Talk:Monty Hall problem

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Distinction and relation

To  clearly distinguish between different directions of view   really could encourage perspective and perceptibility of the article.
In treating the sources, different aspects should not indistinctly and confusingly be mixed with one another:

• The 2/3 "overall" probability, being an unchangeable and unvarying constant, not modifiable by Morgan.
  And also Morgan not even varying the probability of every definite event, as long as
• no unexpected (if needed be assumed) "additional information" whatsoever about the actual constellation could eventuelly be flashing up (potentially revealed by some assumed hypothetic bias of the host, e.g.),  and by that providing for a new "condition" then, telling more about the actual  constellation only, varying the  "2/3"  to max. 1/1 (in 1/3 of cases) and to min. 1/2 (in 2/3 of cases), confirming the overall 2/3.
  

This is made far too little clear, evident and obvious.  And the article suffers from that, from the beginning to the end. Important: To clearly distinguish different viewing directions of the sources could really help this article, elucidating what it's all about.  Regards,   --Gerhardvalentin (talk) 11:37, 13 July 2010 (UTC)

Especially: The MHP often is misinterpreted as being "a problem solvable by mathematics only". But the real problem is: Unclear and differing arbitrary suppositions and unclear assumptions yield to confusing dramatics. By clearly showing and distinguishing first the different underlying assumptions/points of view, above all, the specific answers/response will be obvious, then.

E.g. Morgans assumption of an extremely biased Host, in one third of the games offering a clear 1/1 chance by switching, and, by doing so, in that third of the games showing the contents of all three doors altogether, then. - Such an assumed host's telltale-bias obviously never having been intended in the "Ask Marilyn" column. --Gerhardvalentin (talk) 11:19, 16 July 2010 (UTC)

Any "hypothetic telltale-behavior", allegedly "telling a more accurate picture of the current situation", unable to show that switching ever can be worse, never was of assistance for answering the fundamental question of the MHP, but just remains a crooked "hypothetic assumption" and a meaningless illusion. An illusion not addressing the MHP, but just addressing students in prob.theory. Without any sound relevance for the MHP. The article broadly suffers from that crooked illusion and from "passing it off as the MHP". Please help to improve and to distinguish facts from not helpful hypothetic assumptions and not helpful illusions. Gerhardvalentin (talk) 15:53, 21 July 2010 (UTC)

Sure, he could. If he can communicate things to the contestant, why can't he just shake his head, meaning 'no - don't do it', when he offers the switch, when the contestant has chosen the car? It makes as much sense as the contestant being aware of any 'host bias' on a game show. Glkanter (talk) 16:59, 21 July 2010 (UTC)

Right. But the topic in the "Ask Marilyn" column obviously and evidently did not provide for any such "additional telltale information" given away by the host's behavior. Quite the contrary: the "Ask Marilyn" column clearly and without any ambiguity implying a strict "no telltale behavior/bias whatsoever". That's one thing. Even more important: No host's crooked bias will ever be enough to "make switching worse" in any game. Thus, considering a "biased host" is a totally worthless suggestion indeed, as switching never is disadvantageous: You should switch. The article suffers from not distinguishing relevant views on the fundamental question from a worthless endeavor, "passing it off as the MHP".   Regards, Gerhardvalentin (talk) 12:04, 22 July 2010 (UTC)

I'm afraid those 'worthless endeavors' are reliable sourced. They should be included where appropriate, per Wikipedia policy, without editorial comment. They should not be used to disparage other viewpoints, however. Glkanter (talk) 13:30, 22 July 2010 (UTC)

Of course not, you're right. But the article should, from the beginning to the end, make clear what it's all about, clearly distinguishing the various aspects. The article is nebulizing e.g. by misapplying the "conditional" on the "number" of the door opened, instead on the underlying "assumed host's bias" when he opens that door. Not accounting for (assuming an unbiased host) opening of a door means "NO new condition". But presenting excessive math homeworks in probability theory, instead of just clearly showing all respective different underlying assumptions. And lacks to present the clear result of any trial:
Assumed initially all three doors are equally likely, the guest cannot strictly improve the overall success-rate by staying instead of switching, in each case separately. The probability for the car behind the second door, assumed "no host's bias whatsoever", is 2/3, and even with an assumed "extreme host's bias" will always be at least 1/2, and she'll win unconditionally on average 2/3 of the time.   Regards, Gerhardvalentin (talk) 14:28, 22 July 2010 (UTC)

You're preaching to the choir, I'm afraid. Glkanter (talk) 15:29, 22 July 2010 (UTC)

Thank you, I'm hoping "not only to the choir", too, commending the words of Falk: "the correct solution depends on explication of underlying assumptions". And progress in the effort to clarity in dealing with statements of the sources is noticeable. Especially words of the last days were a relief indeed and will help to avoid the vague blur of differences of perspective, and I appreciate: "cannot be considered as 'obvious' and simply skipped for the reader, but they require a careful explanation" . . ."each solution needs to be precise regarding what question it actually answers and under what conditions" . . ."the exact question it answers and under which conditions or assumptions. . .The issue is about not describing the exact circumstances and their meaning to the reader" e.g., and I friendly welcome this change. Regards, Gerhardvalentin (talk) 21:46, 31 July 2010 (UTC)

Rick and Glopk, one hypothetical question, I'd like to know what you think: In case some remark would have been added to Marylin's answer or even to the well-known statement of the problem that has been published in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (as per Whitaker/vos Savant 1990):
"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?"
added with the following or a similar remark (for clarification that this was what she meant to be the basis for her answer):
"Supposed/given the car is initially placed uniformly at random and the host chooses a goat door uniformly at random, if he has the choice from two goat-hiding doors, and he always offers to switch ... '
In case this clarification had originally been added by vos Savant, and if she had said: "No news, he always can open one goat-hiding door and it does not matter which one if he has the choice between two goats, because he will chose uniformly at random, so you should switch, yes it always will be wise to switch", would conditionalists nevertheless say "her answer is numerically right, but it does not reflect the conditional problem that asks for the probability P(win by switching AND player initially picks door Y AND host opens door Z)"? I would be glad if you could give your comment to this (my hypothetical) question. Thank you! Regards, Gerhardvalentin (talk) 12:21, 5 August 2010 (UTC)

Gerhard, this hypothetical properly belongs to the Argument page. I have added a comment in a new section entitled "Gerhard's question" therein. glopk (talk) 15:12, 5 August 2010 (UTC)

attempting to see the problem from another angle

I think this requires some clarification. Imho almost all people here in particular the "conditional camp" as well support the simple solution. The issue starts with passing off the simple solution of vos Savant as a complete solution to the conditional problem, that is were Martin is almost alone and that's for a good reason. Note that there is not even much of a disagreement (aside from possible sourcing issues for the moment) that the simple solution can be extended or augmented into a full solution of the conditional problem for the standard MHP (no news and symmetry arguments), but those cannot be considered as "obvious" and simply skipped for the reader, but they require a careful explanation. So far in much of his arguments Martin seems to attempt to provide the careful explanation on the discussion page only to justify passing off the original simple solution as complete in the article and that's basically bullshitting the readers hence a no-go for many other editors.--Kmhkmh (talk) 05:53, 20 July 2010 (UTC)

