Talk:Monty Hall problem

Is there still a consensus?

Twice in the past there has been a clearly expressed consensus to resolve the continual argument about this article by adopting my propose structure yet a minority of editors have managed to prevent this structure from being used.

For those new to the article I repeat it here with some clarification

The section titles are indicative of what should go in the sections. If they are considered too POV I would be happy to change them.

1 The problem

Just Whitaker's statement. It is by far the most well-known problem statement. Although it is very vague, most people seem to understand what it is about

2 Vos Savant's and other simple solutions

Without health warnings. These are off-putting for the general audience and unnecessary because all the relevant points will be covered later.

2.1 Vos Savants solutions
2.2 Other simple solutions
2.3 Media furore

This was what made the problem so notable.

2.4 Aids to understanding

Mainly why it matters what the host knows and helping people to understand the solution and any ways of helping people to see the right answer

2.5 Sources of confusion and the psychological aspects
Why people find this problem so hard.

3 Academic criticism of the simple solutions
3.1 Morgan's paper
3.1.1 K&W formulation

4 More detailed and comprehensive solutions

Including the 'conditional' solutions.

3.2 Other 'probability' sources
3.3 Criticism of the criticism
3.4 Summary of 'The Truth' (essentially that there is no such thing in this case, as per reliable secondary sources)

4 Variants

4.1 Other host behaviors
4.2 N doors
4.3 Quantum version

5 History

Rationale

My rational for this structure is:

• No solution or angle on the problem need be omitted.
• All POVs (as expressed in reliable sources) on the problem are clearly and openly presented. It does not push the simple solutions as the only ones. Weaknesses in the simple solutions are made clear at the appropriate point.
• It is the same format as most good text books and encyclopedia articles - easy first then hard.
• Is is accessible to all levels. The general public can read the simple solutions and then bale out. Experts will quickly skip through the simple bits to see if we have covered all the angles properly.
• It promotes cooperative editing because editors can work on the bits that interest them without having to discredit other editors or sections.
• It has been accepted as the consensus twice before.

I am proposing this one last time. If there is no consensus to adopt then I will leave the page, as many other have done, to a fate of eternal bickering. If there is a consensus then we should implement it immediately and keep the article that way until there is a clearly expressed consensus to change it.

There is some scope for modification but I the items in bold are an essential part of my plan to stabilise the article and avoid conflict. Please indicate your support or otherwise below. Martin Hogbin (talk) 16:26, 10 July 2011 (UTC)

Support

Martin Hogbin (talk) 16:26, 10 July 2011 (UTC)

Richard Gill (talk) 07:23, 11 July 2011 (UTC)

Supporting, with Special Opinion - see below. [Machtindex]

Oppose

• Oppose. Separating the so-called simple and comprehensive solutions (especially with intervening material relevant to both) is a disservice to the reader. There's a clear path to give the reader: run the simulation hundreds or thousands of times, and {2/3, switch} is the winner (even Monty agrees). Give the reader a chance to recognize that any intution about 50-50 has a problem. (Why do so many misinformed editors keep inserting solutions showing {1/2, doesn't matter}?) Then provide {explanations, solutions, demonstrations, what-have-you} that explain why switching works. And that should be all WP:DUE solutions: simple, conditional, game theory. The explanations should be readable and offer insight; sadly, many current explanations/solutions do not. The stated/previous structure is a poor compromise that is more about appeasing the two nominal sides of this debate rather than explaining the MHP to the reader. Glrx (talk) 17:50, 11 July 2011 (UTC)

It seems Girx that you think that what is wrong with Martin's proposal is that it does not address the real problem, which according to you is not the structure of the article, but the poor quality of the explanations of the more complex solutions. So, how could we make the more complex more accessible and appealing to the ordinary reader? Well, that is exactly what I have been arguing for ever since I started working here.

Give "easy" versions of the conditional solutions. Give solutions of the conditional problem which use the simple (unconditional) solution as a bridge. Remove the negativity of "so and so said that the simple solutions are *wrong* and you *have* to do it this way". Just present the solution itself. If you want the general reader to learn about conditional probability you have to take a different strategy! Make the conditional solutions appealing and show that though it costs a bit more work, it also delivers a bit more information.

We are in a better state to do this today, than a year or two ago. There are new "reliable sources" which provide the material; previously they did not exist. So all editors should now go home and "digest" the new sources. Come back here again, when they have brought their knowledge of MHP up to state-of-the-art level.

No need to quarrel now about the structure of the article. Better to improve the separate sections "in situ" and only after that discuss again whether or not a new structure is needed. Richard Gill (talk) 17:40, 17 July 2011 (UTC)

I agree in part. Although there are two nominal camps, there is some broad agreement between them. Neither camp is dismissive of the other. Martin's proposal attempts to service both camps, but it awkwardly separates simple and conditional methods with other material. My goals are different from the two camps: I want clear and simple explanations of how to solve the MHP. My target audience is high school students. To that end, your desire to avoid devisive labels and for "easy" versions of conditional solutions is apt. Ultimately, a variation of Martin's proposal may be appropriate -- but it would involve conditional methods being applied to nonsimple variations.

I disagree with some of your other points, but that's for another day.

The existing explanations of methods need work. The Bayes section bothers me the most because it omits the payoff. Instead of an explanation, it's a plug-in-the-numbers proof with little insight. It doesn't even explain that leaving C and S as variables covers all the cases (and triggers a complicated enumeration that MvS avoided but the DT included). I'd like to see a simple explanation of what Bayes' Theorem does with a simple recitation of Bayes' formula. Then I want an application of that theorem where P{Monty reveals a goat} = 1 -- meaning the probability that the car is behind the initial selection is still 1/3. That insight is the payoff, and I don't want it buried in a huge calculation. The more involved proof can follow. At least, that is where I want to go, but I haven't had the time to write it.

Glrx (talk) 19:50, 17 July 2011 (UTC)

Neutral

As always, I am neutral on the content dispute. If I see evidence of a clear consensus (I have not seen this so far despite various claims that a consensus exists) I will support that consensus. Guy Macon (talk) 07:17, 11 July 2011 (UTC)

Discussion

• The claim that this outline represents "consensus" based on past polls is simply incorrect. Polls do not determine consensus. Per wp:poll: "Remember that Wikipedia is not a democracy; even when polls appear to be "votes," most decisions on Wikipedia are made on the basis on consensus, not on vote-counting or majority rule. In summary, polling is not a substitute for discussion." (emphasis in original) -- Rick Block (talk) 16:58, 10 July 2011 (UTC)

In this case we have been discussing the subject for over two years and not one single person has changed their opinion. A majority is the only kind of consensus we are going to get. Martin Hogbin (talk) 17:25, 10 July 2011 (UTC)

Speak for yourself. Perhaps since I still don't agree with you, you haven't noticed that I've moved from a stance that the (well-sourced) criticism stating that the "simple solutions" address the wrong problem needs to accompany the first mention of these solutions (i.e. "health warnings" are required) to a stance that the article should initially present both simple and conditional solutions in a perfectly neutral fashion without favoring either one (i.e. no health warnings, but no favoritism shown to either simple or conditional solutions). You have apparently not changed your position, which I'll paraphrase as "the simple solutions are the correct way to view the problem, anything else should be presented subservient to these solutions and must not even be mentioned until later in the article because we must first convince the reader these solutions are the one, true way to approach the problem". I am seeking a middle ground between Richard's (a) above (my original stance) and his (c) (your original and apparently still current stance). This middle ground constitutes NPOV - between the opposing POVs of "only conditional solutions are correct (and simple solutions address the wrong problem)" and "simple solutions are correct and sufficient (and conditional solutions are of academic interest only)". -- Rick Block (talk) 18:06, 10 July 2011 (UTC)

I find it odd that Martin was unable or unwilling to work towards a clear statement of his position when I asked so I could take this to content dispute resolution,, and yet is now willing and able to run a straw poll containing what appears to be a clear statement of his position. Now he says "If there is no consensus to adopt then I will leave the page, as many other have done, to a fate of eternal bickering.", yet the eternal bickering is a direct result of his and others failure to to work towards a clear statement of the positions Guy Macon (talk) 07:17, 11 July 2011 (UTC)

Why don't we take to Content Resolution the question whether or not we should organize the article according to Martin's proposal? The two names of positions are then "Martinist" and "Anti-Martinist". The outcome is that one side or the other wins. In defence of Martin, you yourself, Guy, gave Martin the lead in re-forming the page according to that proposal, according to the declared plan that afterwards someone else (Rick?) could take the lead in creating an alternative. Only then, with two alternatives side-by-side, we would decide to go for one or the other.

In case of a Martinist victory, everyone can back get to work on the article with a clear plan which allows everyone to work hardest on the bits they understand best, giving collegial feedback to the others on the other bits. (I for one will then try to sneak a "conditional" solution into the early part of the paper by writing it in such a simple way and without even using the words "conditional probability" that no one can complain. Esoecially since it'll be reliably sourced.

In the case ofan anti-Martinist victory, everyone can back get to work on the article (except that Martin might quit for a while, which I think would be a pity, but maybe he needs the break anyway). There will be a more elaborate conditional solution in the early part of the article but criticism and counter-criticism of the different approaches will be kept for later.

Either would be fine. Either way, we get to move on. Richard Gill (talk) 07:48, 11 July 2011 (UTC)

That would be agreeable to me. I am agnostic as to what terms are used and what the description of each is, as lond as everyone agrees that their side is properly described (I don't want to go through content dispute resolution, get a ruling, and then have someone claim their position and arguments were not included)

Does anyone object to Richard Gill's proposal above? I need an alternative so that the two can be compared, of course. Guy Macon (talk) 09:25, 11 July 2011 (UTC)

Yes I do. See below.

Firstly, I have considered that I might be part of the problem, which was why I took a break from this discussion to see if all the argument disappeared. There was no sign of a consensus emerging and the article got into a greater muddle so I have come back to have one last attempt at resolving this conflict.

Guymacon and others. I must ask you to accept my proposal as a good faith attempt to end the disagreement here. I have no objection to stating my position, in fact I will do so below, but my proposal above is not an attempt to push my POV by stealth but a way to avoid conflict. Please look at my bullet points above and see how many you actually disagree with.

