 8 April 2013 (UTC)

Thank you again, Richard. We are not talking about infamous Morgan, okay? Btw, it's: In Pacem Requiescat (or: Requiescat In Pacem).
"He'll get the car with probability 2/3" is right again, but only partly right. 2/3 only if the host always "selects" a goat to show, but never will select the car. My horizon (tested):

But if the host opens an unchosen door at random,
* then the host happens to show the car in 1/3, and at those events both still closed doors are hiding goats, and
* then the host happens to show a goat in 2/3 only. In that 2/3, switching looses in 1/3 (if the contestant first selected the car) and switching wins in 1/3 (if the contestant first selected a goat AND the host just happened to show a goat.
Such host will just "happen" to show a goat in the subset of 2 in 3 only, and staying:switching is 1:1. Tested in one million. What is wrong? I am not talking about M., but about that "subset". Please help me to understand what you say in giving me some (exact?) reference to your relative papers. Gerhardvalentin (talk) 16:57, 8 April 2013 (UTC)

We are talking about MHP, I believe: the host is certain to open a different door to the door chosen by the player and reveal a goat. I wrote a paper on MHP which appeared in Statistica Neerlandica in 2011, you can find it at http://www.math.leidenuniv.nl/~gill/essential_MHP.pdf There are some more publications on MHP by me at http://www.math.leidenuniv.nl/~gill/#MHP . I'm not going to explain or promote my contributions here. Richard Gill (talk) 20:27, 8 April 2013 (UTC)

R.I.P. is a familiar abbreviation in English. I never saw I.P.R. before. Compare RIP, IPR. Richard Gill (talk) 20:30, 8 April 2013 (UTC)

The famous unintuitive paradox

Thank you, yes, so everything is fully clear now. Your relative papers are based on the scenario that Marilyn vos Savant presented very clearly in saying:

"Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens ...". And in saying:

"... the host always opens a loser." And in saying

"... the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question."

In reporting on the famous paradox, the article should say so clearly enough, just from the beginning. For years it has been my concern that the article never clearly said so. Now Martin's introduction, along with that significant clarification, will help the readers to better understand the hitherto nebulous article. Thank you once more. Gerhardvalentin (talk) 11:32, 9 April 2013 (UTC)

The introduction + the problem sections

To suggestions above, I will break down and catch up in more detail on the camps of representation. That said, that won't help these two sections which, taken together, may be missing the forest for the trees. For example:

In 1990 the same problem was restated in the form shown above in a letter to Marilyn vos Savant's Ask Marilyn column in Parade. There are certain ambiguities in this formulation of the problem: it is unclear whether or not the host would always open another door, always offer a choice to switch, or even whether he would ever open the door revealing the car (Mueser and Granberg 1999).

This doesn't illuminate, rather it's a black hole of (average) reader frustration. Of course the host is not going to reveal the car, what's the drama in that? Such permutations of looking at the problem belong in the esoteric mathematics of the impact of initial conditions and assumptions. The problem is not "ambiguous," it wasn't necessary to describe how the show works because everyone knows how the show works. A scenario such as the host, at random or intentionally, opening the door with the car (OOPS?!?) is simply not going to happen. Ever. Considering that scenario is an intellectual exercise best left to mathematicians. (!)

If I had to sum up the article's current state of identity crisis:

TV viewers are totally clear on how the show works and how the host creates the drama,
(yet) mathematicians rush in with variations on initial conditions and scenario assumptions and all possible variations.

IMO, the article should be about #1 with #2 as interesting reading for the more mathematically inclined. VєсrumЬа ►TALK 21:58, 8 April 2013 (UTC)

Vecrumba, as far as I can see the article *is* about the standard Monty Hall problem - what you call #1. The wikipedia article surveys the enormous literature on MHP and that literature, whether popular, statistical, decision-theoretic, psychological, educational ... is about problem #1. The fact is that there are many ways to approach problem #1. There are "popular" approaches and there are mathematicians' approaches. Different mathematicians prefer different approaches. Different popular writers prefer different approaches. There is not one standard solution which everyone agrees is the right solution. That's why the article is big.

Most readers understand right away that we are talking about #1 but there are always readers coming to the article who need it said explicitly. That's why early on in the article it is useful to emphasize that we are meant to understand that the host always will open a different door to the door chosen by the player and reveal a goat; we understand that the host can always can do this because he knows where the car is hidden. What's the harm in emphasizing this? Especially since almost every publication on MHP starts off by making the same clarification? Richard Gill (talk) 08:52, 9 April 2013 (UTC)

Well, we agree here, so why per my example the extensive verbiage at the outset on alternate scenarios (such as the host picking the car)--which are not part of the problem? That was the point I was trying to make--the divide you perceive in discussing #1 and #2 is not as clear as it could be. VєсrumЬа ►TALK 01:54, 10 April 2013 (UTC)

More comments

Calling it "Introduction" is incorrect, we generally mean lead for the word! Section should be renamed.
A huge problem of the whole article is lack of inline citations! See WP:ILC for help. All the references should be changed to that format.. the sooner the better! It'll also help to verify the sources.
Frankly speaking, this article has multiple issues. There are some books in internet like "Learn javaScript in 21 days", "Learn HTML in 1 week", this article looks like that- "Learn everything about Monty Hall Problem in 1 hour" Here is one suggestion: in Wikipedia cover only the basic points, history, criticism, basic arguments, basic outlines of versions etc. And send/export everything else (specially the sections which (might) deal with "reader's guide", "learner's study material", "beginner's introduction", "answers for cynical readers" etc) to Wikiversity and link the Wikiversity section to this article! --Tito Dutta (contact) 23:43, 9 April 2013 (UTC)

Interesting suggestion. I agree the article has now grown over-weight.

This is what wikiversity has so far on MHP: Monty Hall Lesson on wikiversity.

I many times proposed deleting the brute force mathematical solution with Bayes theorem, in which is written "This requires introducing quite a lot of mathematical notation, which may distract from the essential simplicity of the calculation". Indeed! This section seems to come straight out of many introductory textbooks on probability theory where the writer's aim is to get the student familiar with the notation and rules of calculus, not to gain insight into MHP. The same material is used in the wikipedia article on Bayes theorem for precisely the same pedagogical purpose. However it seems that this formalistic approach is still the favourite solution of some editors. Richard Gill (talk) 06:16, 12 April 2013 (UTC)

Also turning up by invitation.

I think Titodutta raises a very good point: What should a Wikipedia article about the Monty Hall Problem be like? Should it describe the problem, its history and its relevance, or should it attempt to convince the reader of the correct answer in as many different ways as possible in the hopes that one will stick? The latter strikes me as unencyclopedic.

The key points for a Wikipedia article to cover seem to me to be:

The history of the problem
A standard statement of the problem
The common assumptions
The standard answer
The controversies surrounding the answer (Marilyn's column, the popular response, the observed fact that the correct answer is unintuitive)
A small number of proofs (maybe as many as four to give samples of intuitive and rigorous versions of Bayesian and frequentist arguments)
A summary of the significance of the problem for cognitive psych
Variations on the common assumptions and the fact that they produce different answers

I like the "Introduction" section, and wouldn't object to its content being kept in the relevant sections, but its inclusion is, in my opinion, a step in pursuit of the wrong goal - that of convincing readers of the correct answer rather than informing readers of the nature and significance of the Monty Hall problem. Rmsgrey (talk) 09:56, 10 April 2013 (UTC)

These too are interesting suggestions. But are the many different solutions which are presented in the article an attempt to convince the reader of the correct answer in as many ways as possible? I think they simply represent an attempt to be encycopedic. There are very many solutions in the huge literature on MHP and the article covers, as far as I know, all notable solutions. Each major paper on MHP does the same - presents a range of different ways to approach MHP.