Well I obviously can't speak to whether any solution is complete or not. For me it has been mainly an issue with the flow of the article. I think that most of my issues are covered by the current layout. I imagine Martin will have something to say about his feelings on the solutions being complete or not. But I most likely wont be able to add anything constructive to that particular debate. I simply feel that leading with the simple and the psychology and then delving into conditional is the best layout. I also don't think there should be an attempt to minimize or discredit either view, other than pointing out what sources do so in the context of that particular source's opinon or view. Colincbn (talk) 06:26, 20 July 2010 (UTC)

The flow of the article is another point of conflict in particular between Martin and Rick, though that imho a less important issue (personally i don't really care much about the order/organization). Nevertheless it has been a constant point of conflict with no compromise in sight so far.--Kmhkmh (talk) 06:48, 20 July 2010 (UTC)

There is no special conflict between myself and Rick. There are two groups of editors here, one group which includes myself, Glkanter, Gill, Gerhard, and others, supports giving more prominence to the simple solutions; the other group, which includes Rick, Nijdam, KmhKmh, and Glopg believes the simple solutions are incomplete or answer the wrong question and that this must be made clear to all readers. Rick and I are at least willing to discuss the subject in a civil manner rather than hurl insults or edit war. All other editors are welcome to join in the discussion.

If there is a general feeling that this is a topic more suitable for the /Arguments page I am happy to continue the discussion there. All I would ask in return is that editors who wish to express an opinion on the subject do so there rather than waiting for a proposed change to the article to appear on this page. To put this another way, if you want to discuss this subject on the /arguments page then please do so, if you do not want to discuss the subject then please accept the result of that discussion.

I think the current state of the article is a reasonable compromise and follows a logical order, from simple to complex. I have a few minor issues with it. I would like to remove the 'health warnings' from the simple solutions section and improve the 'Sources of confusion' section by replacing an incomprehensible paragraph based on Falk with an agreed conclusion on the subject perhaps from other sources. Apart from that the article is looking much better in my opinion. Martin Hogbin (talk) 08:35, 20 July 2010 (UTC)

There is no right or wrong question, but each solution needs to be precise regarding what question it actually answers and under what conditions. The issue is from my perspective (might not be true for nijdam or rick) not about the prominence of the simple solution but being about fuzzy about the exact question it answers and under which conditions or assumptions.--Kmhkmh (talk) 11:20, 20 July 2010 (UTC)

I guess we agree on most things then. We all agree that the simple solution applies in the unconditional case but I assert that it also apples in the totally symmetrical (car placed uniformly, host chooses a goat door uniformly) case. We have a reference (Falk) to support this view. Perhaps something on this could be added. I also believe that, because we are asked to answer the question from the player's SoK, we must treat the host's goat door choice as uniform, thus the symmetrical case and simple solution apply to the conditional problem. Many editors agree with me. Martin Hogbin (talk) 18:45, 21 July 2010 (UTC)

Nobody disagrees with the fact, that the simple solution can be used (or "extended"). The issue is about not describing the exact circumstances and their meaning to the reader in favour of a (somewhat deceivingly) simpler looking explanation. I.e. you cannot simply give vos Savants argument or variety of it that solves the unconditional problem and then state it solves the conditional one as well without further explanations. I.e. stating explicitly which principle/assumption is used and why that forces the conditional probability to be identical with the unconditional. Doing otherwise is exactly what I meant with "being fuzzy" and it's a disservice to readers.--Kmhkmh (talk) 18:20, 31 July 2010 (UTC)

Could you explain how your approach works with Selvin's 2 letters to the American Statistician? This is where the problem is first presented and named The Monty Hall Problem. Selvin puts forth the scenario that the contestant on a game show has chosen box B, and the host has shown that box A does not contain the set of car keys. He solves it in his first letter with a simple solution. Has Selvin provided the 'wrong' answer to the question that he himself has just asked? How can that conceivably make sense? Glkanter (talk) 19:27, 31 July 2010 (UTC)

+ to Colincbn, + to Kmhkmh and + to Martin.

"What matters for reaching a correct solution to many probability problems is often not only the given information, but also the manner by which it has been obtained" (Falk)

"the correct solution depends on explication of underlying assumptions" (Falk)

"whereas the correct answer (assuming an unbiased warden) is 1/3" (Falk)   Regards, Gerhardvalentin (talk) 21:08, 21 July 2010 (UTC)

Glkanter's changes

I've reverted Glkanter's changes. There was no discussion on these while the article was protected and nothing has apparently changed. The image is still WP:OR. And deleting (as opposed to editing) the introductory paragraph at the beginning of the "Solution" section is not the prevailing consensus. -- Rick Block (talk) 13:39, 29 July 2010 (UTC)

New section

An anonymous user added the following new section. It's completely unreferenced, and as far as I can tell is simply WP:OR (including the section title).

The easiest way of proper understanding the problem

I have 150 honey-bees: 149 workers and one queen. The queen does not fly. I divide the lot into three boxes, 50 bees each. Now I choose one box at random. The chance that it contains the queen is 1/3. Now I transfer the bees from the two other boxes into a forth one, with some small stones on the bottom: the chance it contains the queen is 2/3. Now I cut the hole through the lid* - and the bees begin to fly out. When half of them flies out, the probability that the queen is still in is 2/3. Even when 99 bees leave the box the chance the queen is still inside is 2/3.

*The still more convincing way: I open the lid of the forth box and hoooosh! Plenty of bees flies out in panic. I do not see the bottom of this box - and I do not know the number of bees that left it, or perhaps return - but the probability the queen is stil in, remains unchanged: 2/3.

Important remark: it does not make any difference whether one throws away (from the fourth box) 50 workers or 50 stones - as long as you throw out only "non-queens".

I've deleted it from the article but have copied it here for comments. -- Rick Block (talk) 13:15, 31 July 2010 (UTC)

Let's Suppose...

Let's suppose that the simple solutions to the MHP provided by Selvin, vos Savant, Devlin, Rosenthal, Adams, and many others were determined to be wrong by a consensus of editors on this page.

Would that then give as the 'authority' to espouse that POV in the article? Despite the fact that none of those reliable sources had renounced their papers?

Of course not. So what's the point of arguing which side is right? Talk pages are meant for discussions about editing the article, not as a tutorial on Probability. Both sides are reliably published. We're to present the reliably published sources in a manner the reader can comprehend. Which does NOT include warning the reader that he is being misled by certain reliable sources. As that would be a violation of NPOV. Glkanter (talk) 17:46, 31 July 2010 (UTC)

There are in fact many more sources giving simple solutions. Martin Hogbin (talk) 20:03, 31 July 2010 (UTC)

Of course. And, to underline the need for the Wikipedia NPOV policy, we editors can't even agree on which sources are on which side of the dispute. We're as divided over Rosenthal's paper as we are over the MHP itself. For what benefit? The article has to be NPOV. Each of us has typed, directed at another editor, 'Your opinion is not what is important. It's what the reliable sources say'. And yet, we argue endlessly over which side is right, and how to best phrase in the article that one side considers the other wrong. Glkanter (talk) 20:11, 31 July 2010 (UTC)

I do find the Rosenthal argument rather hard to believe. How is it that a source which literally says of the simple solution 'It is actually correct' can be understood to say, 'It is actually incorrect'? I think we need some more uninvolved opinions. Martin Hogbin (talk) 20:18, 31 July 2010 (UTC)

For what purpose? How does any such 'opinion rendered' change either the content of the reliable sources, or Wikipedia policies, most notably, NPOV? Glkanter (talk) 20:20, 31 July 2010 (UTC)

Removal of simple solution disclaimer

Glkanter's removal of the 'health warning' in the simple solution section is not vandalism but removal of a POV misinterpretation of Selvin. Selvin says nothing about one solution represting the case where the player chooses before the host opens a box.