I do not like the idea of having two versions. Firstly, I must repeat, this is not my opinion vs Rick's opinion, this is a plan to avoid conflict by adopting a totally neutral approach. Either there is a consensus for my plan to avoid conflict or there is not. If there is, let us do it, if not, I give up. If you all really cannot see that the structure above separates 'vos Savant and the simple brainteaser with media furore' from 'an interesting and instructive problem in statistics/probability/philosophy/decision theory', in a way that, avoids endless POV arguments, and is in accordance with WP policy and the writing of a good encyclopedia article for a wide audience then I despair of the WP process. Martin Hogbin (talk) 09:37, 11 July 2011 (UTC)

Martin, I am not just assuming good faith on your part, it is a conclusion based upon abundant evidence. I also don't believe for a second that you are part of the problem. All available evidence suggests that you, Rick, Richard, Gerhard, Mathindex, Girx, etc. all just want to improve the article. This is simply an intractable good-faith difference of opinion concerning content (AKA a content dispute), with nobody at fault. Guy Macon (talk) 17:40, 11 July 2011 (UTC)

another digression

Special conditions of Machtindex I agree basically with the excellent suggested structure. However, I find it VERY incomplete. The recent viewpoint on the MHP as THE MONDEE GILLS GAME (MGG) embeds the MHP in combinatorial games. In the classical Whitaker/vos Savant setting this is the most primitive version of a hide-and-seek game played on a 3 x 4 playboard with four chips (prize, contestant,checked door, contestant) which is solved by dominance. The game has complexity level one point higher than tic-tac-toe. All difficulties connected with conditional probabilities, and with probabilities at all disappear at once by this approach. To give you idea what I am talking about: think of 3x3 tic-tac-toe, where the first player makes a move |x|-| and then covers two fields by matchboxes, so that the second player does not see where x stays (say, on the second player's first move). The MHP-MGG is much of the same level of difficulty, albeit with nice "switching combinatorics". Once the problem has a combinatorial solution by the dominance, which in the MHP-MGG instance is there, all probability solutions with arbitrary asymmetric priors and arbitrary payoffs for finding the prize at door a,b,c for premium A,B,C, whichever this A,B,C is probability law, or money rewards, or smth else are all banal. The probabilistic interpretation of A,B,C is posted on the arxiv.org. THE DOORS is downloadable from there. Would you come up with idea of playing tick-tack-toe by making random legal moves? My constructive suggestion is to include The Mondee Gills Game as a subsection in the most elementary part of the story. And in more complex part with antagonism of seeking and hiding players. At present there are presentations undesrandable by 5 year old kids who are capable of understanding the rules of the game: choose door, then hold or switch. What Host is doing is completely irrelevant, he is a dummy player, which only makes the situation more interesting. The Bayesian "machinery" is an excellent way to exercise the conditional probability However, it is beaten completely by the mentioned combinatorial approach. The 20 yrs-dispute conditional-vs-unconditional is a SHAME for the probability community. Leaving big traces of it on the Wiki page only says how isolated are some probabilists from other mathematical developments, in the first turn in computer science, game theory, machine learning, operations research and combinatorial optimisation. The Mondee Gills Game as a 2-person 0-sum was solved by Olle Haggstrom in his textbook Streifzuege... translated from Swedish. Complete game-theoretic discussion with tree presentation and information sets is available: THE MONTY HALL PROBLEM IN THE GAME THEORY CLASS for your downloads from arxiv.org The informational aspect of the problem (Revealer Host) is addressed in the case of four doors, and posted on the arxive.org. You may download THE UNLUCKY DOOR. The latest write-up of THE MONDEE GILLS GAME: STRATEGIC DOMINANCE FOR DUMMIES is available upon request. The first yet unpublished version of THE MONDEE GILLS GAME with discussion of the controversy is available upon request. The earlier version of the work THE MONTY HALL PROBLEM: SWITCHING IS FORCED BY THE STRATEGIC THINKING is available from arxiv.org. These are the latest. Add Richard's Stat Ned paper, add Olle Haggstrom's book. Many Wiki articles are based on 3-4 publications, ckeck the article Robbins' problem. Look how Benford's Law page is organised: nobody struggles there on the occasion of much more notorious phemomenon of the first digit. And everybody who wants to promote knowledge has their space. The basics of the game-theoretic, optimisational and combinatorial approaches to the MHP HAVE BEEN CREATED. Only blind cannot see this, sticking in 20yrs old nonsense. If we are here to bring knowledge to laymen instead scholastics then revert the moves in the discussion first SWITCH/HOLD the door then OPEN the door. Trust me: the moves are exchangeable: this is a theorem which is equivalent to the optimality of always switching.Machtindex (talk) 08:58, 11 July 2011 (UTC) I love the MGG approach but it has to become part of the mainstream literature on MHP before wikipedia can incorporate it in a big way in an article on MHP. You really must carefully study what wikipedia is! See WP:ABOUT and WP:FIVE. Please also try to learn how to do wikipedia markup. Richard Gill (talk) 10:19, 12 July 2011 (UTC)

Martin's POV

I am sure you all know this already but I have been asked to state my POV. My POV is that, taking into account the original context and the intended audience, the simple solutions to the MHP are perfectly correct. The 'conditional solution' is a minor academic diversion (since retracted by its originators) of interest to neither the general public or real experts in the subject, but of some value in teaching elementary conditional probability theory. Martin Hogbin (talk) 09:44, 11 July 2011 (UTC)

Conclusion by Martin

There is a small majority for my structure but not enough that I could claim a consensus. I guess this argument will now continue indefinitely, or at least until enough new editors join to produce a consensus. This will not help the article improve. Martin Hogbin (talk) 09:16, 17 July 2011 (UTC)

Or you could try my solution -- Going through the process of Wikipedia Content Dispute Resolution. Wikipedia has a process for resolving this sort of content dispute. Why not give it a chance? Guy Macon (talk) 21:38, 17 July 2011 (UTC)

Would it help to disambiguate "the" question?

It is my perspective that there are two distinct interpretations of what the question is asking. (This is different from the binary "switch or not" vs. the quantitative "what are the odds" distinction.). These have sometimes been described as (a) asking for a strategy independent of the door opened, and (b) asking for a posterior decision given an open door. More fundamentally, if somewhat abstractly, I think of these as stipulating that (a) "say, door #3" means treat the doors as undistinguished, and (b) "say, door #3" means either door might as well be distinguished. The problem, as I see it, is that some solutions to "the" MHP can be considered superfluous digressions for answering MHP(a) or insufficient for answering MHP(b) without justifications or "health warnings".

My preference would be to distinguish at the outset, nonjudgmentally, between two interpretations of the question, rather than treating solutions as different opinions about the answer to life, the universe, and everything. Advantages of this approach could be (1) presenting simple solutions as unreservedly good answers to a simple question, (2) shedding light on why conditional analysis is important, for the benefit of readers who are unfamiliar with it, and (3) mitigating disputation by reducing occasions for criticizing or justifying solutions. Disadvantages of this approach could be (1) the distinction may be too subtle for some readers to follow, (2) the relevance of some material does not fall exclusively under a single interpretation, and (3) exacerbating disputation by bifurcating treatment of "the" MHP.

Is there any chance this approach could lead to a way forward? ~ Ningauble (talk) 18:11, 10 July 2011 (UTC)

In my opinion, it definitely could lead to a way forward. In particular, it seems extremely promising as a way of getting to the short name and descriptions that are needed before this can be submitted to Content Dispute Resolution. Good thinking! Guy Macon (talk) 20:54, 10 July 2011 (UTC)

I agree that there are two interpretations of "say Door 1". Vos Savant even confirmed her intention: the numbers are meant from the outset to be irrelevant. In fact Whitaker's actual words, which were revealed in Morgan's response to Martin's correction note last year, did not specify any numbers but explicitly talked about "switching" or "not switching" in general.

But I disagree that one can distinguish between looking for (a) a strategy, (b) a posterior decision. Under the notion "strategy" is included strategies which could involve specific door numbers. And looking at posterior probabilities - which one can perfectly well do in advance - is a way to fix a strategy. In fact, the good reason to investigate the conditional probabilities iis because it guarantees that you'll find an optimal strategy this way. However if you are happy enough with the incredible gain of 2/3 overall as compared to 1/3 overall, then there is no reason to hurt your brain trying to figure out if you could do better still. Alternatively, come up with a simple reason why 2/3 overall can't be beaten, and then you needn't look at conditional probabilities at all. Theorem: A constant strategy is optimal if and only if the conditional probability favors this strategy in each individual case.

It is such a pity that the standard probability and statistics literature never says why it's a good idea to look at conditional probability. It's even sadder that the more unthinking writers even say you must. Richard Gill (talk) 07:17, 11 July 2011 (UTC)

In exemplifying that some have characterized a difference between strategy and posterior decision (I might also have mentioned a difference between unconditional and conditional) I did not mean to define the difference I am raising thusly, but to contrast with what I believe is a more fundamental and useful difference in ways of reading the question, as distinct from ways of approaching an answer. There is a real difference between stipulating undistinguished doors and concluding, when considering distinguished doors, that they might as well be considered undistinguished.

I think your remarks about strategy support my view that some of the dichotomies that have characterized disputes here are inapt. I am suggesting differentiating between (a) a question about undistinguished doors and (b) a question about distinguished doors. Do you think this would be a useful way to present (a) simple solutions to a simple problem and (b) deeper analysis of a subtler problem (some of which, interestingly, points back to the simpler problem)? ~ Ningauble (talk) 13:10, 11 July 2011 (UTC)

The MHP was initially intended as a simple puzzle. There is a long-standing convention in simple puzzles that all necessary assumptions are made in order to keep the puzzle simple. Thus "say, door #3" should be taken to mean 'another door'. We know, in fact, that this was exactly what vS meant when she added it. Everything else is an academic diversion, interesting in its own right, but not in the spirit of a simple puzzle. Neither is it in accordance with the principle expounded by Seymann that when an expert answers a question from a layman they should strive to find out what the questioner actually wants to know, rather than stick strictly to the exact wording.