Also interesting is your interpretation of the purpose of the "Introduction" section. We only added this solution recently and we did that because many editors complained that it took so long to find out from the article what the standard answer to MHP is: switch. It was not put there in order to convince readers of the correct answer. It was merely there to inform the reader as early in the article as possible of the central content of the literature which the article is surveying. Richard Gill (talk) 06:23, 12 April 2013 (UTC)

Okay, my assessment is that the article is very good, but it is somewhat repetitive and occasionally argues the point rather than stating it. For example, the section "Refining the simple solution" seems to show that simple solution suffice, but the later "Criticism of the simple solutions" seems to contradict this, and both are stated in Wikipedia's voice. This is what I would think is most important for the article to achieve, leaving aside history and other relevant non-mathematical content:

Explaining why the always-switching-player has a 2/3 winning chance.
Explain when a conditional analysis is necessary, perhaps using a variation where the conditional analysis has a different numerical answer.
Explain the symmetry arguments and the assumptions under which simple solutions suffice to answer "what should you do having seen X and Y" type questions.

As far as I understand, there seems to be almost universal agreement on the assumptions that make simple solutions suffice to answer the conditional question, and the dispute is only about what question the original formulation was asking and whether a solution that does not make symmetry assumptions explicit can be called a solution. If that is so, one should simply explain the assumptions and let the reader decide what is "intuitive" and what needs to be made "explicit" to count as a solution. Or have I misunderstood what the disagreements are all about? Vesal (talk) 17:19, 14 April 2013 (UTC)

Monty_tree_door1.svg

Monty_tree_door1.svg shows every possible outcomes if door 1 contains the prize. The current description of that page is "Tree showing the probability of every possible outcome if the player initially picks Door 1". This is clearly not the case, as the tree diagram still applies to the case if the player initially chooses chooses door 2. I don't understand why User:Gerhardvalentin reverted my edit. I am reverting his reversion. Fangfufu (talk) 22:59, 13 April 2013 (UTC)

The diagram is intended to show every possible outcome which can occur given that the player initially picks door 1. The car can be behind door 1, 2, or 3. Depending on where the car is, the host may open door 2 or 3. You could of course draw a similar diagram if you took a different approach: considered the location of the car as fixed and the choice of the player as random. But both the text in the article referring to this figure, and the annotation of the diagram itself, show what is supposed to be fixed, what is random, in this case. I've reverted the text again! Richard Gill (talk) 12:23, 14 April 2013 (UTC)

Well, in that particular diagram, you can only get the car, if you pick door 1 in the end. Car at door 1 is the only fixed condition. I can pick door 2 or door 3 as the first outcome on the tree, and that diagram still makes sense. Personally I think the diagram/article combination in its current state is poorly worded. You are a mathematics professor, so I am not going to revert your changes. But I really think the diagram would be better if it is worded differently. Fangfufu (talk) 17:41, 14 April 2013 (UTC)

Have you read the words written above the diagram? "Car location:" above "Door 1", "Door 2", "Door 3". "Host opens" above "Door 2" or "Door 3". And in the text "By definition, the conditional probability of winning by switching given the contestant initially picks door 1 ... These probabilities can be determined referring to ... an equivalent decision tree as shown to the right". And also in the text: "Assuming the player picks door 1 ...". I think it is clear from text and diagram together what the writer intended to be fixed and what is variable. Perhaps the problem is not the diagram, but the text in the article which is illustrated by the diagram. Richard Gill (talk) 06:16, 15 April 2013 (UTC)

Citation style?

Someone has complained that inline citations need to be used, but harvard references are a form of inline citation, so I will remove the complaint from the article. However, I will ask here whether key contributors here would object if someone (certainly not me) volunteered to change the citation style to regular footnotes? Vesal (talk) 18:28, 12 April 2013 (UTC)

No, go ahead it seem to have been a rather stupid tagging. The article overall well sourced and if somebody misses a source for an individual claim he/she can use the "citation needed"-tag.--Kmhkmh (talk) 20:16, 12 April 2013 (UTC)

I think there was a proposal to change to sfn some time ago, but it did not get consensus. I have no problem with Harvard references, and in this article it may be appropriate to continue to use them. HR allows the reader to know the source without looking at the footnote.

I have no problem with converting the HR to use {{harvtxt}}; that would involve using citation, cite with ref=harv, or a hardwired CITREFauthoryear.

Glrx (talk) 02:38, 13 April 2013 (UTC)

I agree, we need more inline citations, and Harvard style - Author(Year) hyperlinked to a complete bibliographic reference - is informative and user friendly. The list of references at the end of the article is pretty complete, I think. So it is mostly a question of cleaning them up and putting in the links. Richard Gill (talk) 07:26, 16 April 2013 (UTC)

Vos Savant sources

The literature list of the article specifies three of Vos Savant's columns "Ask Marilyn" in Parade magazine: September 1990, December 1990, February 1991. And also a little known follow-up November 2006 (no reference to it in the text!). Yet according to Rosenhouse's book "The Monty Hall Problem", and to Marilyn's website [5], and to Marilyn's book "The power of logical thinking" (1996), there were four articles in 1990 and 1991. So we are missing one of the four basic sources. Unfortunately the two sources which reproduce all four (Marilyn's book, and Marilyn's website) don't give the dates.

The very first article (September 1990) has Whitaker's problem statement and Marilyn's first solution - she just states the 1/3, 2/3 answer and describes a million door version for insight. The second article (December 1990?) has a little table of three possible locations of the car (player's choice is door 1). The third article (February 1991?) extensively emphasizes the standard rules of the game (till then, they were implicit; actually, given her solutions so far, they are obvious). The fourth article (??? 1991) reports on all the simulation experiments done around the country.

We could just cite book and website and state that these reproduce her four columns from 1990 and 1991 without distinguishing one from another. Richard Gill (talk) 16:09, 16 April 2013 (UTC)

Ha! Found the fourth reference: July 1991. This meant that a number of other references had to be changed (old MvS 91b became 91c). (My source is Granberg and Brown, 1995). Richard Gill (talk) 16:56, 16 April 2013 (UTC)

The misty article about a paradox without expounding the paradox

Thank you all. Imo the article is incomplete without the intended paradox as per the standard version, so the intro should say so. According to your proposal I am trying to say it very short, but complete. Gerhardvalentin (talk) 10:24, 10 April 2013 (UTC)

Gerhard, I think 'chosen' is the wrong word to use. I understand what you mean but I think it would be better to say that he is required, by the rules of the game, to show a goat. How about:

'The above answer depends on the rules of the game requiring the host to always reveal a goat. This means that, in 2 out of 3 cases, when the player has originally chosen a goat, the host must reveal the remaining goat thus ensuring that a player who swaps must win the car'. Martin Hogbin (talk) 13:36, 10 April 2013 (UTC)

Yes, splendid. I just cited the source (the host chooses) and will change that, following your proposal. Thank you, Gerhardvalentin (talk) 13:50, 10 April 2013 (UTC)

As to he most common intuitive answer: Suggestions to say it clearer? Gerhardvalentin (talk) 14:29, 10 April 2013 (UTC)

In my opinion the Mlodinow material doesn't belong here at the beginning of the article; at least, not so extensively. Wikipedia is an encyclopedia, not a text book or other pedagogical tool. It's not the primary purpose of the article to convince people of the right answer. If we think that there are good places for gaining intuitive understanding of MHP we should refer the reader to those places. Tastes differ. I think Rosenthal is a better source of intuition on MHP than Mlodinow. Other people will have yet other favourites.

I also don't see why it is necessary to hammer away on the point that the 1/3 : 2/3 answer depends on the "rules of the game". Anyone who is able to understand any of the arguments for 1/3 : 2/3 can see for themselves that the reasoning depends on the standard rules being in force. Marilyn said herself that the overwhelming majority of her correspondents (and the majority initially disagreed with her solution!) understood exactly the implicit problem assumptions, and in any case, her solution made perfectly clear that she had made these assumptions.