Selvin does, however, approvingly quote Monty Hall's response to his first (simple solution letter) on the subject Monty says, "Oh and incedentally, after one [box] is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. "

Note that it says that after one box is seen to be empty, the chances of the car being behind the originally chosen box remain what they were in the first place, one out of three. The simple solution is thus agreed to be valid for the case when the player chooses after a box has been opened.

Once again, some editors are basing their opinions what they would like the sources to say rather than what they actually do say. Martin Hogbin (talk) 18:41, 1 August 2010 (UTC)

The section that I have just removed is pure OR. No source that I am aware of sets out to solve the problem on the basis that the player chooses before the host opens a door. Selvin, the quoted source, does not mention this possibility. Martin Hogbin (talk) 20:11, 1 August 2010 (UTC)

I've changed the text to what I think was a less controversial version much closer to what we worked out some time ago (before Gill's changes). -- Rick Block (talk) 22:33, 1 August 2010 (UTC)

@Glkanter - Per WP:NPOV: "The neutral point of view neither sympathizes with nor disparages its subject, nor does it endorse or oppose specific viewpoints. It is not a lack of viewpoint, but is rather an editorially neutral point of view." Saying what sources say is not a violation of NPOV pretty much no matter what the source says. On the other hand, deleting what sources say because you don't agree with them is POV. -- Rick Block (talk) 03:48, 2 August 2010 (UTC)

Rick, the text that I deleted did not reflect in any way at all the quoted source, it was pure OR. Martin Hogbin (talk) 09:57, 2 August 2010 (UTC)

Your latest text is no better. The quoted source is Selvin, This says nothing about, 'outcomes of always switching to the outcomes of never switching', or, 'not taking into account the knowledge of which door has been opened' or 'mathematical sources' or 'explicitly addressing the probability in the example case where the host opens Door 3 after the player initially chooses Door 1'. [Martin]

Martin - you previously agreed to very similar wording, see Talk:Monty Hall problem/Archive 17#Why Isn't The Door 1 and Door 3 Pairing a Subset of 'Always'?. I'd be fine with another version of this, such as:

There are multiple approaches to solving the Monty Hall problem all giving the same result—that a player who swaps has a 2/3 chance of winning the car. Most popular sources present solutions based on simple probabilistic reasoning. A different approach to solving the problem, commonly used in mathematical sources, is to treat it as a conditional probability problem.

The version I added to the article is essentially what this morphed into following several edits. The citation to Selvin in the current version is specifically the sentence that says he presented both types of solutions. The other content (in the current version) is descriptive of what is contained in the following major sections. References for this content are in those sections (this is the same sort of approach used in the lead section of articles - since this is summarizing content that is elsewhere the references are not repeated). -- Rick Block (talk) 15:00, 2 August 2010 (UTC)

How about having the Rosenthal comment everyone? Martin Hogbin (talk) 12:23, 2 August 2010 (UTC)

Your paragraph above is fine with me (apart from maybe some minor quibbles about words) , but not in the simple solution section. It is more of an introduction to the solution section as a whole. Martin Hogbin (talk) 15:34, 2 August 2010 (UTC)

Edit warring needs to stop

We've got one user already blocked for edit warring on this article. I see there has been an attempt at mediation but it has apparently not achieved the desired result. I would remind everyone editing here that the only exception to WP:EDITWAR is the reversion of obvious vandalism, and that users who have not crossed the line of WP:3RR can still be blocked for edit warring. The only thing edit warring will accomplish is periods of page protection and/or blocks for all users involved. Keep that in mind please. Beeblebrox (talk) 21:50, 2 August 2010 (UTC)

In my humble opinion, we are still waiting to begin Formal Mediation. Two brief attempts ended when the Mediators resigned. Glkanter (talk) 10:10, 7 August 2010 (UTC)

Falk comment

The Falk comment was replaced, with an edit summary quoting the talk archive. The version replaced seemed to be the original version rather than anything discussed on the talk page.

I have replaced the text with Rick's proposed version from the talk archive quoted. I still do not agree that this is a consensus version but it is better than the original. I may propose some changes later. Martin Hogbin (talk) 12:44, 3 August 2010 (UTC)

What was there was the version we ended up with after our extended discussion about this - from Talk:Monty Hall problem/Archive 20#Another proposal for Falk's point (in the show/hide box). What you've changed this to is an incomplete version, omitting the last sentence - which is what we discussed at length. I don't understand your intent here. Do you want to have the same discussion again? -- Rick Block (talk) 15:49, 3 August 2010 (UTC)

Sorry if I have made a mistake (I missed the box) but what was there looked like the original version to me. I think the whole thing is rather cryptic anyway and we never reached a final conclusion. No one else seemed interested in commenting so there is hardly a consensus to have anything.

What would you like to have? Martin Hogbin (talk) 21:22, 3 August 2010 (UTC)

I would like you not to play dumb. The article was locked. I suggested we work on this paragraph on the talk page, editing it in place in the show/hide box as if it were in the article. What we ended up with is what I added to the article after it was unlocked. What you changed it to is what the starting proposal was, minus the last sentence. At least Gerhard, Glopk, you, and I made changes to the starting proposal, all of which you dropped (in addition to the last sentence). We had worked on this for a while, specifically the last sentence - which I ultimately suggested we reword to be much closer to what Falk actually says using essentially a quote. The discussion is archived at the link I provided above. Your last comment was that you think the wording that is essentially a quote might be seen to conflict with the rest of the paragraph. My response was that this wording is directly used by Falk and is the entire point of this section of her paper and, hence, this paragraph in the article - so if it does conflict it is the rest of the paragraph that should be changed. You didn't respond to this, which I took as tacit approval. Please change the paragraph back to what we ended up with. If you're unhappy with it we can discuss it more. -- Rick Block (talk) 10:44, 5 August 2010 (UTC)

Sorry Rick, I am not playing dumb, I really was. Please feel free to change the wording to whatever version you think was nearly agreed between ourselves. I do not think anyone else cares. Martin Hogbin (talk) 13:28, 5 August 2010 (UTC)

I've restored it to what we discussed. Assuming you're still going to be unhappy with the "not because of the former that the latter is true" bit, here's the direct quote from her paper - where she's directly talking about the Three Prisoner's Problem, but this section is talking about the general principle of "no news" which she has already said applies to MHP as well: "In the original version of the problem, the warden can always truthfully name at least one of Tom and Harry, and the probability of pardon for Dick does not change. Yet, as we have seen, it is not because of the former that the latter is true." As I've previously said, if you insist we can cite her book where she repeats this phrase verbatim (including the italics for "and" and "not because") while talking specifically about the MHP [1]. -- Rick Block (talk) 14:09, 5 August 2010 (UTC)

I think my complaint would be that it is partially because of the former that the latter is true. In other words the forme is a necessary but not sufficient condition. I would be happy with any for of words which makes this clear, as Falk does elsewhere in the paper. Martin Hogbin (talk) 15:17, 5 August 2010 (UTC)