You can always add complications to a problem if you try hard enough, and there is no reason that we should not follow those sources which have extended or dissected the MHP but please, after the simple puzzle has been simply resolved. Martin Hogbin (talk) 16:14, 11 July 2011 (UTC)

Notwithstanding that my own preferred reading of the problem is MHP(a) (i.e. that "say, door #3" not only means some other door, but stipulates that the choice makes no difference), I must still acknowledge there is a substantial body of literature that addresses MHP(b). I think that distinguishing between these two readings of the problem would help to delineate between the part of the article that is of most interest for readers who are curious about the "simple puzzle", as you call it, from the part of the article that is of interest for readers who would like to explore further mathematical considerations. Furthermore, I think it more than satisfies the aim of WP:TECHNICAL because it does not treat "simple" solutions as mere glosses, but treats them neutrally and objectively as real solutions to a simple riddle.

Do you think that differentiating between these two interpretations of the problem could facilitate your aim of treating "academic diversions", as you call them, after the simple puzzle has been simply resolved, by delineating where one leaves off and the other begins? ~ Ningauble (talk) 17:38, 11 July 2011 (UTC)

Yes, of course we must distinguish between what you call MHP(a) and MHP(b) but in the correct place, which is after the simple solutions have been given. Many readers will have no interest at all in the difference between the two problem, especially as (for the standard problem) they both give the same result. There are several rationales for giving the simple solutions (without health warnings). One is, as you state, that the question can be taken to mean MHP(a) (this is why we should not give the K&W problem description at this stage) another is that even for MHP(b) the simple solutions are perfectly correct by the application of a simple, obvious, and intuitive symmetry. In the symmetrical case the conditional probability must be equal to the unconditional one by the law of total probability. Falk refers to this logic as impeccable. Although we do not state the symmetry, my guess is that most readers intuitively assume this. Just ask anyone whether they think it makes any difference whether the host opens door 2 or door 3.

My suggestion is therefore, first give the simple solutions to the puzzle (without let or hindrance) then cover all the rest of the MHP openly and clearly according to reliable sources and including the criticism of the simple solutions. Why would anyone not want to do that? Martin Hogbin (talk) 18:06, 11 July 2011 (UTC)

Well, Rick has explained many times why he thinks this creates a bias. I believe that at the moment he is in a minority but the fact remains this is *the* dispute which has been going on for years. First we had mediation (2x) without success, then arbitration, which led to the removal of two editors, a welcome influx of new ones, and the article is "under probation". We now have a "probation officer" (Guy Macon) and everyone is doing their best to behave in a civilized way. In my opinion Guy is doing a splendid job. Now, he proposes we move to "content dispute resolution", but first we would have to agree what *is* the content of the content dispute. So my question to Guy is, again: can't we take Martin's proposed framework as the issue of content dispute, so that it can be settled one way or another, whether or not we follow this proposal? After further wrangling, involvement of yet other new editors, there will be an outcome. Either Martin or Rick will be highly disappointed with it. But at least we get closure on this question, for a while. Then everyone who wants to can concentrate on constructive improvement of the article within the constraints of the framework proposed by Martin or of one proposed by Rick.

BTW, I would vote for Martin's outline. I think he gives cogent reasons why this is an apropriate way to structure all this material on wikipedia. He explains clearly why it *is* a compromise aimed at achieving broad consensus. In particular, it does not actually strongly favour his own personal point of view on MHP.

Ningauble, I'm glad that you agree about the inaptness of some dichotomies which have been floated here. And I believe that the distinction between your MHP(a) and MHP(b) is actually one of the deeper reasons why many ordinary people find a conditional approach a meaningless diversion. In the preprocessing step (before proceeding to formal reasoning) they simplify to problem (a), whether following cues in Marilyn's wording or following their own instinct. From a subjectivist point of view, their instinctive move is justified. One can just as well argue beforehand, as afterwards, that door numbers can be neglected. See e.g. Georgii for a text book presentation following MHP(a), see my notes on my university home page. Unfortunately, this observation is not prominent in the literature. Richard Gill (talk) 10:51, 12 July 2011 (UTC)

To distinguish "did the host open door 2 or door 3" refers to some suspected underlying asymmetry, for in the asymmetric case the odds (given the host opened door 2 and given he opened door 3) could slightly differ from 2/3, nevertheless switching will still have an overall 2/3 probability. But as long as such asymmetry and its direction is not known for sure at the outset but can only be considered as suspected to exist for this one game given, it makes no sense to use it, especially as it can't change the decision to switch. It remains what it is, a less important side aspect that could be mentioned also, later on. What the sources say: In short, as long as you don't know for sure about existing asymmetry, to distinguish between door 2 and door 3 is not worth the trouble. But I would like to see a conditional formulation, clear and in odds form, already in the beginning also, just as an aid to understand that the odds are not 1/2 in general, but that in general the odds are 2/3. Gerhardvalentin (talk) 15:38, 12 July 2011 (UTC)

Gerhard, if you don't know for sure about existing asymmetry, in particular if your knowlege about possible asymmetry is neutral to the direction of asymmetry, then for a subjectivist (probability represents your state of knowledge) the host is equally likely to open either door when he has a choice. For such a subjectivist the problem is symmetric and hence by symmetry the door numbers can be thrown away from the start. That is what Georgii does in his book. A frequentist has a harder life. The sources do not say what you say they say. They do not say "to distinguish between door 2 and door 3 is not worth the trouble". Richard Gill (talk) 09:28, 14 July 2011 (UTC)

Right, but that was just "in short" what they say (being not just textbooks to teach conditional probability theory). Reliable sources on MHP say:
The actual numbers of door chosen and door opened are irrelevant to deciding whether to switch or stay.  –  Or:
Simple plus symmetry: by symmetry the probability that the car is behind door 1 cannot depend on whether the host opened door 2 or door 3. The unconditional probability was 1/3. Therefore the two conditional probabilities are equal to 1/3 too. Reference: Bell (1982).
And other sources on the MHP that don't even pay regard to whether #2 or #3 was opened, leaving associated unecessary assumptions aside. Gerhardvalentin (talk) 13:44, 14 July 2011 (UTC)

I do not mean to make a big deal of indistinguishable doors in a way that implies simple solutions are only relevant to one reading of the problem. Notwithstanding the many and various reasons one may interpret, assume, hypothesize, or conclude that the doors are undistinguished, or choose not to, the fact of the matter is that this choice characterizes a sharp and fundamental distinction between different ways of expressing solutions, and underlies much of the contention over correctness of solutions. I believe it would greatly clarify the article for the widest possible audience to explicitly indicate this difference rather than meandering from one solution to another without providing this context.

What I would like to see is a main section on simple solutions prefaced by a very brief statement of what is being stipulated or assumed (not a health warning). Concluding the section by observing that considering distinguishable doors leads to the same conclusion, and that some approaches treat it as a distinction that does not make a difference for reasoning to reach that result, would be a good segue into subsequent sections on less "simple" treatments.

Honestly, I think it would make the article clearer; and I do not think this widely noted distinction is original research, nor that using it to provide context creates structural bias for or against different approaches to the problem. ~ Ningauble (talk) 15:18, 13 July 2011 (UTC)

I agree with your conclusions, Ningauble. I also agree that simple solutions are not *only* relevant within one particular reading of the problem. A glance at the sources shows that this is not true! Improving your overall success-rate from 2/3 to 1/3 is a good reason to switch, however you read the problem. And it only requires the assumption that your initial choice has chance 1/3 to hit the car. Richard Gill (talk) 09:28, 14 July 2011 (UTC)

I agree too. The simple solutions have many diverse justifications, however, that is not the main reason we should have them first. The main reasons that we should do this are firstly, they are simple, and secondly, they are by far the most well-know and notable. The fact that the simple solutions do have many justifications makes the decision to put them first an easy one.

Once you go beyond the simple solutions and look at the philosophy of the subject and different possible interpretations you lose the interest of many readers. This is a separate subject which in my opinion deserves to be treated properly and in depth. Attempting to give a half-baked analysis of the philosophy of this problem at the start is a waste of time for everyone.

The only correct solution to any probability or statistics problem is the one that provides the questioner with the information that they wished to acquire. Martin Hogbin (talk) 09:13, 17 July 2011 (UTC)

To summarize responses thus far (If I have misunderstood anyone's position the fault is entirely mine, and I welcome the opportunity to be corrected.) —

• Guy believes the idea has merit, but considers it a matter of dispute. (Branding my suggestion as disputatious is not unreasonable: I broached it with some trepidation that I might be pointing out the elephant in the room.)
• Richard and Martin, after some clarification, are agreeable to the idea. (The amount of clarification needed probably indicates that it would take considerable effort to find suitable wording and implement it in the article.)
• Gerhard prefers to have an explicitly conditional formulation included in the beginning, and notes that some sources leave associated unnecessary assumptions aside as irrelevant.