Why not just say, in the introduction, "it matters that the host knows where the car is and therefore can always reveal a goat", citing anyone you like. Why it matters will become clear in the article. Richard Gill (talk) 06:56, 12 April 2013 (UTC)

+1 Nijdam (talk) 07:49, 12 April 2013 (UTC)

I agree too. The introduction was intended to be a short simple basic solution together with an short explanation of why it matters that the host knows where the car is (always reveals a goat). Martin Hogbin (talk) 17:03, 12 April 2013 (UTC)

Okay, but whatever you write, please check it with someone with reasonably fresh eyes. This article is extremely prone to grandmothering, i.e., explaining things that only makes sense to those who already understand it. Personally, I was at first quite perplexed by the fact that if I'm playing the Monty Hall game, and after Monty offers me the choice he says, "Boy was I lucky to not reveal the car!", somehow my probability of winning by switching is then only 1/2, and yet superficially the situation is identical to that of the knowing host. I find that quite fascinating, so please don't think it is super-obvious to everyone. Thank you, Vesal (talk) 18:10, 12 April 2013 (UTC)

+1 to Vesal, and "no" to Richard's "Why it matters will become clear in the article." Because in that cumbersome article it is not only about "always can", but in fact and in the first line it is about "... and always intentionally does". It's about Mlodinow's "The host is fixing the game." It's about the paradox that is so counter-intuitive. And a great "yes" to Martin's "always reveals a goat". Because it is of utmost importance to say that just in the beginning of that regrettably but unnecessarily cryptic article that lengthy shows where the paradox may eventually not arise.

And by the way, on the topic of "combining": Wikiversity says about the paradox being so counter-intuitive and the host's resp. the computer's role in "helpfully eliminating" a loosing door resp. a loosing box: The original guess plainly has a 1/3 probability of being correct. The paradox arises principally because it does not seem as if the computer is imparting any useful information by opening an empty box. However there is a 2/3 probability of the prize being in one of the two boxes that were not guessed, so after the computer helpfully eliminates one of them, the 2/3 probability remains attached to the remaining unopened box, which is therefore twice as likely to contain the prize.

And, btw I fully agree with Guy, imo it's long overdue that new editors are taking the reins. Gerhardvalentin (talk) 12:07, 13 April 2013 (UTC)

So what needs to be said is "Because the host knows where the car is hidden he always can reveal a goat behind a different door to the door chosen by the player; we assume that he always does do this ". And Gerhard, you are citing the "combined doors" argument, now as formulated by wikiversity: this solution is silently assuming that the information which other door is opened is irrelevant to the question whether or not the car is behind the door originally chosen by the player. Richard Gill (talk) 12:50, 13 April 2013 (UTC)

Combined, silently assuming, yes. Because which door actually has been opened, be it #2 or #3 *never* "can" give us sound justification to insolently redefine the scope of those three scenarios of 1/3 each (lucky guess:wrong guess:wrong guess) to other values (subject matter for teaching conditional probability theory, but completely irrelevant to the decision).

MvS intended to present an (already well-known) counter-intuitive paradox. Unfortunately, the article gives a damn about the intended paradox, but pages over pages in the first line presents literature about misapprehension and misinterpretations including inglorious MCDD's pushing unknown scope as "actual value". Perforce wrong if it differs from 1/3 : 1/3 : 1/3, though all of that could be said within a short section about historical inability to "see" the clean paradox arise.

"Because the host knows where the car is hidden he always can reveal a goat behind a different door to the door chosen by the player; we assume that he always does do this." ? No. We "know" that he never will show the car. Not only because MvS later clearly said so, but because this *IS* the best-known main item which lets the "paradox" arise. Gerhardvalentin (talk) 13:59, 13 April 2013

I don't see any difference between "know" and "assume". So OK, let's write "know". But let's keep it short and sharp. This little introductory section is not meant to give the complete answer, and I think it is also wrong to write it following just one particular writers' point of view (Mlodinow). We are writing an encyclopedia not a text book. Our purpose is to inform, not to teach.

I've shortened the text - reduced it to just pointing out the key rules of the game, not the pedagogical explanation. And changed the names of the first two sections. Richard Gill (talk) 12:48, 14 April 2013 (UTC)

If you don't want to explain the role of the host's knowledge here, we should simply remove the talk about the host. Now, we have have a sentence spelling out completely obvious things which are only significant to people who already know why they are significant. I understand that you are offering this as a compromise, but please review this section now and tell me what is the point of that sentence? What does it convey to the reader? Vesal (talk) 16:40, 14 April 2013 (UTC)

The point of the sentence is to make it absolutely clear, right at the beginning of the article, what the Monty Hall problem is about, and hence what the article is about. I agree, for many readers Whitaker's quoted words are enough. For them, the sentence is superfluous. But for some readers Whitaker's words are not enough. We know such readers exist because we frequently have wikipedia editors who think that a different problem has been stated. There are even several publications on Monty Hall problem (but I think not very good ones) which start off "on the wrong leg" so to speak by solving "the wrong" Monty Hall problem instead of the right one. The sentence is for this special minority. Richard Gill (talk) 06:29, 15 April 2013 (UTC)

Okay, I accept this, but it is still somewhat disappointing. You fully acknowledge that these sentences are not written for the average reader, but intended to hold people with certain POV at bay. Things like this are inevitable on Wikipedia, I get that, but what is your own opinion about the effect this has on readability for the general reader? Do you think most people are in a position to understand the point of these sentences given how early they appear? You have since slightly improved the wording, but you moved it to the lead, so it is even more critical that you ask yourself what the non-minority will make of them. Vesal (talk) 16:23, 16 April 2013 (UTC)

It's not a question of holdng some people with a minority POV at bay. It's just a matter of making it clear to everyone what MHP is about. For some readers, some information will be superfluous. Note: we are not saying anything controversial. We are following many, many reliable sources. It *is* important to know the "rules of the game". And the answer to the problem ("switch") depends crucially on those rules. Just about every source says so, and we simply repeat this "encyclopedic" information. Richard Gill (talk) 06:07, 17 April 2013 (UTC)

Grandmothering

Vesal, I have not heard that term before but we are all very aware here of the problem. The MHP is notoriously refractory; it is extremely hard to convince most people who have not come across the problem before of the right answer yet, once it is understood can be hard to work out what the original problem was.

The 'Introduction' section was intended to address that very problem, the argument being that, if you do not believe the correct answer, the rest of the article is completely wasted.

One problem is that there is very little in the published literature on how to convince people of the right answer. Krauss and Wang make some progress but no one has come up with an argument that convinces most people on sight; such a thing may be impossible.

My personal opinion is that there are three stages to fully understanding the problem. First, accepting that the correct answer might not be 1/2, second, understanding why it is 2/3, and finally understanding how the host's use of his knowledge is important.

Most people have their own preferred solution, which they tend to believe must convince everybody else. As you seem to be a relative newcomer to this problem, perhaps you could tell us what is was that helped you understand the problem. Martin Hogbin (talk) 09:49, 14 April 2013 (UTC)

Initially, I found it useful to imagine many doors, but I must of let myself be too easily convinced because when I read the sentence here in the introduction saying that that an ignorant host would not achieve the same effect, I was slightly surprised. I got to step 3 by imagining a computer simulation where the cases in which Monty accidentally reveals the car are discarded. Now, I'm not sure the introduction section should explain intuitions, but something along the lines of what was there should maybe be in the Carlton section? Vesal (talk) 16:40, 14 April 2013 (UTC)

The paradox

Editor Richard Gill likes to avoid naming the core of the world-famous paradox: "1:2" correct, and the common first intuitive answer "1:1" simply wrong. In contrast to editor Richard Gill, who today eliminated the dominant explanation, the relevant literature is clearly expounding the dominant explanation of the famous "paradox": Henze says that the host is to eliminate a goat. Period. And Henze says that in 2 out of 3 the prize will be behind an unchosen door, so switching will give the prize automatically in this 2 out of 3.

And Henze cites MvS: If in 2 out of 3 the prize is behind an unchosen door and the host opens one of them, then we automatically know with probability 2 in 3 the actual location of the car. So this *is the rule* to enable the paradox to arise.

Henze says that in 2 out of 3 the host will definitely show *the second goat* and in this 2in 3 switching automatically gives the prize. So 1:2 and not 1:1. It is important to say this very clearly and to clearly show the alternative. It was in the article, but Richard Gill removed it. Henze clarly recommends University S.D. that shows why a host who opens one of the two unchosen doors at random, be it goat or car, reduces the 1:2 to the value of 1:1 (as per most common intuitive answer). Henze's tip to plainly see the difference between "1:2" (host intentionally shows a goat) and "1:1" (host shows a goat per chance) is important. Richard Gill removed such indication. Is Henze wrong? I oppose against eliminating the core of the unintuitive paradox and leaving the article misty. I am against Richard Gill's opinion that the article will anyway be showing each and everything later.