Is your complaint that you think this sentence is not accurately reflecting what the source says or are you partially disagreeing with the point being made? In both this paper and her book she emphasizes this point with italics - this sentence as it currently appears in the article is virtually a direct quote. There's plenty of context in the surrounding paragraph that makes it clear what else is required for the initial probability to remain unchanged, i.e. the third sentence and the sentence immediately following the one you seem to dislike. -- Rick Block (talk) 14:53, 6 August 2010 (UTC)

I think Falk does not make herself particularly clear in that quote. It seems that we all, you, me, and Falk all agree on the facts, which are that the fact the host reveals a goat when it is known that the host must reveal a goat is not, on its own, 'no news' (there because there might be 'news' in the door number the host chooses). On the other hand, the fact that the host must reveal a goat combined with the knowledge that he must choose one uniformly does mean that when a goat is revealed there is no news and the probability of the car being behind the originally chosen door cannot change. I do not think your proposed Falk quote makes this clear, although Falk does elsewhere in her paper. Could we have something like, 'it is not just because...the host must also choose a goat door randomly...'. We should be intelligently summarising the agreed content of the paper rather than picking a poorly worded quote. Martin Hogbin (talk) 16:02, 6 August 2010 (UTC)

I think Falk makes herself perfectly clear and she repeats this in her book as well (just follow the link above, italics in the original): "Truly, Monty can always open one of the two other doors to show a goat, and the probability of door No. 1 remains unchanged subsequent to observing that goat, still, it is not because of the former that the latter is true. The probability of winning the car by sticking with door No. 1 remains unchanged due to a specific combination of priors and likelihoods characterizing this problem." It seems to me that you disagree with this point and are trying to change it to something far less damning of the simple solutions that rely on this argument. Falk is not being in the least unclear. X and Y are both true, but X does not imply Y. -- Rick Block (talk) 18:42, 6 August 2010 (UTC)

I agree (and so does Falk) that just X does not imply 'Y' but X is an important part of the implication of Y. Should we not make that clear? We all (including Falk) agree on the facts, why not present them clearly and unabiguously. Martin Hogbin (talk) 19:06, 6 August 2010 (UTC)

Should quotes from sources that are not speaking of the MHP be included at all? I can see that leading to endless trouble... Making a correlation, and paraphrasing the content from a source on a different problem sounds like a violation of WP:Synthesis to me.Glkanter (talk) 16:22, 6 August 2010 (UTC)

The Falk paper is primarily about the 'Three Prisoners Problem' although is does briefly digress to talk about the MHP. The proposed quote is, however, referring specifically to the TPP and thus to insist on quoting it directly here is not justified.

Please read this entire section. The paper is about both the MHP and the TPP, and the quote in question is repeated in her book, talking specifically about the MHP. -- Rick Block (talk)

Aids to understanding

The 'Why the probability is not 1/2'subsection seems to have been changed without consensus to something that does not seem to aid anyone's understanding. Does anyone object if I change it back to its original, simpler version? Martin Hogbin (talk) 22:02, 3 August 2010 (UTC)

If there is no objection, I propose to revert this to the long-standing consensus version. Martin Hogbin (talk) 13:25, 5 August 2010 (UTC)

Which version are you talking about reverting to (just for clarification)? -- Rick Block (talk) 14:19, 5 August 2010 (UTC)

Say this [2] version. Martin Hogbin (talk) 15:36, 5 August 2010 (UTC)

Section order

I strenuously object to Martin's claim that his preferred section ordering is "consensus". The history here is he made this change (some time ago), Nijdam reverted it, he made it again, I reverted it, he made it again and I let it stand rather than edit war (assuming mediation will resume at some point). Nijdam moved the conditional solution earlier in the "solution" section and Martin has reverted this change. I strongly prefer Nijdam's ordering - but suspect we won't actually reach a consensus on our own. -- Rick Block (talk) 18:58, 5 August 2010 (UTC)

I dunno. 2 reverts already today. It's quite clear to me who is doing the edit warring. And claiming ownership. And ignoring NPOV policies. And encouraging OR. Glkanter (talk) 19:04, 5 August 2010 (UTC)

The current order has been stable for some time now and represents a reasonable compromise between simple and probabilistic solution supporters. Anyone who gets as far as wanting to understand about host bias will undoubtedly have understood the reason why the answer is 2/3. The current order is the logical order used in most text books and WP article. start with the simple and move on to the complex. Starting up this old argument serves no useful purpose except to default on a reasonable compromise Martin Hogbin (talk) 19:13, 5 August 2010 (UTC)

The correct solution should in fact be the first one. I can live with the position directly after the simple "solutions", but don't want it to be hidden in some sense!!Nijdam (talk) 22:47, 5 August 2010 (UTC)

@Martin - Our last discussion about this is archived at Talk:Monty Hall problem/Archive 16#Aids to understanding moved. You had previously tried this change and it was reverted before this. You tried it again just before this archived discussion and I reverted it. The archived discussion does not indicate a consensus and you went ahead and reverted my revert. Rather than edit war, I let it sit. This order has been "stable" only because you were willing to edit war to preserve it. -- Rick Block (talk) 01:28, 6 August 2010 (UTC)

This is a compromise of sorts, but one based on good practice in text books and WP articles. If you remember, you adjusted the heading levels to reduce the 'space' between the simple and probabilistic solutions. I accepted this without argument.

It is only Rick and Nijdam who seriously object to this topic order. Kmhkmh has said several times that he accepts it and Glopk has not commented. On the other hand several editors strongly prefer it.

Whatever you may think, it is not my desire to hide the more complex solutions, I just want to make sure that they do not obstruct less mathematically able or less interested readers. Many readers will simply not be interested in the complex solutions, you may feel that they should be because they are 'correct', but the fact is that some readers will just want to confirm that the correct answer is 2/3 and understand why this is so, making no distinction between the conditional and unconditional cases. I do not object to educating readers about the potential conditional aspect of this problem, but we cannot do this against their will. There will of course also be readers who want to understand the problem more thoroughly; the current format offers no impediment to their doing so. Martin Hogbin (talk) 09:09, 6 August 2010 (UTC)

The current order has been stable since 1 June. Can we please discuss here to get a consensus before changing. Martin Hogbin (talk) 17:10, 6 August 2010 (UTC)

Martin, nice to hear, you won't in some way keep the correct solution out of sight for the readers. Yet practically you do, by placing it far away in the article. I want every reader to see what the problem is with the so called simple solution. That's why I actually prefer even to place the correct solution first. But it's fine with me to have the simple explanations first, as long as the right solution comes immediately thereafter, so no one will be kept in ignorance, or given the wrong idea about the solution. I can not think of any objection to this, other then keeping it out of sight.Nijdam (talk) 17:17, 6 August 2010 (UTC)

I strongly object to moving the section on the probabilistic solution further down the article. Just like Nijdam, I can live with having it following the "simple" solutions, but not after the "aids to understanding" section. I feel that doing otherwise would do a disservice to the readers, by presenting logically incomplete solution as the whole story and ignoring reputable sources that say they are, in fact, incomplete. glopk (talk) 17:22, 6 August 2010 (UTC)

Then let us get some outside opinion in on just the section order. No content change. Martin Hogbin (talk) 17:45, 6 August 2010 (UTC)

As I have said many times before I feel that the article should progress from easy to difficult math. I am not a mathematician and the first time I read the article, long before I ever read this talk page, I had to skip over the conditional section because it made no sense to me. I think people forget that in most scholarly papers where the MHP is discussed they use the simple solution. Now before you jump to disagree with me please note I said "most scholarly papers" not "most mathematics papers". The MHP is used extensively in psychology and sociology and even neurology. In fact if it was not for the psychology of the MHP the problem would not even exist as such. If you count all the non-mathematics papers they are by far the more plentiful. I have never seen a non-mathematics paper that discusses or uses the MHP that does not describe the simple solution.