Before trying to draft specific language, I would appreciate more feedback on this idea from other folks who still watch this article. I don't want to proceed with a draft that fails to take other perspectives into account, or to spend time on details if there is little chance this approach could lead to a way forward. Thanks. ~ Ningauble (talk) 16:23, 22 July 2011 (UTC)

What source or sources are you intending to use for the proposed discussion? As I see it, the main problem is that the sources presenting simple solutions essentially never say what problem they're exactly addressing (a or b). One might assume they're addressing (a), but since these sources don't make it clear it becomes vanishingly close to wp:synthesis for the article to make this clear on their behalf. As I've said repeatedly, there are plenty of sources that distinguish (a) and (b) nearly all of which say in one way or another that vos Savant's wording of the question implies (b). Rather than attempt to make this distinction early on, I think a more neutral approach is to present both a simple solution and an entirely accessible conditional solution as two different approaches without trying to favor either one. Both of these approaches are extremely common and (IMO) both should be presented very early on in the article. The discussion of the subtle differences between these approaches can come later, but IMO it's distinctly not NPOV to present the simple solutions at length without any mention whatsoever of the equally common conditional approach (particularly since there are numerous sources that explicitly prefer the conditional approach). Presenting both, as equally valid approaches, lets those readers who are thinking of the "unconditional problem" see a simple solution (these readers will presumably gloss over the conditional explanation) and but also lets those readers who are "misled" by the problem statement into thinking of the door 1/door 3 (conditional) case see a solution that pertains specifically to this case. Presenting them as complementary, not antagonist, approaches is IMO more NPOV and likely to be more convincing as well. -- Rick Block (talk) 19:45, 22 July 2011 (UTC)

Eliminate anything that is completely irrelevant to the decision asked for, but is just suitable for the mathematics class-room. I prefer to show the Bayes' rule in odds form just at the beginning also, easy to read and understand to anyone. Show the remarkable and controversial history later. Cut out the conditional solutions part of the page and have a link instead to where it belongs, to Bayes' theorem. Most of all: the article is about the famous paradox and the relevant decision, not just a lesson in conditional probability theory only. Gerhardvalentin (talk) 09:05, 23 July 2011 (UTC)

Rick asked "what sources?" I have told him so many times ... The text-book by Georgii, the encyopedia article on www.StatProb.com by one R.D. Gill. StatProb is hosted by Springer and published by a consortium of all the major national and international societies for statistics and probability, both professional and academic. Read the sources! Read the latest authoritative tertiary literature! Richard Gill (talk) 18:59, 23 July 2011 (UTC)

The question was for Ninguable, who seemingly expresses concern about how to cite this discussion in the very beginning of this thread. I have read the textbook by Georgii and your encyclopedia article on StatProb - and the original paper by Morgan et al, and Gillman's column, and the textbook by Grinstead and Snell, and the paper (and book) by Falk, and the paper by Puza et al., and the paper by Eisenhauer, and the paper by Carlton, and the paper by Rosenthal, and the paper by Lucas et al (and many, many more). There is no question that some authors who know what they're talking about approach the problem by examining the result of a predetermined strategy to switch. The question is whether the vast majority of popular authors who present "simple" solutions claim to be saying that the probability of winning by switching to door #2 given you've initially chosen door #1 and have seen the host open door #3 is 2/3 (with the decision point being standing in front of a closed door #1 and closed door #2 and an open door #3 showing a goat, i.e. after the host opens door #3), or whether they're only claiming with a predetermined strategy of switching the probability of winning the car is 2/3. The usual problem description makes it clear we are to consider the door 1/door 3 case but the usual simple solution ignores this case (and instead considers only predetermined strategies) - and it is this "bait and switch" that makes the simple solutions so hard for most people to swallow. As Eisenhauer puts it "what could and should have been a correct and enlightening answer to the problem was made unconvincing and misleading". I'd rather not have the Wikipedia article repeat this mistake. -- Rick Block (talk) 20:00, 23 July 2011 (UTC)

Dear Rick,

(1) You seem consistently to misunderstand me and hence misrepresent what I'm saying. I am *not* talking *only* about predetermined strategies. I am talking about *all* strategies. Strategies determined in advance, strategies determined on the fly. Strategies which always recommend "hold" or "switch" independently of the door numbers the player will see, or strategies which allow different actions in different circumstances. Strategies which use coin tosses and strategies which are deterministic. Also included is the strategy "calculate conditional probabilities after the host has opened a door and decide on the basis of that conditional probability". Also included is the strategy "choose door 1 and when a door is opened think about it for a while and make up your mind what to do". Whether the calculation or the thought is done on the fly, both possible calculations (or thoughts) can (for the purposes of argument) be thought of as done in advance. The decision moment makes no difference. The solution I describe tells us what to do in all situations hence also in the situation that 1 was chosen and 3 was opened. The moment in time at which the decision is made is totally irrelevant. One may in advance consider what might possibly happen in the future and what one would then decide. A solution method for the "1,3" case becomes a solution of every case, by permutation of the numbers. Vos Savant said "say", by way of example.

(2) I am not aware of any popular author who presents a simple solution and who moreover claims to be saying "the conditional probability of winning by switching to door #2 conditional on the event that you've initially chosen door #1 and have seen the host open door #3 is 2/3". Of course a section on simple solutions mustn't contain obvious untrue statements, at least, it would not be helpful for the reader to include obviously untrue statements.

(3) I too have read all these sources, and a great deal more. I've corresponded with the authors of several key sources and written a few myself. Richard Gill (talk) 20:22, 23 July 2011 (UTC)

1) I know perfectly well what you're talking about.

2) Most, if not all, popular authors at least implicitly say "the conditional probability of winning by switching to door #2 conditional on the event that you've initially chosen door #1 and have seen the host open door #3 is 2/3". Would you like quotes?

Here's vos Savant The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door. To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger. She doesn't use the words "conditional probability" here, but she's clearly saying the conditional probability the pea is underneath the shell you originally chose must be the same as it's prior probability regardless of which shell is turned over.

Vos Savant's three shells are indistinguishable. Her argument is correct for three shells. But anyway, we are not going to copy every solution in the literature. We are going to give generic solutions which are widely found in the literature. And be careful that what we say in every case is both true to the originals and mathematically correct. There is no need to present any arguments which are wrong! Richard Gill (talk) 19:21, 24 July 2011 (UTC)

Without Bayes wrong from the outset? – Rick, Falk says indeed "if this bias exists and you know about that bias". But read the sources: *Whether or not you know about her bias* is completely irrelevant, you will have learned *nothing* to revise your decision to switch to the offered "door #2". Even the greatest difference in "conditional probability" can nor will ever be of disadvantage for switching, and switching now and here gives you the maximum benefit possible. Quite another issue is teaching and learning conditional probability theory in a maths classroom, although without any relevance to the correct decision asked for, and never a sine qua non for the MHP. Show it where it belongs, by a link to Bayes' theorem. Gerhardvalentin (talk) 08:49, 24 July 2011 (UTC)

Not exactly in the same class as "popular" authors, but here's Devlin: Suppose the doors are labeled A, B, and C. Let's assume the contestant initially picks door A. The probability that the prize is behind door A is 1/3. That means that the probability it is behind one of the other two doors (B or C) is 2/3. Monty now opens one of the doors B and C to reveal that there is no prize there. Let's suppose he opens door C. Notice that he can always do this because he knows where the prize is located. (This piece of information is crucial, and is the key to the entire puzzle.) The contestant now has two relevant pieces of information: 1. The probability that the prize is behind door B or C (i.e., not behind door A) is 2/3. 2. The prize is not behind door C. Combining these two pieces of information yields the conclusion that the probability that the prize is behind door B is 2/3. You tell me, is Devlin saying the conditional probability the prize is behind door B is 2/3 or not?

That was Devlin, a mathematician, trying to give a conditional solution, and doing it wrong. As he later admitted, he had skipped a step. It is easy to fix using symmetry. This has got absolutely nothing to do with your argument. It does illustrate however (a) that conditional solutions can be simple, and (b) even great mathematicians can slip up occasionally. So fir God's sake please let's only put correct arguments in the article. There's no point in copying known mistakes, or claims which are not only false but which have been refuted (such as Rosenthal's, or Snell and Grinstead's claims that Whitaker is asking for a conditional probability.) Richard Gill (talk) 19:36, 24 July 2011 (UTC)

3) With all due respect, you are but one of the many thousands of professors of probability and statistics. There is no guarantee whatsoever that your POV reflects the dominant thinking in the field - in fact, since you're actively publishing in this area one might assume you're attempting to pursue novel or otherwise interesting lines of thought that are nearly by definition not the dominant thinking in the field. The point of the article is not to reflect Richard Gill's thoughts about the MHP - but rather to represent "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources". That you apparently disagree with Morgan (and presumably Gillman, and Grinstead and Snell, and Rosenthal, and Eisenhauer, and Falk [who is a psychologist], and others) distinctly does not mean that we can or should ignore the POV these authors (as well as the many, many others who simply present a conditional solution without commenting on "simple" solutions) express. -- Rick Block (talk) 01:35, 24 July 2011 (UTC)

Thanks for your respect. I published some mathematical details which are common knowledge in my field, and which seemed to me potentially useful for editors here since they helped defuse the conflict by offering bridges and clarification and simplification, but which unfortunately seemed not to exist in convenient citable form in print. You would rather ignore all this and instead bother the reader with a conflict which is now past history and of no particular interest to the general reader. Too bad for those who come to Wikipedia hoping for information which is clear and up to date and attractively presented. Richard Gill (talk) 19:49, 24 July 2011 (UTC)

This is in reply to Rick's initial response in this thread, which raises several issues. Pardon the lateness of my reply.

1. In my non-specific request for sources, I actually had one particular question in mind. It is more specific than the matter of some sources treating the doors as distinguishable and others omitting or disregarding such distinctions. To wit: some time ago I briefly searched for sources that address the use of "say" in mathematical discourse to signify that an arbitrary distinction is being introduced for the convenience of concrete exposition but that it is expressly to be understood that the situation is symmetric with respect to the choice being distinguished. Although I have a shelf full of textbooks that exemplify this common usage, I did not find sources that discuss it in reference to the semantics of MHP, and I have not found any source that gives an authoritative definition for this façon de parler. I would be grateful if someone could find such sources.
2. As discussed with Martin and Richard above, it is not my intention to presumptively ascribe reasons why some sources do not distinguish doors, but to identify as "simple" those that do not do so, by initially indicating simply that this may be stipulated or inferred.
3. I very much do want to treat these as equally valid approaches in a complementary presentation. I believe that mixing different approaches together without clarifying this distinguishing characteristic creates just the sort of "he said, she said" back-and-forth that is eschewed by WP:STRUCTURE. (In the current revision of the article,[1] e.g., using the reading indicated in point 1 above, interposing the K&W formulation of the problem between vos Savant's question and answer casts the latter in a prejudicial light because K&W substitute an expressly probabilistic uniform distribution in a temporal context for vos Savant's a priori symmetry used for a combinatorial solution. Some read it differently.)
4. In pursuit of the WP:TECHNICAL approach for putting the most understandable parts of the article up front and writing down a level, what I want to do is replace subjective assessments of which approaches are "simpler" with the objective characteristic of not treating the choice of goats as a distinction that makes a difference. That symmetry is a fundamental measure of simplicity is attested by many sources in the literature of science and mathematics, including foundational ones of historic significance. There should be no need to cite or discuss that literature, but I believe this is a sound, objective principle for organizing the article. It is this symmetry which makes the truly simple solutions almost as intuitive as the 50:50 fallacy.
5. Regarding the accessibility of conditional solutions, I lean toward the view you expressed earlier (much earlier), though you may have intended hyperbole, that "the absolute probability of a layperson understanding a conditional probability analysis is near 0." This is not to deprecate conditional approaches themselves, but to argue that it is better not to include them in the opening sections. Contributors here are exceptionally knowledgeable, and we would do well not to underestimate the mathematical illiteracy of the general public which, at least in the US, is widely documented.
6. Readers can indeed be confused and misled by interpreting the problem in a way that is inconsistent with one or more of the solutions. My intent is to remove that confusion by providing context for the solutions that identifies approaches to the problem which they address.