Mlodinow explicitly names the same facts as Henze. It is about the unintuitive difference 1:2 to 1:1, it is about the paradox. I am against removing what reputable sources say, we should not hide it permanently. I controvert that peekaboo. Gerhardvalentin (talk) 21:55, 14 April 2013 (UTC)

Gerhard, I do not think that anyone, including Richard, is suggesting that the random host version, in which there is no advantage in switching, is part of the standard problem. We all agree that in the standard problem the host always reveals a goat and the player gains by swapping. I think we also all agree that the difference between the two versions is important and should be brought to the attention of our readers in the introduction. I do not think though that we can add much more in the introduction, which is intended, as its name suggests, to be a brief introduction to the right answer and the fact that it matters what the host knows. More detailed explanation belong, in my opinion, later in the article. Martin Hogbin (talk) 22:39, 14 April 2013 (UTC)

I do not want to avoid naming the core of the paradox. The question is, where should this be done, and how should this be done? Richard Gill (talk) 06:55, 15 April 2013 (UTC)

I think the problem here is that we are rewriting the wrong part of the article. The lead currently contains some information which is not very important at all (references to older paradoxes in probability theory, reference to a logical classification of paradoxes including the word "veridical"). This kind of academic detail belongs in the body of the article (and indeed, it is all repeated there again, except for the reference to Quine's taxonomy of paradoxes). If someone can reproduce Henze's or Mlodinow's insight in the lead, that would be great. The lead should be a self-contained summary of the main content of the whole article. Richard Gill (talk) 06:52, 15 April 2013 (UTC)

The article MHP is about the historical striking after-effects of MvS's attempt in trying to present a clear unintuitive "paradox", some even say that it had already been known for centuries. The paradox "arises" if one does not care closely enough about the role of the host in opening an unselected door, given that in 2/3 the contestant's first choice was a goat.

As per the most common intuitive answer, chances are simply 1:1 and not more, if in that 2 in 3 the host opens "a door" and luckily shows a goat. Whereas chances are 1:2 only if in that 2 in 3 the host intentionally shows the second goat.

Reliable sources like Henze and Mlodinow frankly call a spade a spade in showing this key, thus everyone knows what the intended counter-intuitive "paradox" is about. And this should clearly be shown in the article, and it should precede the "problems" of historical misunderstandings and misinterpretations. Martin and Vesal succeeded in briefly but clearly represent these simple facts. It was in the article as follows:

The above answer depends on the rules of the world-famous paradox, requiring the host to always reveal a goat. This means that in the 2 out of 3 cases, when the player originally has chosen a goat and switching wins, the host must reveal the remaining second goat, thus ensuring that a player who swaps wins the car in every such case. However if, contradictory to the rule of the standard paradox, the host just opened a door at random and revealed a goat simply by chance, the overall outcome staying:switching would have been reduced from 1:2 of the standard iparadox to only 1:1 as per the common intuitive answer, because half of the potential winning cases are wasted when the host can accidentally reveal also the car. Mlodinow says: The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated. But the host is fixing the game. (Mlodinow 2008)

Richard removed it. I suggest to re-enter that unobvious fact, like Henze and Mlodinow frankly calling a spade a spade. To re-enter it before further explanations on historical issues and "problems". In order to make the article comprehensive for the reader. Gerhardvalentin (talk) 18:51, 15 April 2013 (UTC)

I had two problems with your text, Gerhard (and I wasn't the only one): (1) it was in the wrong place (2) your English is hard to read, you write long and complicated sentences with many repetitions and use of unusual words. So I shortened it to what I think are the essentials (and notice that I left the quote from Mlodinow that one must not overlook the actions of the host). I shortened it to the reporting of facts, not opinions. The beginning of the article proper should (I think) just summarize the problem and summarize a well-known intuitive solution and then go on to survey systematically (encyclopeadia fashion) the literature on MHP. I changed the names of the first two sections accordingly.

Before the article proper starts, there is the so-called lead (some people call it a lede, a phrase coming from newspaper typography). The lead is the place where there should be a summary of the main content of the whole article. At the moment there is specialistic detail in the lead, which belongs in the body of the article (and is in the body of the article - prehistory, a philosopher's typography of paradoxes, ...). If Henze, Mlodinow or whoever have a short summary of the problem and the right answer and why it matters that the host knows where the car is, that is what needs to go in the lead - in our own words.

I've made a first try. Richard Gill (talk) 12:27, 16 April 2013 (UTC)

Richard your cleaning helped. But for the readers, it was of avail if "The problem", also following those strained and varied considerations of Cecil Adams (+probably others, too), would intrepidly and frankly read as follows. Especially because supposedly not every reader is a Cecil Adams.

The problem

The most well known statement of the problem is:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. 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? (Whitaker, 1990, as quoted by vos Savant, 1990a)

The answer is that you should swap and that doing so gives you a 2/3 chance of winning the car. A simple explanation is that players initially have a 2/3 chance of picking a goat and those who swap always get the opposite of their original choice. So players who always swap have a 2/3 chance of getting the car (Carlton 2005).

The above answer follows the rules of the world famous intended paradox, requiring that the host in any case reveals a loser, and is to offer a switch to his other unchosen door. So when the player originally picked a loser in 2 out of 3, hence the prize is hidden behind the host's doors, switching will definitely win. This is because the host has no choice then but to reveal the remaining second goat and to offer a switch to the car. This ensures that a player who swaps wins the car in any such 2 in 3. However if, contradictory to the standard rule and outside the intended paradox, the host just opened a door at random and revealed a goat simply by chance, the overall outcome staying:switching would have been reduced from 1:2 according to the intended paradox, to only 1:1 as per the common intuitive wrong answer. Because half of the potential chances to win by switching are wasted when the host can accidentally reveal also the car. Leonard Mlodinow says: The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated. But the host is fixing the game. (Mlodinow 2008). The 2/3 - 1/3 answer depends crucially on these (standard) assumptions about the host's behaviour.

_____________________

Why not boldly say so, Adams was not afraid to tell about his learning curve and to say it that way also, in his unabashed column. He finally realized that the host in effect says: you can keep your one door or you can have the other two doors, one of which (a non-prize door) I'll open for you." He finally realized that the two different doors are but not 1:1. Basing on Adams' learning curve, it would be good to present not only variants of unknowns but, just from the beginning and following Adams, the clean intended standard paradox also. Adams did not hide it, so why not telling about in the article, also. We should not forget about the readers. Gerhardvalentin (talk) 15:24, 17 April 2013 (UTC)

Again, what you write is (I think) too long for the lead, and your language is flowery and polemic, not factual and encyclopedic. You are trying to argue with the reader, to convince them, not just to communicate information. Mlodinow, or Cecil Adams, or whoever, can write in such a way: they are writing from their own point of view, and they want to convince. Wikipedia isn't a newspaper opinion column, it isn't a popular text book.

I think that the best way to (continue) to improve the article now is (a) making sure we everywhere have careful and complete citations and references and (b) cut out some of the dead wood: I think that at least two sections later in the article are superfluous for various reasons (repetition, out of place, specialistic) and one is in the wrong place. What you want, Gerhard, is somewhere near the beginning of the body of the article a kind of "total solution" explaining everything in just one way. You seem now to like a version of the "combined doors argument". Unfortunately, this argument is logically incomplete. Mlodinow and Adams don't even notice that. Devlin (professional mathematician) did, fortunately. Something which doesn't bother popular readers but does bother everyone who is trained to dissect arguments very carefully and notice where the arguer is jumping to conclusions. Richard Gill (talk) 08:17, 18 April 2013 (UTC)

The lead

From WP:LEAD:

The lead section (also known as the lead, introduction or intro) of a Wikipedia article is the section before the table of contents and the first heading. The lead serves as an introduction to the article and a summary of its most important aspects.