But that is besides the point. Most WP readers will not be holders of degrees in advanced maths or statistics, we should therefore try to present the page in a way that is accessible. That in no way means leaving out the conditional solution. It simply means we should order the article so that those who just want to understand the basics of why the MHP is called a paradox, like I did, will not have to skip that part and go searching for something they can understand that clarifies why the answer is not 1/2, like I did. —Preceding unsigned comment added by Colincbn (talk • contribs) 18:49, 6 August 2010 (UTC)

Again, this has to stop

As Nijdam as said before, this has to stop. This article has been held hostage for quite some time primarily by two editors. One, Martin Hogbin, has repeatedly shown himself to be astonishingly incompetent in the area of expertise affecting the article, but has managed to drag on the discussion by exploting the goodwill of Rick Block and other editors, and drawing on a large reservoir of sophistry when the goodwill was nearing exhaustion. I won't comment on the other, Glkanter, other than note that his history of abusive behavior has already resulted in a bans and wiki-fights with valuable contributors to the Wikipedia project. I hope that the mediation process can be restarted soon, or that further escalation be initiated. glopk (talk) 17:28, 6 August 2010 (UTC)

The fact is that there have been four long term editors who have tried to exert ownership over this page to propagate their own POV on the subject despite what the majority of sources say. I came is response to an RfC claiming those same issues of page ownership years ago. Since that time, and even before it, there has been a steady stream of editors who have come here and argued against the page owners, only to be driven away, exhausted by consistent tag-team tactics. If you like we can go back through the talk history and count them. Glkanter and I have been the only ones to have stayed the course but we represent a suceession of dissenting editors.

All I have done is to propose what seems like a reasonable compromise, to change the subject order to that found in most good text books and WP articles, the simple stuff first then the more complex. Nothing has been removed or glossed over. As the article has remained in this state since the start of June, I assumed that there was a, maybe fragile, consensus to keep it. I was just about to propose a truce, since we now have something that I thought was just about acceptable to all when Nijdam started editing again.

If you want to get more outside opinion please let us do so. There is no need for anyone to join in the fight over what is the 'correct' solution. All we need is opinion on the best section order, with no change of content whatsoever. Martin Hogbin (talk) 17:44, 6 August 2010 (UTC)

As I said, sophistry. "Nothing has been removed or glossed over" - this in the same day that Glkanter has removed twice the very two paragraphs that are the only reason for the conditional solution to exist in the article. "I assumed that there was [..] a consensus to keep it" - where "it" is your own favourite version that you had edit-warred on, and temporarily prevailed by exploting Rick Block's and other editor's patience. "No need for anyone to join in the fight over what is the 'correct' solution" - after having piled for days wrong equation after hare-brained mathematical argument in the Arguments page, and having been proved wrong time after time. All in the vane effort to show that the "simple" solutions without disclaimers are all there is to the MHP, and that the reputable sources that say otherwise are not worthy of consideration. Honestly, I am having a harder and harder time assuming good faith (there is a limit to how much confusion and ignorance one can assume). glopk (talk) 18:14, 6 August 2010 (UTC)

I agree.--Kmhkmh (talk) 19:10, 6 August 2010 (UTC)

Kmhkmh, several times you have said that you thought a change in section order was logical and acceptable to you. Do you want the diffs? Martin Hogbin (talk) 19:26, 6 August 2010 (UTC)

I was agreeing with glopk's assessement, which doesn't state anything about the order of sections. I don't really care much about the order of various sections (as I indeed have stated several times), but I do care about an accurate wording in the various section including the simple solution and so far I haven't seen you suggesting an accurate wording being in line with the sources and that's going on for over a year now.--Kmhkmh (talk) 19:55, 6 August 2010 (UTC)

I was only talking about my revert of Nijdam's change of section order (maybe Glopk was talking about more than that). I think there is a (fragile) consensus to keep it as it is now. If we could all agree to that we could discuss the detailed wording of sections later. Martin Hogbin (talk) 20:07, 6 August 2010 (UTC)

Glopk, I was (obviously) not referring to Glkanter's edits but to Nijdam's change of section order. Regarding Rick's part in this, after I made the change, near the end of May, he proposed changing the section levels to reduce the perceived negative impact of the move on the prominence of the 'Probabilistic solution' section. I agreed with that and it was done. I took this as a sign that some sort of agreed compromise had been reached. Martin Hogbin (talk) 19:26, 6 August 2010 (UTC)

As I said before, the only reason why I started (restarted as I soon learned) this discussion on the MHP is because the so called simple solution, which is no more then a way of understanding, is INCORRECT. Some sources have recognized this, fortunately.There is no source calling the conditional appraoch incorrect (how in the world would that be possible). So the conditional approach is at least impeccable, undiscussed correct. It should be first in the article. If the simple explanation has to come first, any reader should be aware of its shortcoming. I noticed that the MHP is a favorite problem, especially in the not exact area. Almost everywhere there the simple explanation is taken for granted, used as a solution, which it is NOT. What do most people know about this area. Should wikipedia leave them in the dark? I say NO. Even in more exact areas, the MHP is dicussed over and over and students come to consider the simple explanation as a solution. Especially they should take notice of the right solution, but not only they, everyone should know what the right solution is (even Devlin!). So ther is no consensus, what ever that may mean. And Martin, really, you are in no position to judge here.Nijdam (talk) 22:34, 6 August 2010 (UTC)

I believe Martin thinks his section ordering is a better match for the guidelines suggested at wp:Make technical articles understandable. My response has consistently been that I believe this creates a structural POV, as described at wp:Neutral point of view#Article structure. Martin pooh-poohs this, claiming that his intent is not to push a POV but merely to make the article more accessible to the non-expert. I sincerely believe we can't have an argument this heated for this long UNLESS it's fundamentally a POV issue. Martin denies intending to push a POV, but if in the interests of making the article "more accessible" a structural POV is created favoring the simple solutions then his intent doesn't actually matter - the end result is the same whether we got there because of POV pushing or not.

I've argued before, and I still think it's the best approach, that we should treat this conflict as a POV issue. I've also said before that it's odd to do this with a "math" article, since math is (at least usually) more or less objectively correct or objectively incorrect. If we treat this as a POV issue it would mean:

1. We completely cease any attempt (at least on this page) to convince other editors that our opinion is correct and that their opinion is incorrect. Our opinions don't matter - the opinions of sources matter.
2. The various POVs should be identified in the article. This is not weasel wording but using careful phrasing to attribute who thinks what to the various sources that hold the various POVs.
3. The article must not endorse any of the POVs, meaning none should be placed in a position of primacy to the others.