In the parable of the blind men and an elephant it would be original research to suggest why each person is inspecting the elephant from a different perspective; but the story would be incoherent if it didn't explain that this is exactly what they are doing — that would be ignoring the elephant in the room. ~ Ningauble (talk) 20:21, 24 July 2011 (UTC)

1) Krauss and Wang have a very short discussion of the meaning of "say": "Although, semantically, Door 3 in the standard version is named merely as an example ('Monty Hall opens another door, say, number 3'), most participants take the opening of Door 3 for granted and base their reasoning on this fact." They go on to explain that this assumption (that it is specifically Door 3 that has been opened and not Door 2) makes the intuitive solution (what are called "simple solutions" in this context) inaccessible. Whether "say" is meant as an example or not, the context of the problem (a game show, with 3 numbered doors) makes it contextually clear the doors are distinguishable. The use of "say" might very well imply the solution is meant to be the same regardless of which door the player initially picks and which door the host opens, but this is a subtly different argument than that the doors are indistinguishable (if the doors are indistinguishable then the solutions must be symmetrical, but not necessarily vice versa). Once again, what I'm suggesting is that we discuss this topic later in the article, but present both "simple" and conditional solutions early on.

2) Implying or directly saying a published "simple solution" is based on indistinguishable doors if the source does not say so itself or if there isn't some secondary source that says this is WP:OR.

3) I fail to see how presenting two solutions, both of which say the answer is the probability of winning by switching is 2/3, can create a "he said, she said" back and forth. Again, my suggestion is to present both sorts of solutions while remaining absolutely and completely neutral about whether one is better than the other. Interposing a discussion that says some solutions assume the doors are indistinguishable and that some don't, and providing an "indistinguishable context" for the solutions that follow (without basing this on a specific secondary source) is at least as biased as the current content (which is sourced to a highly reputable source). Per my comment above, since the game show context implies the doors are distinguishable, another interpretation is to justify the simple solutions based on "full symmetry" which is what the K&W formulation ensures.

4) The "standard" formulation ensures symmetry, making the "simple" solutions and the conditional solutions complementary. Stating the K&W conditions, forcing symmetry, and ensuring the simple and conditional solutions have the same numeric answer is consistent with the bulk of the literature, and seems like a far simpler approach than trying to divide the solutions based on whether the solutions implicitly assume symmetry (which, btw, many conditional solutions do).

5) The quote you refer to was before I'd spent three (!) years writing and rewriting conditional solutions. My current belief is that an approachable conditional solution is in all likelihood easier to comprehend than the "simple" solutions since it does not require the reader to shift from the mental model of picking door 1, seeing the host open door 3, and then deciding whether to switch while standing in front of a closed door 1, closed door 2, and open door 3 showing a goat. I have no empirical evidence of this assertion - but there is plenty of evidence that convincing people the "simple" solutions are correct is extraordinarily difficult. My guess is that these solutions are "simple" only to people who have already "seen the light", and that many proponents of these solutions have forgotten how incredibly counterintuitive these solutions seemed when first encountered.

6) My suggestion is to provide solutions that would appeal to both of the major ways the problem is interpreted. This is both NPOV and likely to be more convincing. I have of late focused my arguments here almost exclusively on NPOV, but I truly think presenting both simple and conditional solutions very early in the article will help rather than hinder most readers. -- Rick Block (talk) 05:01, 25 July 2011 (UTC)

1) I tried to differentiate between saying "indistinguishable" and "undistinguished," the difference being whether it is a distinction that makes a difference. Saying "say" is commonplace mathematical parlance for indicating that it makes no difference; but I acknowledge that the general public ought not be expected to interpret it thusly.

2) I was not proposing to say that such solutions are based on or justified by indistinguishable doors, but that they do not employ distinguished doors. There is a big difference, a difference between original synthesis and objective description. I had not intended to apply this description to borderline cases, but only to clear-cut examples such as the symmetric combinatorial table of vos Savant (who has said, separately, that it makes no difference) and the "combined doors" approach that expressly undistinguishes them by lumping them together.

5) Having spent several years thinking about the problem, you have a very advanced understanding. Reflecting on what you thought of conditional approaches before spending so much time thinking about them may offer a better perspective on making the article readily accessible to readers approaching the subject from a position of ignorance. (More about this in anecdotal remarks below.)

Before I close, I would like to share some anecdotal thoughts about explaining MHP to the uninitiated. Being based on personal experience they have absolutely no relevance for adding information to the article, but I offer them as food for thought anyway.

Having used MHP to entertain and educate several groups of teenagers and young adults, the "explanation" that I have found most effective is isomorphic to the "combined doors" approach. Even adults present are unsure of conditional logic unless they have had some college coursework that covers it. The approach I find most persuasive, after folks have brainstormed a while, is to suggest thinking in terms of "my door" and "Monty's doors." Somehow, the combination of grouping the doors and personalizing them in a possessive sense prompts many people to think carefully about the implications of selective evidence, and this often leads to correct and logically sound answers. Based on this experience, I use MHP as a "teachable moment," not for probability per se, but for thinking about selective evidence in terms of the set selected from and the feature selected for. This does tie to conditional probability, but bringing it up makes people's eyes glaze over. Concluding the session by asking for other examples of selective evidence that can be misleading, and discussing what the evidence really says, can produce entertaining results.

Of course none of this can be used in the article, but as background perspective on making the article understandable it may serve to complement other suppositions that have been offered about how the uninitiated understand explanations of the problem. ~ Ningauble (talk) 18:08, 25 July 2011 (UTC)

Conclusion: Based on the mixed reactions to my original question, I conclude that the answer is no, there is no chance this approach could lead to a way forward. Thank you all for your feedback. I am done here: there is no elephant. ~ Ningauble (talk) 18:11, 25 July 2011 (UTC)

Thank you for your brave attempt anyway. Martin Hogbin (talk) 22:04, 25 July 2011 (UTC)

Why is the MHP said to be a probability puzzle, and not just the question to switch or not

To Martin Hogbin and everybody in the forum. Could you please answer one simple question: Why the MHP is a probability puzzle, and not just the question to switch or not. If somebody finds the query strange, think of the chess. Playing white or black pieces is decided by tossing a coin. Should we appeal to the probability theory to find winning strategies?RocksAndStones (talk) 15:48, 18 July 2011 (UTC)

Good question, RockAndStones, see The Monty Hall Problem is not a Probability Puzzle (It's a challenge in mathematical modelling) or The holy grail of Monty Hall studies, saying: ". . . Therefore the 2/3 success-chance of always switching can't be beaten. I would call this a proof by coupling."

But, aware of two doors having double chance, some said that you "have to consider that – as to the actual game – some slight additional information could be revealed by some conjecturable special method of the host in opening one door whenever he got two goats to show. Thereby, in each and every game he could be signaling additional info, e.g. that actually the odds on the host's second still closed door are "1" (and switching will win for sure in 1 out of 3 cases), or that the odds could be at least 1/2 likewise (in two out of three cases), according to their casual technical assignment. And of course they are free hereupon to do a lot of unprofitable conditional probability calculus, but that never will nor can proof that staying could ever be better than to switch. They misunderstood the question "to switch or not to switch", utterly ignoring that the 2/3 success-chance of always switching just can't be beaten. Gerhardvalentin (talk) 16:45, 18 July 2011 (UTC)

Added: One Auto, Two Goats  –  And they (Morgan et al.) even said that it is incorrect to reason: "As the car originally is equally likely to be behind each of the three doors, the combined chance of doors 2 and 3 altogether to hide the car is 2/3". They say this is a false reasoning, because "after the host has opened door #3 it is evident that the likeliness of the car to be behind door #3 was not 1/3, but it was zero":

That "AGG, GAG, GGA, each point having probability 1/3" is not a solution to the stated conditional problem is apparent in that the outcome GGA is not in the conditional sample space, since door 3 has been revealed as hiding a goat." – That's what "they" say, all of that completely irrelevant to the decision asked for, and completely irrelevant to the given fact that "the switch here and now-decision" never can be beaten by any other decision. Gerhardvalentin (talk) 08:37, 19 July 2011 (UTC)

RockAndStones, is there reason you have asked me in particular?

I believe that it is important to point out that not only should you switch but that your probability of winning doubles if you do so. I am not sure if that answers your question. Martin Hogbin (talk) 17:30, 18 July 2011 (UTC)

The question is whether you should switch or not. The answer is based on whether or not the probability the car is behind door 1 is the same as the probability the car is behind door 2 (after the host opens door 3). Answering the question based on the probability means the problem is a probability puzzle. As it turns out (with the usual interpretation of the problem), the probability the car is behind door 2 (which naively appears to be the same as the probability it is behind door 1) is actually twice the probability it is behind door 1 - so you should switch.

There are both direct and indirect ways to see this.

An indirect way is to imagine what would happen on (say) 300 shows if no player switched (about 100, or 1/3, of these players would win a car) vs. if all players switched (if 100 win by not switching, then 200, i.e. 2/3, must win by switching - all 100 players who would have won a car by not switching lose, but all 200 players who would have lost by not switching win if they switch). Note that this includes players who initially pick door 1 who switch to either door 2 or door 3 (depending on which door the host opens) - but if 2/3 of the players who pick door 1 and switch to either door win, then 2/3 of the players who switch to each door should as well (everything else being equal).