The lead should be able to stand alone as a concise overview. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points—including any prominent controversies. The emphasis given to material in the lead should roughly reflect its importance to the topic, according to reliable, published sources, and the notability of the article's subject is usually established in the first few sentences. Apart from trivial basic facts, significant information should not appear in the lead if it is not covered in the remainder of the article.

The lead is the first part of the article most people read, and many people only read the lead. Consideration should be given to creating interest in reading more of the article, but the lead should not "tease" the reader by hinting at content that follows. Instead, to invite reading more the lead should be written in a clear, accessible style with a neutral point of view; it should ideally contain no more than four paragraphs and be carefully sourced as appropriate.

I think we should rewrite the lead. Drop the references to the prehistory of MHP and Quine's taxonomy of paradoxes ("veridical"). Include an authoritative concise solution including why it matters that the host knows where the car is located. Richard Gill (talk) 15:42, 15 April 2013 (UTC)

I have boldly rearranged the material in the lead, and added more of the solution and an argument for the solution. I didn't remove "veridical" and the prehistory yet. Someone else might like to do so. Richard Gill (talk) 12:34, 19 April 2013 (UTC)

Reasons for Notability of this Word Problem

The MHP should be called a word problem, not a probability puzzle, because it does not define any probabilities. The popularity of the MHP is its main notability.

Its paradoxical aspect arises from its ambiguity as a word problem. It is not notable, or even paradoxical, in probability theory or logic theory.

Presumably, its popularity arises from its entertaining choice of words, its paradoxical qualities, and its context of the Savant claim to maximum intelligence: so these reasons should be noted.

The MHP may be notable if most people who argue for no advantage to switching provide the wrong justification: but is this supported by a reliable source?

Further notable would be if it were also reliably established why most people make this error. For example, it seems plausible that most people suspect Monty Hall to be malicious (making the offer only when they guess the car location correctly), but do not articulate this in their conscious explanations (for reasons of politeness, including the advantages thereof).

The MHP may also be notable if the underlying effect can be generalized to a principle that commonly occurs in real life situations. For example, do people commonly unconsciously underestimate the kindness of others, and thereby make disadvantageous decisions? DRLB (talk) 17:36, 17 April 2013 (UTC)

There are many sources which make clear that most people give the wrong answer to the problem even when it is stated to them in the most unambiguous form possible. And people do tend to use probability in their (wrong) answer: the car is equally likely behind either closed door therefore I would't switch. You can already find this in Marilyn vos Savant's original four columns in Parade, 1990-1991: the debate with her readers is about probability, not about misunderstanding the "rules of the game". Further support is given in the paper by Kraus and Wang, who are cognitive scientists or psychologists and who analyse why people's (probabilistic) intuition goes wrong and how people do eventually come to understand that their initial answer was wrong. And *understanding* doesn't come from a formal mathematical analysis or from a simulation experiment: it comes from visualizing the problem in a different way. The problem is a trick problem, of course. The build-up story makes you strongly visualize three doors, one (number 3) open with a goat, two (numvers 1 and 2) still closed. The visual detail makes you forget how it came about that exactly doors 1 and 2 are still closed. It was very asymmetric, it depended on the sequence of two person's actions, where the host's action is influenced by his knowledge of the location of the car.

Does this mean that the lead should give more references and more explanatiom of the notability? Richard Gill (talk) 06:33, 18 April 2013 (UTC)

On the one hand: if people have the exactly same error rate with Bertrand's box paradox, or perhaps the Three Prisoners' Problem, then I would agree that error would be caused purely by error in probability reasoning, and I would suggest that each problem is equally technically notable to probability theory. At some point, various re-wordings of the same logical problem are no longer notable, unless they discovered independently, or become more widely known, which may be the case MHP. Also, the reasoning error common to each problem probably deserves its own article since it is common to three other articles, although maybe that is OR. On the other hand: if the visualizations in MHP, as you suggest, or even something else about its wording, cause more frequent errors in reasoning in MHP than the other two logically-equivalent problems, then MHP is also notable not just for the common erroneous probability reasoning effect, but also for the additional effect of interpretation of words (and perhaps hidden assumptions about the rules of the game). DRLB (talk) 15:06, 18 April 2013 (UTC)

Regarding effect of the wording: read Kraus and Wang! They systematically investigate this through careful experimentation. Notability: vos Savant made the problem world famous. Every major newspaper in the world carried articles about the controversy which raged in her column. For a year. Though now 23 years have passed, the shockwaves are still reverberating. Follow-up articles every few years; most recently just one or two years ago "Die Zeit" had an article reporting a completely new kind of proof (completely non probabilistic, but decision theoretic instead) by Sasha Gnedin. MHP features prominently in every popular book on brain-teasers and every popular book on understanding probability and statistical reasoning. All this simply didn't happen with the Three Prisoner's Problem. And Bertrand's three boxes is purely of academic and historical interest. So: the three doors problem became part of folk-lore, part of pop culture. Richard Gill (talk) 06:08, 19 April 2013 (UTC)

DRLB, this is not a word problem. The exact question is indeed not precisely stated but this is not the reason that most people get it wrong. The ambiguity, however, is a point resorted to by those who do get it wrong. We have several sources showing that most people do understand Whitaker's statement to have the generally accepted meaning, but still get the answer wrong. Martin Hogbin (talk) 22:02, 19 April 2013 (UTC)

So the reason for DRLB's comment is that he gets it wrong? And the reason for Martin Gardner's following statement in the New York Times 1991 was that he got it wrong: The problem is not well-formed, unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero.? And the same for Persi Diaconis: The strict argument would be that the question cannot be answered without knowing the motivation of the host.? And for Monty Hall himself: If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.? And what about plenty of sources, who claim the 2/3 solution, but don't understand the problem? And why should people make assumptions which are only necessary for the 2/3 solution, when claiming the 1/2-solution? And what do you think about the woman who said: I should switch, because the host is not screwing me? And about the teacher in the film 21, who says to the student who claims the 2/3 solution: But the host may screw you.? And about those sources, who say now that of course a real show does not run with the rules which lead to a 2/3 solution? The only reason for the everlasting discussions about MHP is that publicists propagate a combination of task set and solution which is wrong. And if we resolve the little wording problem - which should be done within minutes, not within decades - we see that the whole furor is not based on a paradox but on a joke.--Albtal (talk) 21:33, 20 April 2013 (UTC)

Of course it is always possible to invent bizarre rules (this is often done by people who got the answer wrong and want to show that they got it right really) but vos Savant says that nearly all of her respondents understood the question to have its standard form (that the host always reveals a got behind an unchosen door and always offers the swap). Krauss and Wang also confirm that most people see the problem this way. The problem is interesting because, even when the problem is put clearly and unambiguously, most people still get the answer wrong, and not only that they persistently refuse to believe the right answer. Martin Hogbin (talk) 22:37, 20 April 2013 (UTC)

These are never-ending discussions. Why do they keep arising? Because Whitaker's words are quoted out of context. Out of context, they are ambiguous. The context includes Vos Savant's answer, and that answer makes the intended interpretation completely unambiguous.

I propose we add Vos Savant's answer, quoted literally, to both the lead to the article and to the body of the article.

It's amusing to compare Vos Savant's so-called quotation from Whitaker's letter with the actual text, which is quoted in Morgan et al's response to Hogbin and Nijdam. That perhaps belongs also somewhere in the article! Richard Gill (talk) 10:54, 21 April 2013 (UTC)

I think it is an excellent idea to quote vos Savant's answer in the body of the article. I think putting it in the lead is too much. Martin Hogbin (talk) 13:21, 21 April 2013 (UTC)

I added Marilyn's original answer, and also, in the history of the problem, Craig's original question. Richard Gill (talk) 11:45, 25 April 2013 (UTC)

Dead wood

How about removing some dead wood from the article? "Less is more"! I propose deleting the sections "Bayes theorem" and "From simple to conditional by symmetry".