We've made progress before on drafts I've floated of a combined NPOV solution section, for example the one in the show/hide box at /Archive 16#A genuine attempt at compromise..

@Martin - would you be willing to work on a combined solution section that presents both unconditional and conditional solutions in an editorially neutral manner? If not, would you be willing to abide by outside opinions (perhaps from Wikipedia:Neutral point of view/Noticeboard) about whether the structure you're advocating is NPOV? -- Rick Block (talk) 03:28, 7 August 2010 (UTC)

I have made a suggestion below which may help with reaching agreement. Let is find out how much we agree first. Martin Hogbin (talk) 10:42, 7 August 2010 (UTC)

Cease fire in place for two weeks

Article is now protected from editing for the next two weeks due to continued edit warring. Some things for you all to keep in mind:

• It is impossible for only one user to edit war, so there's plenty of blame to go around on that count
• Since there was already a very clear warning posted, I would have been equally justified in blocking everyone who participated in the edit war
• For the record I don't know or care who is right or wrong as far as the facts. It is not relevant as all parties who edit war are equally guilty.
• Please limit the discussion here to discussing the article and not each other. Lay off the incivility and personal attacks

Beeblebrox (talk) 03:21, 7 August 2010 (UTC)

I doubt there will ever be total agreement between the regular editors here over what is really a very small and uninteresting matter of mathematical philosophy. On the other hand, the page is not that far from agreement as it is.

We need some way to break the deadlock here other than just plain discussion, which has completely failed. In over two years not one person has changed their mind. Mediation has also failed and, in my opinion, is not likely to be successful if re-started.

Maybe we could have some sort structured RfC where uninvolved editors are asked to make some decisions for us based on an agreed summary of the issues. It may be beneficial if these editors are not interested in the maths and could therefore contribute imaginatively in other ways. From what I can see now there is very little disagreement about the strict mathematical facts, so these can be presented as an agreed position to incoming editors, who could propose the best way to present these facts based on WP policy and the sources. In anticipation of at least some support for this idea, I will start an 'agree facts' section below. Martin Hogbin (talk) 09:47, 7 August 2010 (UTC)

Apparently that suggestion is disruptive so I will drop it. Martin Hogbin (talk) 18:17, 7 August 2010 (UTC)

Way to go guys!

Article locked again. And here I thought Israel-Palestine was a heated topic area :) One quick question, while not much else is happening: Player picks Door 1, Monty knows where the Car is, opens Door 3 to reveal a Goat. Player now takes the option to switch to Door 2, increasing win probability from 1/3 to 2/3. What if there are four doors, Player picks 1, Monty opens 3&4 to reveal Goats, Win probability goes from 1/4 to what by switching to Door 2? ADD: Ok it's 3/4 right? But wow it's hard to grasp this when for example there are 1,000,000 Doors, Player picks and the odds are one in a million he gets it right, Monty opens 999,998 Goat Doors then the Player is left looking at two Doors, so the incredible 999,000/1,000,000 win probability by switching is a function of the Player's original pick? Play nice. Respectfully, RomaC ^TALK 03:41, 7 August 2010 (UTC)

I think your example is in the article. But what I really want to point out is that this collective of editors is improving!!! Our last article protection was imposed for a full month, with a couple days off for 'good behaviour'. Little did the unprotector know, the same discussion had move to the 'arguments' page. This protection is only two weeks! Finally, PROGRESS!!! Glkanter (talk) 04:09, 7 August 2010 (UTC)

In either the 4-door or 1,000,000-door case is the door the host leaves closed picked randomly from among all losing doors if the player has initially selected the car, or not? If yes, then the answers are 3/4 and 999,999/1,000,000. If not, then both answers might be anywhere from 1/2 to 1. For example, in the 4-door case, if the doors are laid out 1-2-3-4 left to right on the stage and the host always starts at door 4 and opens 2 doors right to left skipping the car, then if the host leaves door 2 closed you have a 1/2 chance of winning by switching - but if the host leaves door 3 or door 4 closed you have a 100% chance of winning by switching. The preference doesn't have to be absolute for it to affect the chances - the point is that the player's chances of winning by switching are (always) a function of how the host decides which door to leave closed in the case the player chooses the car. If the host has a probability p of leaving the remaining door closed (where p is a probability between 0 and 1), the player's chances of winning by switching are 1/(1+p) - regardless of how many doors there are! In the 4-door case, if the host exhibits no preference then p=1/3 and the player's chances of winning by switching are 3/4. If p is higher for the specific door that's left closed, say 1/2, then the player's chances are only 2/3. If p is lower, say 1/4, then the player's chances are 4/5. So long as the car is randomly placed at the beginning, the player's chances are always between 1/2 and 1, so switching is never worse. And, on average across all players, switching wins with probability (n-1)/n where n is the number of doors - but the chances for any given player don't have to be the same as the chances for all other players and always depend on how the host decides which door to leave closed if the player picks the car. If the host has no preference among the n-1 doors the player didn't pick, p=1/(n-1), so 1/(1+p) works out to (n-1)/n. -- Rick Block (talk) 07:00, 7 August 2010 (UTC)

And there, RomaC, is perhaps the major point of contention as we edit the article, in all it's Glory. I referred you to a reliable source (not technically, since it was Wikipedia, but you get the idea). Rick gave you the answer, in his own words. It's the difference between pointing the reader of the article, perhaps with a brief summary, to the reliable sources, versus turning the article itself into a primer on Probability. If you've read Nijdam's recent comments, and other's, they feel the article must present the 'right' solution prominently, and equally prominently explain how the 'simple solutions' are flawed in their eyes, and in the eyes of some reliable sources. But the simple solutions come from reliable sources, too. So some of the editors, currently mostly Martin and I, plus countless others over 7 years who grew weary from the stalemate, feel that what Nijdam and Rick advocate violates the NPOV policy. Equally as important, we believe the reader is being distracted from his quest to understand a simple puzzle and it's paradox: why it's 1/3 & 2/3, instead of 1/2 & 1/2. And, here we are, the article is protected a second time. Glkanter (talk) 07:20, 7 August 2010 (UTC)

An idea to allow more independent editors to contribute

The purpose of this section is to state as clearly as possible the mathematical and other facts that are unanimously agreed by everyone here.

I will start to list them below, the idea is that if anyone challenges them they must be removed until an agreed position is reached. If no agreement can be reached then we say nothing about the subject. A certain degree of discipline will be required if this approach is to help us. Please put nothing other than unanimously agreed facts in the 'Agreed acts' sub-section. Comments and discussion should go in the section below but remember, the rule is simple, if one editor disagrees then it does not go in. (Obviously we will need to cope with obviously crazy people who may turn up).

I have three sections below, 'Agreed facts' and 'Candidate facts' and 'Rejected'. I propose the following rules:

Anyone can propose a candidate fact but, only if no one objects, can the statement be moved to 'Agreed facts'.

Anyone can move a statement from 'Agreed facts' or 'Candidate facts' to the 'Rejected' section.

The only way to deal with a rejected fact is to change it so that everyone will agree and then propose it as a candidate again.

I suggest that agreed facts are not signed. I ask everyone to stick to their principles but not to be obstructive. There are many subtle points to be made but we do not need unnecessary awkwardness. Also please no, 'You rejected my suggestios so I will reject yours' tactics. This section is to show just how much we do agree.