A direct way is to compute the conditional probability that the car is behind door 2 given that the player initially picked door 1 and the host has opened door 3. The host must open door 3 if the car is behind door 2 (this happens 1/3 of the time) but chooses whether to open door 2 or door 3 if the car is behind door 1 - assuming this is an even choice then the host is opening door 3 when the car is behind door 1 1/6 of the time, i.e. if you pick door 1 and see the host open door 3 you have a 2-1 advantage by switching (the conditional probability the car is behind door 2 is 1/3 / (1/3 + 1/6) ). -- Rick Block (talk) 18:41, 18 July 2011 (UTC)

MHP - thought of as a decision problem - can be completely solved without any probability. A player chooses a door and will switch or not switch when the host opens either of the two other doors. For instance: "choose door 2, hold if door 1 is opened, switch if door 3 is opened" (there are 3x2x2 possible strategies - 3 initial choices, hold or switch when the lowest numbered of the other two is opened, hold or switch when the highest numbered of the other two is opened). To any strategy which involves "hold" in some situation there is a strategy of always switching which does at least as well, wherever the car is, and whatever the host does. For instance, in my little example: with the given strategy, if the car actually were behind door 3, the host would be forced to open door 1, the player would hold, and not get the car. But the player who used the strategy "choose door 3 and switch, whatever the host does" also doesn't get the car if it's behind door 3 but does get it in every other situation, whatever the host does.

So, a little bit of thought in advance shows that whatever one should do, it makes no sense ever to "hold". The only question is "which door to choose first".

People who are interested in probability can go on and say more if they want to, but if you just want to know what action to take, the answer is "switch". References: recent articles on arXiv.org by A.V. Gnedin. Approach: using the notion of dominance from game theory.

If you want to add some probability to all this, one might for instance be prepared to assume that the car is initially equally likely behind every door. We see that however one chooses an initial door, if thereafter one switches one gets the car with probability 2/3. This can't possibly be improved since any strategy which sometimes plays "hold" is case-by case worse than an always switch strategy. Consequently, every conditional probability that the car is behind the other door given your first chosen door and the door opened by the host must be at least 1/2. Nothing was assumed here about how the host picks his door to open. If we are also given that he does this with equal chances either door, when he has a choice, then the whole problem is symmetric and the chance of winning by switching can't depend on door numbers. The conditional probability of winning by switching is trivially equal to 2/3 since winning by switching is statistically independent of the door numbers one observes in any particular case (by symmetry).

All in all, one needs no probability at all to solve the decision problem, and after that, if one is interested in probabilities, and if probabilities are available - because more information has been given - then one can get all the probabilities one likes without calculating conditional probabilities at all.

Unfortunately it will be 10 years before this approach is in all the standard text books, so in the meantime wikipedia readers are going to have to be told the difficult ways to solve the problem, and the article will be at least 20 times longer and 20 times less accessible than is necessary. Richard Gill (talk) 20:02, 23 July 2011 (UTC)

Preparing for Wikipedia Content Dispute Resolution

As has been discussed before, I want to take the longstanding difference of opinion about how this article should be written to Wikipedia Content Dispute Resolution and get a ruling so we can inform one side or the other that the consensus of the wider Wikipedia community is against them and that they are not going to get what they want.

For this I need a short description of the differences for the other editors to consider and rule on. So far I have failed to get that, and thus am stalled. So I am changing what I am asking for. I am now asking for any one involved editor to write up a short description of the dispute, and I plan on taking that to Content Dispute Resolution. I no longer care whether anyone else agrees that the description properly describes their position. Everybody had a chance to write up their own short description and failed to do so, so I am moving forward with whatever I have.

I am giving everyone 14 days to write up the short description of the conflict. If I get more than one, I will take a straw poll to see which one I will take to Content Dispute Resolution.

If, after 14 days, I still have nothing, I will follow the example of so many others who have tried to mediate this dispute, call my effort a failure, and quit trying.^{Got the first short description of the conflict, so this option is now precluded.} Guy Macon (talk) 08:54, 23 July 2011 (UTC)

Thank you, Guy Macon. In my view, the controversy is that some insist in presenting only the narrow one-sided view of a solitary source that gloatingly forgot about the famous question "switch or stay", but replaced it with the totally irrelevant question:

After the host has opened "door 3" (not just door Y), what's in maths education the conditional probability of the still closed "door 2" (not just door X)? Cause there could be a slight difference to the "2/3" probability, but within the fixed range of at least "1/2" to "1". And what circumstances could have caused such slight difference?  see The dispute.

This narrow view shows just an irrelevant mathematical side aspect. Repeat: irrelevant, because it never has nor had any influence to the decision that the famous question is still asking for. Readers want to solve the paradox. The fruitless remarkable and controversial history can be shown later in the article. Gerhardvalentin (talk) 15:09, 23 July 2011 (UTC)

I think it would be useful to present the dispute in a way that makes it accessible to as wide a range of user as possible. We do not want to re-run all the same mathematical/philosophical arguments with new people. If we can present the dispute in a way that enables people with no interest or ability on mathematics to contribute it would be an advantage. Martin Hogbin (talk) 16:55, 23 July 2011 (UTC)

Rick's response

The dispute is whether the article should primarily satisfy

1) Wikipedia:Make technical articles understandable), with an initial, extended section focusing exclusively on "simple solutions" that makes no mention of any other solution approaches, in particular the approach using conditional probability. All other approaches will be relegated to later sections of the article intended for experts only. This structural outline (but not the content aspects) are shown in this version of the article.

oder

2) Wikipedia:Neutral point of view, with initial sections of the article addressing the most common interpretation of the problem using various approaches specifically including both simple and conditional solutions. The version of the article following the May 2008 FAR (this version) was more or less along these lines, although the "Solution" section in this version of the article arguably expresses a bias in favor of the conditional approach.

I'd be willing to create a version of the article that better exemplifies the #2 approach if anyone thinks this might be useful. -- Rick Block (talk) 17:29, 23 July 2011 (UTC)

That is an attempt to reword the original argument in a way that gives preference to your POV. There is no battle between making technical subjects understandable and NPOV. The simple solutions win on both counts. They are simple, and they are the most notable with the most sources. The rest is a sideshow. Martin Hogbin (talk) 18:12, 23 July 2011 (UTC)

LOL. I assume you're suggesting #1 comes across as biased - however, isn't this exactly what you've been saying for nearly two years (would you like diffs of your own words)? Guy asked for short descriptions of the dispute. If you don't like this one, write your own. -- Rick Block (talk) 19:15, 23 July 2011 (UTC)

I agree that there is no conflict at all (or need be no conflict at all) between Wikipedia:Make technical articles understandable and Wikipedia:Neutral point of view. All editors should concentrate on trying to make the solutions that they understand best or like best as understandable as possible. The perceived conflict between "simple" and "conditional" would then melt away. It's a red herring. The simplists should be aware of the concepts of conditional probability and take care not to write statements which are mathematically speaking false. The conditionalists could think about presenting conditional solutions which build on the simple solutions. Richard Gill (talk) 20:31, 23 July 2011 (UTC)

I worry that, unfortunately, this is preached to deaf ears. The self-evident principle that a simple approach should not be based on – mathematically speaking – false statements, is contradicted here voluntarily and knowingly. Seeing the words simple approach, they insist to put at the same time the words "incorrect argument, that *because* the probability to hit the car in the initial choice of door is 1/3, hence switching gives the car with probability 2/3."  –  That, and only that is what they accept as "the simple approach" or as "the simple solution", and nothing else. As one participant already has repeatedly pointed out in this discussion. See also The dispute.  Gerhardvalentin (talk) 22:20, 23 July 2011 (UTC)

Re: "That is an attempt to reword the original argument in a way that gives preference to your POV", assuming that this is factually correct, I say "Good!" You all had your chance (and still do - for the next two weeks) to create a short description of the content dispute suitable for Content Dispute Resolution that is more to your liking. If the best description I can get is biased, too bad. We can only hope that the outside editors who will be asked to settle the dispute will be able to see past any bias. Or that someone else will write a less-biased description of the content dispute. Guy Macon (talk) 00:23, 24 July 2011 (UTC)

Just to make sure there is no misunderstanding, I am not saying that the description is or is not biased. I am neutral on that question. I am saying that I don't care if it is biased or not. We are going to CDR (Content Dispute Resolution) in two weeks with whatever (as determined by a straw poll) is the best description of the conflict. Note that it doesn't have to be just one; I can present two or even three competing descriptions to CDR if that makes sense. Guy Macon (talk) 00:40, 24 July 2011 (UTC)

I am happy to go along with your approach and I may attempt to to write something that will enable a a wider range of users to contribute. However, I would like to remind you that I originally came to this page as an outside editor to resolve a content dispute. It is quite clear that we will never reach complete agreement unless we all take the view of the original 'page owners'. Martin Hogbin (talk) 09:39, 24 July 2011 (UTC)

As far as I am concerned, your efforts to resolve the content dispute have been very helpful. So have Rick's, Richard's, etc. In my opinion the lack of agreement is not because of any deficiency in anyone's behavior or arguments, but because the content dispute really is is intractable. When editors disagree on content and cannot make progress toward agreement despite good-faith efforts to do so, that's where CDR comes in. Somebody is going to make an argument that the majority of outside editors agree with, and this page will then be edited according to that consensus. Somebody is going to fail to convince with their arguments and that person is simply going to have to accept the fact that they are not going to get what they want. I really don't know which position will prevail - both sides seem to have valid arguments to me. Guy Macon (talk) 17:38, 24 July 2011 (UTC)

Martin's proposal for a content resolution question

Should this article treat the MHP principally as an undergraduate exercise in conditional probability or should it treat it as a simple, well-known, probability puzzle that most people get wrong but which was correctly and simply solved by vos Savant and many other sources and also include a full discussion of all other aspects of the problem for the more specialist reader? Martin Hogbin (talk) 09:35, 25 July 2011 (UTC)

This would be a fine description except for one thing - absolutely no one is arguing that the "article treat the MHP principally as an undergraduate exercise in conditional probability". Other than that one teeny little detail (i.e. that the description of one side of the conflict is an absolute and utter straw man), it's a fine description of the conflict. -- Rick Block (talk) 05:45, 26 July 2011 (UTC)

You insist on giving the undergraduate exercise in conditional probability equal or greater prominence than the either the simple solutions or the wider picture. You want to centre the article around one narrow POV. Martin Hogbin (talk) 08:18, 26 July 2011 (UTC)

Martin. both you and Rick have POVs. As Wikipedia:NPOV tutorial says, "Everybody has a point of view.[...]your view is just one of many possible views that might be reasonably held." I strongly believe that both of you are, in the words of Wikipedia:Neutral point of view, "Editing from a neutral point of view (NPOV), [which] means representing fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources." Your proposed content resolution question is, in my opinion, not suitable. It describes Rick's POV in pejorative terms and described your POV with glowing terms. Please try to describe the content dispute in a fair and unbiased manner. You will have ample opportunity to make your points in your CDR arguments.