The first (Bayes theorem) is all about notation and formalism, not about understanding, not about MHP. It duplicates a section in the wikipedia article on Bayes Theorem, where MHP is used as an illustration of the theorem. A link is enough, and the explanation "MHP is often used un introductory probability and statistics texts to illustrate Bayes Theorem".

The second (from simple to conditional by symmetry) contains a rehashing in more formal terms of what has been said in words elsewhere in the article and is a relict of the days of the big fight between supporters of "simple solutions" and those of "conditional solutions" (it was intended to make a bridge behind these two kinds of approaches).

I think the section on simulations should be moved to the end of the list of simple (popular) solutions. It's important and currently out of place.

And there is lots of work to be done adding inline references (hyperlinks) Harvard style. Richard Gill (talk) 09:04, 18 April 2013 (UTC)

Yes, the odds formulation is far more insightful! There really isn't much lost if you remove that routine computation. However, you may consider creating a subarticle with Solutions to the Monty Hall problem, so that less important arguments could be moved there. Vesal (talk) 12:31, 18 April 2013 (UTC)

I think subarticles are not approved of by Wikipedia. But for a start, these sections could be moved to the article's subsidiary talk page, "Arguments". Richard Gill (talk) 06:10, 19 April 2013 (UTC)

OK, so shall I delete the sections I mentioned, and move the simulations section to between the simple and the conditional solutions? Richard Gill (talk) 17:15, 21 April 2013 (UTC)

I have done it. Richard Gill (talk) 20:14, 21 April 2013 (UTC)

Bayes' theorem: Unencyclopaedic?

It would be great if the concerns about this section could by spelled out. It's not entirely obvious (not even by symmetry) what is wrong with this section. There are many things, such as MOS:YOU, that can bother a highly sensitive person, but it would be stupid to address concerns that are only in my own mind. Vesal (talk) 21:29, 23 April 2013 (UTC)

"...does have a big impact on the chances for doors 2 and 3 themselves!..."
"...tells us that we have to be very careful..."
Do these examples sound like encyclopaedic text? Th4n3r (talk) 22:30, 23 April 2013 (UTC)

I deleted the offending sentences (they seem quite superfluous). And changed "we" to "the contestant". Improved some more of the text and removed the "clean up (unencyclopaedic)" tag. Anyone who thinks more clean up is needed, put the tag back if you like, but please also give further specific suggestions (or do it yourself). Richard Gill (talk) 06:14, 24 April 2013 (UTC)

As per MvS we know that the host reveals a goat but never the car, and just illustrated that "somewhere else" in The problem section. Gerhardvalentin (talk) 10:50, 25 April 2013 (UTC)

It's simple really

This is a trivial statistical problem surrounded by a great deal of confusion.

When looking at 3 doors, the odds of picking a winner are 1 in 3.

When one door is opened, you now have 2 doors with no data at all as to which is the winner, thus the odds are now 0.5 for each door. Thus it makes no difference whether one switches doors or not. Statistical probability figures are nothing more than reflections of the data they're based on, and opening one door changes the available data. Missing that point is where so many are going wrong.

Other points where people go wrong on this are:

A computer simulation proves nothing because we don't know what was programmed into it - the output is merely a reflection of the views of the programmer.

The IQ and experience of a mathematician proposing an answer also does not establish it to be true, everyone makes mistakes. It seems to be a popular myth that high IQ and specialist subject knowledge immunise people from that.

In the end the problem is simple O level stuff. 82.31.66.207 (talk) 11:35, 29 April 2013 (UTC)

You just showed the weakness of the lousy article. Although there newly is the last paragraph in section "The problem", reading:

Leonard Mlodinow says, "The Monty Hall problem is hard to grasp, because unless you think about it carefully, the role of the host goes unappreciated." {Mlodinow 2008) The correct answer 2/3 - 1/3 is based on the premise about the host's behavior. The host knows which door hides the car, so the host will always reveal a goat and never reveal the car. If the player initially selected the door that hides the car (a 1-in-3 chance only), then both remaining doors hide goats, the host may choose either door, and switching doors loses. On the other hand, if the player initially selected a door that hides a goat (a 2-in-3 chance), then the host has no choice but to show the other goat, and switching door wins for sure.

Please read this paragraph and say whether, from your point of view, it can help to understand the clear premise that this world-famous paradox is based on.

And another question: Could it be of help to understand the significance of the premise / rule, if there was added:

However if, contradictory to the premise of the standard paradox, the host just opened a door at random and revealed a goat simply by chance, the overall outcome staying:switching would have been reduced from 1:2 of the "paradox", to only 1:1, as per the common intuitive wrong answer, because half of the potential winning cases are wasted when the host can accidentally reveal also the car.

It would be of advantage to hear your comments. Thank you, Gerhardvalentin (talk) 12:30, 29 April 2013 (UTC)

82.31.66.207: your comment is the natural initial reaction of almost everyone (including myself) on first hearing about the Monty Hall problem. The question which is asked (switch or stay?) is in fact a trick question. You were successfully fooled (like almost everyone, including myself). That is why so much has been written on MHP and why the article is so long and complex. Perhaps you should carry out the simulation experiment yourself, with playing cards or cups, not with a computer, in order to find out for yourself whether switching or staying is the smart thing to do. Richard Gill (talk) 07:51, 2 May 2013 (UTC)

Richard Gill! You cannot simulate (repeat) the content of the letter, because it describes a one-time-game resp. a one-time-situation. A repetition or simulation is possible only if you have rules for it. But there are no rules at all defined in the letter. So you were successfully fooled! But you are in "good" company because many mathematicians were fooled too. --94.68.139.15 (talk) 15:15, 5 May 2013 (UTC)

These are the rules: The host hides a car behind one of three doors. The player chooses a door. The host opens a different door to the door chosen by the player and reveals a goat. The player may now choose either to switch or stay.

Two people can now play this game, e.g. with three cups and a coin. Compare the success rate got by staying every time, with the succes rate got by switching every time. Richard Gill (talk) 09:46, 6 May 2013 (UTC)

Where are the rules from? They are not defined in Savant's letter of the problem. So her answer isn't correct. --94.68.139.15 (talk) 15:14, 6 May 2013 (UTC)

The rules are implicit in the question and answer in vos Savant's first Parade article on MHP in 1990 and they are spelt out explicitly in her subsequent three Parade articles 1990-1991 in which she discussed the problem with correspondents (you can find the four articles on internet, see links in the wikipedia article). More than 30 years ago. Since then a hundred papers and books have been written on the problem. All with these same rules as starting point. Richard Gill (talk) 18:23, 6 May 2013 (UTC)

Richard, you are wrong. Fact is, that in almost all publications the rules which lead to a 2/3 solution are not part of the task set. And as I already wrote, most people now think that the pure fact that the host opens a not chosen door with a goat leads to a 2/3 solution. And even Krauss in 2004 mentions the necessary rules only in a footnote after having fooled the readers. And All with these same rules as starting point? - Look at the actual wikipedia article! And look at thousands of other sources! - You answered with fog and mist to a clear argument.--Albtal (talk) 19:08, 6 May 2013 (UTC)

Kraus and Wang don't specify the rules in a footnote. In a footnote they merely state that they consider the Whitaker question as formulated by vos Savant to be "the standard version" of the problem. On the second page of their article, after the introductory section, they write "These discussions [the media debate 1990-1991] have verified vos Savant’s conclusion that the mathematically correct solution is for the contestant to switch, providing that the rules of the game show are as follows: Monty Hall has in any case to reveal a goat after the contestant’s first choice, and he cannot open the door chosen by the contestant." Richard Gill (talk) 11:09, 10 May 2013 (UTC)

For Krauss 2004 see Krauss/Atmaca with a nice supplement in the task set. --Albtal (talk) 18:43, 11 May 2013 (UTC)

Thank you. This confirms my statement. Kraus and Atmaca 2004 write "Gelegentlich wird argumentiert, dass die Regeln in der Problemformulierung nicht genügend präzise formuliert seien (zB dass nicht klar sei, welche Tur der Moderator offnen müsse, wenn der Kandidat zuerst auf das Auto gezeigt hat). Wechseln ist nur dann die richtige Losung, wenn die Spielregeln lauten: 1. Die Wahrscheinlichkeit, dass das Auto hinter Tur 1, 2 oder 3 steht, ist anfangs für alle Türen gleich groß; 2. Der Moderator öffnet immer eine Tur, die nicht die Erstwahl des Kandidaten war und hinter der sich kein Auto befindet. Es stellt sich allerdings heraus, dass genau diese intendierten Regeln von nahezu allen Versuchspersonen implizit vorausgesetzt werden (vos Savant, 1997; Krauss & Wang, 2003)." In other words, almost all people interpret the rules correctly.