I have proposed some definitions below, just to for the purposes of this discussion. If there are more forms or accepted definitions of these terms I am happy for them to be used here. The aim is not to make any particular point but just to allow us all to be clear what we are talking about.

For example, someone might claim that 'vos Savant's answer was correct but her solution was not', meaning that the probability was indeed 2/3 but the reasoning she used to get it was flawed.

Definitions

• An 'answer' is the numerical value of the probability that the player wins the car by switching their choice of door.
• A 'solution' is a logical series of statements needed to arrive at the answer.

Agreed facts

• A basic description of the Monty Hall problem is given in the lead to this article.

• This discussion only concerns the game with the standard rules, that the host always opens an unchosen door to reveal a goat and always offers the swap.

• That the in the 'standard formulation' of the problem the car is initially placed uniformly at random and the host opens a door selected uniformly at random from those permitted under the game rules.

• The answer to the standard formulation is exactly 2/3.

Candidate facts

• The original problem was posed by Selvin in two letters to the 'American Statistician' Martin Hogbin (talk) 10:26, 7 August 2010 (UTC)
• The most notable and well-know statement of the problem is that by Whitaker, which leaves many details undefined. Martin Hogbin (talk) 10:26, 7 August 2010 (UTC)
• The most notable answer and solution were thos given by vos Savant in Parade magazine. Martin Hogbin (talk) 10:26, 7 August 2010 (UTC)
• That we wish only to discuss here the case where the player must make their choice after the host has opened a door. Martin Hogbin (talk) 10:26, 7 August 2010 (UTC)Nijdam (talk) 11:22, 7 August 2010 (UTC)

• The 'answer' is based on the numerical value of the probability that the player wins the car by switching their choice of door given the situation of the player.Nijdam (talk) 11:21, 7 August 2010 (UTC)

Rejected

• That the player must make their choice after the host has opened a door.

This is not the obvious and clear intent of Whitaker's question. Martin Hogbin (talk) 10:26, 7 August 2010 (UTC)

• The player chooses a door uniformly at random.Nijdam (talk) 11:21, 7 August 2010 (UTC)

Discussion on 'An idea to allow more independent editors to contribute'.

I have started an idea above to help us reach a conclusion. Please give it a try. The aimis that, when we have suffiecient agreed facts, we get some independent editors to help us reach agreement. Martin Hogbin (talk) 10:28, 7 August 2010 (UTC)

You seem to be in a particularly perverse mood at the moment Nijdam. How can you reject that the player chooses a door uniformly at random when it is part of the standard problem formulation? Martin Hogbin (talk) 11:32, 7 August 2010 (UTC)

You better put on your glasses and reread the statement(s) of the problem.Nijdam (talk) 17:59, 7 August 2010 (UTC)

You could propose that we only answer the question with the conditions that the player has chosen door 1 and the host has opened door 3. Martin Hogbin (talk) 11:34, 7 August 2010 (UTC)

What exactly do you mean by, 'Given the situation of the player'. Do you mean based on the player's likely state of knowledge? Martin Hogbin (talk) 11:45, 7 August 2010 (UTC)

To be precise: given the numbers of the chosen and opened doors.Nijdam (talk) 17:59, 7 August 2010 (UTC)

Nijdam, I have refactored one of your moves, with an explanatory note. I am just suggesting we use the term 'answer' to mean the numerical value of a probability not the logical steps needed to reach it. This is just for clarity of expression and is not intended to make any point about the MHP. If you would rather we use more formal terms for these concepts I am happy to do so, all I want to do is let people express clearly and concisely what they mean. Martin Hogbin (talk) 11:45, 7 August 2010 (UTC)

Martin - this whole section has completely the wrong focus. You're asking editors to agree with "facts". If we treat this as a POV issue, the only "facts" that matter are facts of the form "some reliable source (naming the source) says this". Nobody needs to agree with what the source says, only that the source says it. If you want to proceed with this I strongly suggest rephrasing all "facts" in this form, and I'll nominate every paragraph of the entire article as of the last FARC (linked from the article milestones box at the top of this page) as candidate "facts" (at that point, every "fact" in the article was attributed to a reliable source). In addition, I reject any "fact" not attributed to a reliable source in the same sense as if it had a {{citation needed}} tag.

And, frankly, given how close we've come before to an acceptable combined solution section and your apparent rejection of my suggestion (2 sections above) to resume working on such a solution section, it's difficult for me to read this section as anything other than wp:disruption. We aren't starting from scratch, but from a featured article. The primary point of disagreement is your continued insistence (to the point of edit warring) that the article be changed to consist of an extensive explanation of the "simple" solutions, with all criticism of them and any mention at all of the universally accepted conditional approach relegated to a subservient position in the article presented as if this is of academic interest only. If we're not talking about this, we're not talking about the actual issue. -- Rick Block (talk) 16:37, 7 August 2010 (UTC)

You make a good point Rick. There is no point to arguing the math, or which side is right, or what Whitaker really meant. All that matters is the reliable sources. But you seem to enjoy the debates as much as anyone. Even though Wikipedia talk pages are intended for discussions of editing the article, not to debate which source has it right and did they state it just right, or why the simple solutions are wrong or as a tutorial website for readers who are not familiar with the MHP. Eliminate all the pointless noise, and maybe we could get something accomplished. Like the removal of POV and unsourced editorializing, aka OR from the article. Maybe a mediator would even be willing to wade in then. Glkanter (talk) 17:33, 7 August 2010 (UTC)

And I'm not trying to sweep one POV under the rug. I have suggested countless times that the all controversies be addressed together in their own section. Without any refereeing by the editors. Let the reader decide for himself, just as Wikipedia intends it. Glkanter (talk) 17:45, 7 August 2010 (UTC)

Rick, your accusation of disruption was uncalled for. This was a good faith attempt to allow some independent editors to help us decide how to structure the article. I know we are starting from an FA which is why I thought we could agree on nearly everything easily but it seems this will not be so. Continued argument gets us nowhere so I suggested something new. As it seems no one likes my suggestion, let us move on to Glkanter's point about what the sources actually say.

The fact is that many reliable sources give a simple solution without saying that they are incomplete or that they answer the wrong question. On what basis do you presume to overrule these sources. You can start with vos Savant and Selvin. Martin Hogbin (talk) 18:13, 7 August 2010 (UTC)

Exactly. Many reliable sources give a simple solution without saying that they are incomplete or that they answer the wrong question. This is one POV. There's ANOTHER POV expressed by many reliable sources that the simple solutions are incomplete or answer the wrong question. The article needs to address BOTH of these POVs, without endorsing either one - the question is how can we do this? So far, I think I've heard 3 suggestions:

1) Completely ignore this in the interests of making the article more accessible to non-experts, i.e. your preference to focus first and foremost on the simple solutions without mentioning there's another POV. IMO this creates a structural POV effectively making the article endorse the "simple solutions are correct" POV. You apparently don't agree with this. I've suggested we might try to get an outside opinion on this from Wikipedia:Neutral point of view/Noticeboard. You have not responded to this.

2) Glkanter's suggestion that all controversies go in a "controversies" section. If we literally do this then the simple solutions go there as well as the criticisms and we're left with the completely unchallenged-by-anyone conditional solutions. I assume this is not actually Glkanter's intent (which is apparently that the criticisms would go there but not the simple solutions), let alone yours. The problem with putting only the criticisms in a "controversies" section but not the simple solutions is that there are two side to any controversy. Identifying only one side as "controversial" is blatantly POV.