Regarding your earlier comment; "I would like to remind you that I originally came to this page as an outside editor to resolve a content dispute." This is true, but your method appears to involve deciding that one side of the content dispute is right, the other side wrong, and adding one move voice to the argument. How has that been working out for you? Has it resolved the content dispute yet? Guy Macon (talk) 10:10, 26 July 2011 (UTC)

All I have done is produce a proposed statement of the dispute for dispute resolution as you requested. Martin Hogbin (talk) 15:07, 26 July 2011 (UTC)

Gerhard's  p.o.v.

"Solving the paradox":  Is it okay that the article should make evident in the first line that the dilemma / paradox is based on the fact that most of us simply overlook and are missing the "underlying development resp. history", and therefore just are judging "50:50" when we see two still closed doors, one of which must hide the only car for sure, and the other one must hide the second goat for sure, period.  –  Forgetting about, that the first door had been selected as "one out of three", and (remember there's only one car)  that the host's two doors must have double chance in this actual game the famous question is about (a game, that - btw - in exactly this manner supposably never was on real stage before nor thereafter    (...as long as not s.o. appears with  "Double chance??? - you could know a little more closely  than just 2/3, if you just knew  what no-one of us knows,  nor will ever be knowing,  but what,  with ease,  could be assumed... ... ...")

Okay that we should forget about useless conflicts, but should favor a multiple (!) approach in "solving"  the paradox?  Including plausible help to understanding by illuminating and not by obfuscating the paradox?

Together with a short example of Bayes' rule in odds-form, showing the conditional probability, just at the beginning? And later on with a link to the article Bayes' theorem? (that's where all of those ineffective mathematical considerations belong, and not elsewhere.)

Based on reliable actual sources, leaving aside unhelpful arguments about that misinterpreted conflict between "conditional" and "simple", and avoiding any misleading statements which are - mathematically speaking - false statements?

These are my questions, and I support to especially refer to the most actual sources, also. Gerhardvalentin (talk) 16:39, 25 July 2011 (UTC)

Once again: We should show a short example of Bayes' rule in odds-form just at the beginning, just as a help to "convince" sceptical readers. But I am clearly and strictly against the article to remain hyped up and applauded for quibbles which had always been and forever will remain completely irrelevant for explaining the famous paradox and completely irrelevant for the decision asked for (switch or stay), messing about with the readers. As serious sources clearly show that the decision asked for never can nor will depend on conditional probability recommending "don't switch" nor on door numbers, and clearly say that conditional probability theory never was nor will be the unique sine-qua-non necessity to find and to give the only correct answer "switch". Reliable sources that give evidence that to switch forever will produce the maximum benefit and to stay never ever can nor will be any indication. We are to refer to reliable sources concerning the famous paradox, not to other kinds of sources that just are using the MHP as a useful example in quite another discipline, missing that, although mathematically correct, all of this is completely irrespective to the famous MHP and its famous question and to the decision asked for. We just should report on that absurd piffling "historical" conflict. Gerhardvalentin (talk) 13:58, 26 July 2011 (UTC)

Richard's p.o.v.

The conflict arises because editors are using wikipedia principles whose interpretation is highly ambiguous in this case. Neutral point of view? Reliable sources? How does one weight academic writers and popular writers? When explaining a popular brain-teaser to children and old age pensioners, is the point of view of writers of university text books on statistics (anxious to sell Bayes' theorem by means of a fun example) an important guide to organisation of the article? Fortunately, while appealing to wikipedia policies has not helped solve disagreements in this case, we are in a situation where, if we wanted to, we could agree to appeal to another authority: the Truth. <shocked silence>. I know that this is strictly speaking not allowed on wikipedia. But we are only talking about elementary logic and elementary mathematics, agreed?. If we could agree to formulations of solutions which are mathematically and logically correct, then we could easily present simple solutions without telling lies to children, and we easily could explain in constructive (layman's) terms what is gained by doing a more sophisticated analysis. We don't have to quote "authorities" when they write sentences which are not strictly speaking true. There are plenty of "authorities" to choose from. Choose formulations which are true, which don't hide things; no need to tell lies for children. The key to this would be realising that Vos Savant asks for an action, not a probability. We only calculate probabilities in order to guide our decision making. It surely cannot be denied that the fact that an "always stayer" wins only 1/3 of the time, while an "always switcher" wins 2/3 of the time is a good reason to go along to the quiz show determined to switch. It also cannot be denied that this conclusion is attained while only making the assumption that our initial door is correct 1/3 of the time, while a conditional probablity solution requires further assumptions, which are certainly not written down explicitly in Vos Savant's question. Oh well, I am surely preaching on deaf ears, again. Richard Gill (talk) 12:44, 26 July 2011 (UTC)

Probability is completely irrelevant for resolving your dispute, although it is a good thing

You are arguing about the wrong thing. The starting point must be what I know and what do not, and how this affects solution. For suppose I *know* that D1 is winning. Then, obviously, the policy A="choose D1, then stick whichever happens" is the best I can do. But I will not do any worse if I play B="choose D2, then switch whichever happens" (check this!). If I am to some extent unsure that D1 is winning, then B is better than A in any reasonable sense (reasonable means that I want to win). A minor variation is required to discard C="choose D1, switch to D2 if offered, do not switch to D3 if offered". Thus my information about the location of prize (not to say about what the Host is doing) does not matter: notswitching should be discarded. What you are doing on the MHP site is comparable with advising chess-players to decide on the outcome by tossing a coin. What I am saying is a rationale to discard notswitching in a *single* round, if I come to the show once and will never appear there again. With notswitcing discarded, we are left with a choice of a policy out of three. Disputing how to choose one of them is much the same as guessing the roll of a 3-sided die. *Now* thinking of a probability model is a good thing: if some probability model is *assumed* for some reason then *within the frame of the model* to win the prize I should try to minimise the likelihood of the first guess. This can be varied in a usual way regarding known/unknown/known up to a prior, or empirically estimated distribution. This is my POV, you may discard it of course, as switching the POV's is more difficult than switching the doors.RocksAndStones (talk) 15:07, 25 July 2011 (UTC)

Another willing and thoughtful editor gives up in frustration.

The editor Niguable has already left this discusion page. As have many, many others.

Meanwhile, one editor continues to oppose a consensus that contradicts his page ownership.

Another editor makes rulings that there are no coduct violations taking place. This same editor rules that a third editor is not as pious as he, due solely to the length of time of that editor's involvement in these discussions.

Perhaps the article has gained a second Page Ownership violating editor?

75.33.51.159 (talk) 14:34, 26 July 2011 (UTC)

Please do not attack other editors. Comment on content, not on contributors. Personal attacks damage the community and deter users. Please stay cool and keep this in mind while editing. Thank you. --Guy Macon (talk) 08:06, 27 July 2011 (UTC)

To be clear, in writing "I am done here" in a thread above I meant that I was done there, done with the question raised in that thread. I have been here, following this article and occasionally participating, for more than three and a half years. If I have chosen to speak up only on rare occasions when I thought I had something constructive to offer, something potentially more meaty than chewing old bones of contention, it does not mean I am not here, just that I don't have such brainstorms very often. ~ Ningauble (talk) 12:02, 28 July 2011 (UTC)

Perhaps I have misinterpreted the meaning of "There is no elephant." 76.190.251.93 (talk) 13:19, 28 July 2011 (UTC)

Other corrections needed: here is zero evidence of any consensus. Nobody here has posted any "rulings," only observations and opinions. [Comment Deleted] Guy Macon (talk) 19:17, 28 July 2011 (UTC)

Guymacon, I think you challenged me when I made a comment similar to this. If you suspect sock puppetry go through the proper channels. Martin Hogbin (talk) 22:36, 28 July 2011 (UTC)

You are entirely correct, and I do appreciate the correction. Guy Macon (talk) 02:15, 29 July 2011 (UTC)

Ningauble, your views on this subject seem not too dissimilar to mine. Are you willing to discuss this, either here or elsewhere? Martin Hogbin (talk) 22:38, 28 July 2011 (UTC)

"Discuss this" is too nonspecific for me to formulate a response. Regarding the subject of this thread, which is nominally a meta-discussion of my participation here, I have said about as much as I am able to articulate. Regarding discussion venues I think centralized discussion is better. Although I have responded to enquiries on my talk page, I would prefer it not become a fork of this discussion page. ~ Ningauble (talk) 16:01, 29 July 2011 (UTC)

Veritas

Regarding the deletion of this section, the original reverter did not discuss his reasons. Invoking "personal attacks" is a unique interpretation of an edit that doesn't name a single editor. It's equally or more likely that the reverting editor doesn't care to acknowledge the accuracy of the post, and used a contrived excuse to revert it. 166.216.194.35 (talk) 05:58, 27 July 2011 (UTC)

Please remember to assume good faith when dealing with other editors. Thank you. --Guy Macon (talk) 08:06, 27 July 2011 (UTC)

Two ways to look at the MHP

Ninguable (and anybody else) I believe there are two ways to look at the MHP. The first is as a simple probability puzzle. In this case there is a long standing convention to make the necessary assumptions to keep the problem simple (for example that the host chooses evenly when he has a choice and always offers the swap). This is clearly how the problem was intended both by Whitaker and Selvin, as a simple brain teaser. Whitaker did not mention door numbers in his letter to vos Savant so we assume that he did not think the individual doors chosen to be important. Vos Savant unfortunately added the door numbers, intending to clarify the problem. She later recognised this as a mistake. If the problem is taken as a simple brain teaser, the simple solutions are fine.