What you cite here is exactly the footnote which I mentioned above wording: And even Krauss in 2004 mentions the necessary rules only in a footnote after having fooled the readers. And the last sentence of the citation is a joke. --Albtal (talk) 20:04, 15 May 2013 (UTC)

I don't know what publications you consult, Albtal. I have read all the publications cited in the article, and many more, and my conclusion is that for most writers and most readers (especially native English writers and readers) the rules are perfectly well understood, whether spelt out explicitly or not. Some "newcomers" are confused, starting already in 1990 and continuing to the present day, and this continuous process of say 5% of people doggedly misunderstanding the problem leads to a continuous level of 5% pure noise; annoying but one gets used to it. I haven't studied e.g. the German language literature. Maybe it's different there. MHP has a long prehistory. You could say that it's "just another version" of Bertrand's three boxes, more than a century old, and more recently of Gardner's three prisoners, more than half a century old. The difference is that this time, somehow Selvin's 1975 reformulation got talked about in academic circles, see Nalebuff 1987, and then with vos Savant, 1990, it hit the popular imagination. Like some new flu virus jumping from goats to people. Richard Gill (talk) 07:07, 7 May 2013 (UTC)

There are many publications which doesn't spell out explicitly the rules needed for Savant's solution, for example Carlton (2005). I think the article wants to fool the reader by giving a wrong solution (2/3) and insisting on this solution against the mathematical and logical ratio. What a pity! --94.68.156.45 (talk) 20:03, 7 May 2013 (UTC)

Carlton's solutions - he first gives the standard "full" conditional probability solution using a tree diagram and later, as an afterthought, a simple solution to help intuition - both make perfectly clear what he is assuming, if you are familiar with his language and his mathematical tools. He is writing for educationalists, people teaching probability theory, and he is writing in 2005, 15 years after vos Savnat, at a moment when 95% of the whole world knows the standard problem assumptions, the standard mathematicians' solution, and the standard popular solutions. It is amusing that the article uses him to support a particular popular solution whereas his stated aim is to advise teachers to get students familiar with tree diagrams in order to systematically solve this kind of probem by conditional probability calculations. Richard Gill (talk) 08:29, 9 May 2013 (UTC)

I hope you agree that there are many publications which *do* spell out the rules needed for the 2/3 solution. To start with, vos Savant's own four columns on the problem. Before that, Selvin who invented the problem and Nalebuff who further popularized it (between Selvin and vos Savant). I hope you agree that given those rules, 2/3 is the right answer. Most sources give the 2/3 solution. Some are good some are bad, it sees you only met bad ones so far. The wikipedia article spells out the rules and explains the solution which is based on those rules, following the published literature. Pity if you disagree with that literature. Richard Gill (talk) 16:35, 8 May 2013 (UTC)

The wikipedia article doesn't spell out the rules neither in the lead nor in the section "The problem" before saying the 2/3-solution is right. This is fooling the reader and that I disagree with really. --94.68.194.35 (talk) 17:40, 8 May 2013 (UTC)

94.68.194.35, you are quite right the article no longer seems to state the 'standard' assumptions in the lead or in the section 'The problem'. It used to and someone has removed it, I have no idea who or why. It needs to be restored.

Seems to be there now. True, the 2/3 solution is announced *before* the standard assumptions are given explicitly. But the article doesn't say that the 2/3 answer is right. It says that so and so gave the 2/3 answer. Wikipedia doesn't report the truth. It reports what people have said in sources which are commonly considered reliable. So if most reliable sources give a wrong answer, that is what has to be in wikipedia. Richard Gill (talk) 08:34, 9 May 2013 (UTC)

You are quite right Richard it is there, in both places if you look. I have clarified the wording, I hope. I think some way of clarifying the 'standard' problem before we give an answer would be good though. Martin Hogbin (talk) 17:35, 9 May 2013 (UTC)

@94.68.194.35: We should no longer waste our time here.--Albtal (talk) 12:53, 9 May 2013 (UTC)

@82.31.66.207: I think that the following variant of the game which I posted earlier contains all essential points of MHP:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. The host knows the door with the car. Instead of simply choosing a door you have the choice between two "jokers":

1. You may ask the host to open one door with a goat before you choose your door.

2. You may determine two doors of which the host has to open one with a goat before you choose your door.

Which joker should you choose? --Albtal (talk) 10:41, 3 May 2013 (UTC)

I think we have answered this one before Albtal. Martin Hogbin (talk) 23:06, 8 May 2013 (UTC)

Maybe other readers including 82.31.66.207 understand the placement of my comment within this section. --Albtal (talk) 15:40, 9 May 2013 (UTC)

I like this variant as a pedagogical device. Maybe it could be in a section "why the answer is not 1/2". Is it published somewhere?

In this context I refer also to Whitaker's actual question to vos Savant: "I’ve worked out two different situations (based on Monty’s prior behavior i.e. weather [sic.] or not he knows what’s behind the doors) in one situation it is to your advantage to switch, in the other there is no advantage to switch. What do you think?" Here Whitaker refers to a third variant: Monty opened a different door to the door chosen by the player at random and just happened to reveal a goat.

Clearly we need a conditional probability now! It's 1/2, given that the host revealed a goat. If Monty accidentally revealed the car then the player would of course switch to the open door, not to the other closed door, to get the car. The conditional probability of getting the car would be 1. The unconditional probability of winning (following an optimal strategy) is 2/3 = 2/3 x 1/2 + 1/3 x 1.

Amusingly, this (unconditional) 2/3 win chance can also be obtained, correctly, from the "combined doors" solution. If the host opens another door at random and offers the player the opportunity to switch to any door he likes, it is exactly as if the host is offering the player the choice of exchanging the originally chosen door for the two others, combined, one of which he is so kind as to open in advance. This is a beautiful illustration of the fact that the "combined doors solution" only gets the right answer by luck. It misses a step in the argument, which fortunately can be patched in the original MHP. But obviously can't be patched in the Monty Fall problem. Richard Gill (talk) 09:13, 11 May 2013 (UTC)

Should we have a 'Why the answer is not 1/2 section'?

We used to have an 'Aids to understanding section'. We have yet another reader telling us that the answer is 1/2 which, as Gerhard points out, means that the article is still not clear enough. The problem is that everybody has their own pet way of explaining the correct answer and we cannot have them all in the article. An alternative might be to have a separate section on why the answer is 2/3 and not 1/2. This might be a little more relaxed about sourcing (but still within WP rules, of course). Martin Hogbin (talk) 17:35, 9 May 2013 (UTC)

Jef Rosenthal's "Monty Hall, Monty Fall, Monty Crawl" is a good source.