3) My suggestion to collaboratively edit a single solution section until it is scrupulously NPOV meaning that it editorially does not take either side - such as any of the ones I've suggested, for example the one in the show/hide box at /Archive 16#A genuine attempt at compromise.. The controversies need to be explicitly addressed somewhere in the article, but presenting solutions (inclusive of all POVs) first without getting into technical mudslinging seems (to me) like a reasonable structure. Do you honestly believe that this is an impossible goal?

I haven't heard any other suggestions. If anyone has any, please speak up. -- Rick Block (talk) 18:55, 7 August 2010 (UTC)

Proposed NPOV solution section

Rick, there is no reason that a reliable source cannot be in the article without qualifications. Both sides of the issue are reliably sourced. Period. Your's and Nijdam's mandate that the simple solutions must be accompanied by the criticisms is POV. Period. The criticisms will be in the article. In a NPOV way. To discredit one side as you are presenting it, even with other reliable sources, is the NPOV violation. In fact, that is the very definition of POV. Glkanter (talk) 19:10, 7 August 2010 (UTC)

Glkanter - did you read the proposed solution section? Please do so and comment again. -- Rick Block (talk) 19:14, 7 August 2010 (UTC)

No thanks. I prefer to use the existing article as my starting point. I think you can address my concerns anyway. Why *must* criticism accompany the simple solutions, or be used as an intro to the conditional solutions? It's pretty obvious that such an approach, as it is in the current article, favors a POV. As editors, we're supposed to make sure both sides are presented without bias. By the way, I wish you would explain to Nijdam that for Wikipedia purpose, neither side is 'right' or 'wrong'. His recent talk page postings concern me very much on this very issue. Glkanter (talk) 19:20, 7 August 2010 (UTC)

Rosenhouse (sp?) directly criticizes Morgan's paper. Unlike Morgan, who criticizes what is it, 6 'simple solutions', but does not make reference to any but vos Savant's. Rosenhouse is reliable. Morgan admitted 3 mistakes in their paper. Why don't Rosenhouse's comments and Morgan's later comments accompany Morgan's earlier comments as they appear in the article, and serve as a warning that the conditional solution is unnecessary? Glkanter (talk) 19:17, 7 August 2010 (UTC)

Here's the text I'm proposing we work on until everyone is satisfied it is NPOV (comments continue below).

---

Solution

Different sources present solutions to the problem using a variety of approaches.

Simplest approach

The player initially has a 1/3 chance of picking the car. The host always opens a door revealing a goat, so if the player doesn't switch the player has a 1/3 chance of winning the car. Similarly, the player has a 2/3 chance of initially picking a goat and if the player switches after the host has revealed the other goat the player has a 2/3 chance of winning the car. (some appropriate reference, perhaps Grinstead and Snell)

What this solution is saying is that if 900 contestants all switch, regardless of which door they initially pick and which door the host opens about 600 would win the car. Assuming each specific case is like any other, this means a player who initially picks Door 1 and sees the host open Door 3 wins the car with a 1/3 chance by not switching and with a 2/3 chance by switching.

Enumeration of all cases where the player picks Door 1

If the player has picked, say, Door 1, there are three equally likely cases.

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

A player who switches ends up with a goat in only one of these cases but ends up with the car in two, so the probability of winning the car by switching is 2/3. (some appropriate reference, perhaps vos Savant)

What this solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1 and they all switch, of these 300 players about 200 would win the car. Assuming the cases where the host opens Door 2 or Door 3 when the player picks Door 1 are the same, this means a player who initially picks Door 1 and sees the host open Door 3 wins the car with a 1/3 chance by not switching and with a 2/3 chance by switching.

The probability of winning by switching given the player picks Door 1 and the host opens Door 3

Tree showing the probability of every possible outcome if the player initially picks Door 1

This is a more complicated type of solution involving conditional probability. The difference between this approach and the previous one can be expressed as whether the player must decide to switch before the host opens a door or is allowed to decide after seeing which door the host opens (Gillman 1992).

The probabilities in all cases where the player has initially picked Door 1 can be determined by referring to the figure below or to an equivalent decision tree as shown to the right (Chun 1991; Grinstead and Snell 2006:137-138 presents an expanded tree showing all initial player picks). Given the player has picked Door 1 but before the host opens a door, the player has a 1/3 chance of having selected the car. Referring to either the figure or the tree, in the cases the host then opens Door 3, switching wins with probability 1/3 if the car is behind Door 2 but loses only with probability 1/6 if the car is behind Door 1. The sum of these probabilities is 1/2, meaning the host opens Door 3 only 1/2 of the time. The conditional probability of winning by switching for players who pick Door 1 and see the host open Door 3 is computed by dividing the total probability of winning in the case the host opens Door 3 (1/3) by the probability of all cases where the host opens Door 3 (1/2), therefore this probability is (1/3)/(1/2)=2/3.

Although this is the same answer as the simpler solutions for the unambiguous problem statement as presented above, in some variations of the problem the conditional probability may differ from the average probability and the probability given only that the player initially picks Door 1, see Variants below. Some proponents of solutions using conditional probability consider the simpler solutions to be incomplete, since the simpler solutions do not explicitly use the constraint in the problem statement that the host must choose which door to open randomly if both hide goats (multiple references, e.g. Morgan et al., Gillman, ...).

What this type of solution is saying is that if 900 contestants are on the show and roughly 1/3 pick Door 1, of these 300 players about 150 will see the host open Door 3. If they all switch, about 100 would win the car.

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

Car hidden behind Door 3	Car hidden behind Door 1		Car hidden behind Door 2
Player initially picks Door 1
Host must open Door 2	Host randomly opens either goat door		Host must open Door 3
Probability 1/3	Probability 1/6	Probability 1/6	Probability 1/3
Switching wins	Switching loses	Switching loses	Switching wins
If the host has opened Door 3, these cases have not happened		If the host has opened Door 3, switching wins twice as often as staying

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Glkanter - is this proposal editorially neutral (doesn't pick which side is right)? If not, where (what words, exactly)? -- Rick Block (talk) 21:54, 7 August 2010 (UTC)

Let us discuss the sources then

No one wants to talk about what we agree on so let us talk about what the sources really say. Martin Hogbin (talk) 18:43, 7 August 2010 (UTC)

Rather than repeat them (yet again), I'll add the following (Gillman, Grinstead and Snell, Lucas, Rosenhouse, and Schepler, and Rosenthal) by reference, see /Archive 16#What the Sources say. You're well aware of Morgan et al. and Falk as well. -- Rick Block (talk) 19:12, 7 August 2010 (UTC)

vos Savant

• Gives a simple solution
• Does not say it is incomplete or that it answers the wrong question.

Selvin

• Invented the problem and gave a simple solution to solve it.
• Does not say it is incomplete or that it answers the wrong question.

Rosenhouse

• Says of the simple solution, 'It is actually correct...'

Cecil Adams

• Gives a simple solution
• Does not say it is incomplete or that it answers the wrong question.

Behrends

• Says, 'Marilyn was right...his [the contestant's] chances increase from 1/3 to 2/3'

Devlin

• Gives a simple solution
• Does not say it is incomplete or that it answers the wrong question.