But I disagree that "yet another reader telling us the answer is 1/2" means that the article is not clear enough. The whole point of the Monty Hall problem is that everyone thinks that the answer is 1/2. It's a trick question. So of course there will always be new readers who are "tricked" and get angry because they've been tricked. Richard Gill (talk) 09:02, 11 May 2013 (UTC)

Yes, the problem is remarkably unintuitive ('trick question' is not the right terminology) but that does not mean that article must be be. It may be hard, or even impossible, for the article to convince everyone who reads it of the right answer but I do not see why that should not be one of the articles objectives. If a reader does not believe the answer given in the article the rest of the article will be ignored. Martin Hogbin (talk) 09:35, 11 May 2013 (UTC)

I boldly made some small changes to the lead to emphasize the "usual assumptions". The lead has Marilyn's question, quoted literally, but not her answer, quoted literally. Her answer underlines the assumptions implicit in her question. Perhaps it would be beter still if Marilyn's wording was not in the lead at all. Richard Gill (talk) 13:46, 11 May 2013 (UTC)

Savants letter

Cited from Tierney (1991):"And although Mr. Hall might have been violating the spirit of Ms. vos Savant's problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant's wording of the problem with the versions they had analyzed in their articles." So you have to distinguish between Savant's problem and the versions which were created by other authors. The ambiguity of Savant's wording should be explained in the lead of the article. --94.68.139.15 (talk) 18:44, 10 May 2013 (UTC) P.S.:The link of Selvin (1975a) is dead! — Preceding unsigned comment added by 94.68.139.15 (talk) 18:45, 10 May 2013 (UTC)

The Selvin (1975a) link is http://www.jstor.org/stable/2683689 . Perhaps it would be better to have a concise, unambiguous description of the problem in the lead, rather than vos Savant's words. You need to see her answer alongside her problem before you can see what problem she is solving (whether or not you agree with her solution). Why don't you go ahead and make a change? Or propose one on this talk page? Richard Gill (talk) 08:57, 11 May 2013 (UTC)

Selvin, too, does not mention the crucial rule which leads to a 2/3 solution, neither in his first nor in his second letter. In his second letter he only emphasizes that the host knows the position of the keys, and that he chooses between the empty boxes at random, if he has a choice. But for the 2/3 solution to derive it is a necessary condition that he shows a not chosen empty box absolutely independently of the result of the (first) choice of the contestant. But as the host in this game is proceeding tactically all the time: Why should he not base his decision on the result of the choice of the contestant? And if the host, as in Marilyn vos Savant's variant, offers a switch, he will naturally do this if he is not obligated by the rules of the game.

Without these rules the solutions in both of Selvin's letters are wrong, and the crucial error, which is a constant companion of MHP, is hidden in the assumption, that in the case he decides to show an empty box he shows this box with probability 1 if the contestant had not chosen the keys, and with probability 1/2 (normal assumption) if he had. But these probabilities are wrong if the "knowing host" in his decision takes account of the result of the (first) choice of the contestant.--Albtal (talk) 16:40, 11 May 2013 (UTC)

The analysis in Selvin's first paper assumes that the location of the car and the initial choice of the player are independent of one another and completely random. It assumes that whatever the combination of these two, the host opens a different door to the door chosen by the player and reveals a goat. He shows that the (unconditional) probability of winning by switching is then 2/3. The analysis in the second paper assumes furthermore that either of the host's choices are equally likely when he has a choice, and shows that the conditional probability of winning by switching given door chosen by player and door opened by host is then 2/3. Selvin works throughout under the standard rules: the host always will open a different door to the door chosen by the player, reveal a goat, and offer a switch. "Always" means "wherever the car is hidden, and whatever door is chosen by the player". So what's the problem?

Selvin is clearly not a very good writer. But a good reader can easily figure out what he means. The maths is completely transparent.

We obviously read different letters.--Albtal (talk) 19:17, 11 May 2013 (UTC)

Maybe we read different parts. I skim through the introduction and problem statement, and study the solution carefully. I suspect you do the opposite. Richard Gill (talk) 06:26, 12 May 2013 (UTC)

And here is Nalebuff (1987); post Selvin, pre vos Savant; most likely Whitaker heard about the problem from Nalebuff's article: "The TV game show Let's Make a Deal provides Bayesian viewers with a chance to test their ability to form posteriors. Host Monty Hall asks contestants to choose the prize behind one of three curtains. Behind one curtain lies the grand prize; the other two curtains conceal only small gifts. Once the costumed contestant has made a choice, Monty Hall reveals what is behind one of the two curtains that was not chosen. Now, Monty must know what lies behind all three curtains, because never in the history of the show has he ever opened up an unchosen curtain to reveal the grand prize. Having been shown one of the lesser prizes, the contestant is offered a chance to switch curtains. If you were on stage, would you accept that offer and change your original choice?" Richard Gill (talk) 18:55, 11 May 2013 (UTC)

And now Nalebuff, 12 years after Selvin? - I even assume that there were letters to Selvin in 1975 which corrected the rules; as it was in 1990 to Marilyn vos Savant.--Albtal (talk) 19:17, 11 May 2013 (UTC)

Selvin (1975b) is written in response to the letters which he received after Selvin (1975a). Notice that in 1975b he states and uses the assumption that the host makes a random choice when he has a choice, and he computes the conditional probability by straightforward application of the definition. In 1975a he computes the unconditional probability and hence does not need that assumption.

Selvin and Nalebuff are academic writers writing in academic journals for specialists who have no difficulties with the mathematics and the concepts from probability theory. Their readers are all familiar with Bertrand's three boxes and Gardner's three prisoners and automatically solve these problems by an application of Bayes theorem, as it has been done in introductory probability texts for 100 years. The difference is that vos Savant wrote a famous column in a popular weekly family magazine. One might even suspect that she deliberately was extremly brief in problem statement and problem solution in order to provoke reactions. She must have loved the media attention. Richard Gill (talk) 06:26, 12 May 2013 (UTC)

Perhaps she even deliberately gave the wrong explanations. Nijdam (talk) 18:12, 12 May 2013 (UTC)

We know that she instinctively identified the three doors problem with the three cups problem. This means that she instinctively used symmetry up front to simplify the problem, before calculating Bayesian (subjective) probabilities (i.e., also derived from symmetries) relative to the reduced problem. I think that is perfectly legitimate, especially in the context of a popular weekly family magazine. Richard Gill (talk) 09:13, 13 May 2013 (UTC)

The 'We' is pluralis majestatis? Nijdam (talk) 14:15, 13 May 2013 (UTC)

I think this is obvious. She proposed the three cups "simulation" as a way to understand MHP. I expect that most editors here who read vos Savant's Parade articles would see that too (i.e., see tha she proposes to solve three doors by reduction to three cups), that's why I write "we". There may be some lonely exceptions. Richard Gill (talk) 19:35, 14 May 2013 (UTC)

We know that her intent was to present the pure and simple paradox (two doors, but instead of 1:1 now surprisingly 1:2). "This (her simple) solution is actually correct" as per S. Rosenthal in Monty Hall, Monty Fall, Monty Crawl, Math Horizons, September 2008, pages 5{7.) And actually correct as per Falk, because the Monty Crawl assumption e.g. is never applicable to the scenario of the pure paradox, as we do not "know" about any such given one-sided bias. Falk explicitly says that we must "know" about such given bias.

Rosenthal, writing for mathematics teachers, says that the number 2/3 is the correct value but that vos Savant's method of solution is wrong, because, in his opinion, you must compute the conditional probability of winning by switching given door chosen and door opened, not the unconditional probability. He shows that the method is wrong by showing that it gives the wrong answer if you change the problem a bit. Simple logic. But apparently difficult to understand.Richard Gill (talk) 12:20, 20 May 2013 (UTC)

Falk (1992), writing for maths educationalists and cognitive scientists, insists on computing the conditional probability. She criticizes vos Savant's three shells (three cups) solution. Apparently, for Ruma Falk, shells are distinguishable and we might know about host bias. But the point of the three shells analogue is that the player cannot distinguish between the shells (it is inspired by a famous street confidence trick and conjuoring trick). Richard Gill (talk) 12:48, 20 May 2013 (UTC)

The article still does not distinguish clearly enough between the pure paradox where any value deviating from 1:2 must inevitably be "wrong", and quite other (interesting) scenarios where the pure paradox will never arise. In treating deviant imaginable scenarios, the article should clearly say that they never concern the pure paradox.

The section "The problem" should better read "The paradox" and also present the 1:1 result of the forgetful host, just to help the reader to clearly understand this difference, just from the beginning. Gerhardvalentin (talk) 17:23, 13 May 2013 (UTC)

"Paradox" is maybe a better word than "problem". I disagree that the article is unclear about this. But go ahead and boldly make it better, if you think you can. Richard Gill (talk) 19:38, 14 May 2013 (UTC)

Okay, I just tried to do that. Please help to improve the wording. Gerhardvalentin (talk) 11:51, 15 May 2013 (UTC)