The other way to look at it is as a serious question. Suppose you were a probability consultant and a client asked you Whitakers question. The solution is now much more complex. You would have to start by asking your client a whole bunch of questions to find out exactly what they wanted to know. These would include things such as, 'Do you want an answer from the perspective of a player on the show?', 'Do you consider the door numbers to be significant?', 'Do you want to know the best strategy to win?', and many more.

Do you agree that these are the two ways to look at the problem? Martin Hogbin (talk) 23:56, 29 July 2011 (UTC)

No, Martin, I disagree. There are as many faces as one can imagine. There is a situation which can :be modelled and discussed, under assumptions required for particular solution. I am very :surprised that nobody in this dispute came up with the solution like "If you know where is the :prize just pick the right door". This problem will *never* be done, and new people will come with :a frish look, very different from the persisting stagnation promoted by the majority of the :editors of this page. The dispute about "correct formulation" of a losely posed problem is a :ridiculous scholastics. In my POV, you disregard my words and that of some editors (notably :Richard Gill) willing here to promote clear views and structure comparable with any serious :mathematical article. The paradox itself that "it is not 50:50" is completely resolved by vos :Savant's argument irregardless of her further opinions about the door labels. If you as most of :practitioners of the MHP promote the probabilistics views, then do not forget to explain to a :laymen what *is* the probability, which is by no means primitive, especially if you appeal to :conditional probs. I had an option to discuss the issue with algebraists, they were slow to grasp :the things this way. The combinatorial viewpoint, which was explained in this discussion several :times is left by you without comment and attention. What does it mean? You did not take care to :look in the argument, or is it so exciting that it is best to wait and see how :other people will :react? You know that the host (as door opener) is a dummy player, he can neither

help by signals nor cheat you by clever door-opening. His behaviour is irrelevant, and this :implies everything you wish to achieve with conditional probs. Please respond if you see the point.

And if you are interested to see new faces of the problem: nothing can be easier. Just open Olle :Haggstrom's textbook Streifzuege..., look at the game matrix, and explain how the structure of the :polytope spanned on the rows (columns) reflects in the 2/3 game value. This is just one of the :faces a quality mathematical article is expected to have. My largest disagreement, however, is the :modus operandi. Instead of having 5-10 drafts to put them together this discussion continues :(very interesting and exciting) farce.RocksAndStones (talk) 06:44, 30 July 2011 (UTC)

Yes, of course there are very many faces to the MHP. Our job is to explain all this to an audience of widely varying interests and abilities. There is an unwritten assumption for mathematical puzzles that you take the problem in the simple way that it was intended. Sometimes, such as with the Two envelopes problem this step is not so easy because the informal language used to describe the puzzle is not capable of defining a precise problem. However, in the case of the MHP there is a simple interpretation of the problem that was undoubtedly the one intended by Selvin, that we know was the one originally intended by Whitaker, and was formulation addressed by vos Savant in her answer. That is where we should start in our article. It is the simplest and most common formulation with the simplest and most common solution.

After the puzzle is out of the way, I completely agree with you. You first have to ask what is even meant by 'probability'. Are we talking about subjective/Bayesian probability where our answer is based on the the information know by a particular individual (such as the player), or are we talking about an objective/frequentist meaning of the word where we envisage a repetion of some process and we consider the relative frequency of certain outcomes? These interpretations may, in the end, result in the same answers but serious discussion is not possible without agreement as to the exact subject of interest.

My point is that a few long-term editors insist that one specific interpretation of the problem and one specific solution to that problem is 'the right one' and that this fact is so important that we can't even talk about the simple solutions without mentioning it in some way. As an answer to a simple mathematical puzzle the simple solution is just fine; as a realistic solution to a real world problem the simple solutions are deficient in dozens of ways. Why give just one of these ways undue prominence? We should discus all these issues in a scholarly manner, after we have answered the simple puzzle Martin Hogbin (talk) 10:16, 30 July 2011 (UTC)

Martin, OK though no answer to my direct quest. Let me explain to be on sure side:

under no circumstances is there a way to win (sometimes, perhaps depending on the history) three :::*cases* out of three. And you have a strategy to make 2 out of three. This is the whole puzzle,

both conditional and unconditional, if you prefer these terms. Everybody willing to attribute :::probs to the cases can do it -- this is secondary, and probs need not be equal. Moreover, :::instead probs you can attribute money value to the cases: all what is needed is additivity.

Of course, the "simple solution" (in my terms : you can make 2 out of three) is the one which :::must stay first, to explain that 50:50 is illusory. This is what everybody understands.

Then we should proceed to discussing why the constant-action always-switching policy is optimal, :::and in which senses it is optimal.

Thus, I see a consensus: you, me, Richard and perhaps Ningauble. Apparently Rick Block and many :::others *love* struggling and will struggle till the end of the world, resolving ambitions on the

level of secondary school math. If we can agree now to draft a reasonable write-up: let us do :::it. If the remaining editors do not want to cooperate -- we can arrange article 3-Door-problem :::in a way we find optimal, leaving some others to computerise their comments to instruct each new :::naive visitor why one needs to multiply 1/3 and 1/2.RocksAndStones (talk) 12:39, 30 July 2011 (UTC)

I agree with you that, 'Of course, the "simple solution" (in my terms : you can make 2 out of three) is the one which :::must stay first, to explain that 50:50 is illusory'. I also suggest that it is a very bad idea for individual editors to complicate this solution with their own pet formulations of the problem or personal interests. The simple solutions stand alone as the answer to a simple puzzle.

After that, there is plenty to discuss. Martin Hogbin (talk) 13:52, 30 July 2011 (UTC)

Citation: "a few long-term editors insist that one specific interpretation of the problem and one specific solution to that problem is 'the right one' and that this fact is so important that we can't even talk about the simple solutions without mentioning it in some way". If you mean conditionalist's approach (that one must for some reason address the problem of odds in the famous situation), then this approach will be gradually pushed in the corner, as it is only suitable for undegraduate probability texts. A layman (in wide and positive sense of the word) cannot understand that the odds depend on the behaviour in the situation when there is a freedom of choice of the door-opener, thus for the general public this approach is a dead-end. Moreover, the conditionalism adds nothing to the dilemma, as the assumption of coin-tossing host is superfluous. Have you constructive suggestions how to proceed, as it is impossible to convince somebody not willing to get convinced.RocksAndStones (talk) 15:59, 30 July 2011 (UTC)

Martin has many times posted a fairly detailed proposal of how the article should be structured. Rick is strongly opposed. I think a majority of presently active editors are for Martin's proposal, if only so that we can say goodbye to the conflict and get to work. That's why I proposed that Martin's proposal be the topic of official "content resolution". Guy seemed to ignore my proposal, I don't know why. He wants names for the two positions, but why not just call them Martinist and Rickist? There will be a decision made by a bunch of wikipedia editors without any particular knowledge or even interest in this particular problem, but who have some kind of authority based on long time work for wikipedia which has been valued by the community. One or more people will be dissappointed, one or more will be pleased, but the main thing is that the thing is settled for a while. Richard Gill (talk) 07:35, 31 July 2011 (UTC)

Martin's plan is incomplete, but it certainly can be taken as a base. Dividing the editors in parties, as suggested above is not a good idea. I am convinced that when the process will start moving the opposition (if any) will dissolve in the air, or will take the normal working attitude. In particular, Rick with his 4 yrs experience in answering amazing laymen questions will certainly be pleased to provide invaluable help, M I right, Rick? As for Guy Macon, his love to order (exhibited in correcting punctuation and indentation of some sloppy mathematicians) might be a huge support too. Now the question is how to settle this technically. We need a draft and (old and new) illustrations. Unfortunately, I am a LaTex man so somebody, not me, should take care of the pictures. Guy Macon, could you help?RocksAndStones (talk) 10:40, 31 July 2011 (UTC)

I'm willing to provide whatever help I can - however I strongly disagree with Martin's outline for reasons I've stated numerous times (specifically that it violates WP:NPOV). As I read the sources, there is a bright line between those that answer the question of whether a preselected strategy of switching is better than a preselected strategy of staying, and those that answer the question of whether a player having initially chosen Door 1 and then having seen the host open Door 3 should switch to Door 2. Starting the article with an extended section based on those sources espousing the former view without any mention whatsoever that other views even exist (because we assume our readers are too stupid to understand the difference) strikes me as endorsing one view at the expense of the other and amounts to a willful violation of one of the fundamental principles upon which all Wikipedia content must be based. Not only are both of these views well represented by highly numerous reliable sources (making this an NPOV issue), but nearly all people initially reading the problem (97% per Krauss and Wang) clearly focus on the latter view. Of course, you can attempt to change the reader's focus from the specific case to the consequences of a preselected strategy and once accomplished this makes the problem "simple" - but effecting this change of focus (particularly without directly discussing it) is far from easy.

As I've also said numerous times, it's not only obviously more in keeping with NPOV but I think more convincing as well to present solutions addressing both questions early in the article (without insisting that one or the other view is "more correct"). Martin's approach asserts the "simple" view is most correct (as it is presented first and without any qualification). The text in the solution section as of the 2008 FAR version of the article arguably says the conditional approach is more correct. We could eliminate any hint of bias with a completely neutral transition between these approaches, i.e. something like "Another approach is to determine the conditional probability of winning by switching given the player has initially selected Door 1 and the host has opened Door 3. Referring to the figure above ...". -- Rick Block (talk) 17:48, 31 July 2011 (UTC)

Rick, you say "As I read the sources, there is a bright line between those that answer the question of whether a preselected strategy of switching is better than a preselected strategy of staying, and those that answer the question of whether a player having initially chosen Door 1 and then having seen the host open Door 3 should switch to Door 2." This bright line which you see is a shining bright red herring. You see it, a few authors see it, but a lot of people and a lot of writers don't. And it is not what we are talking about! This is not the important distinction! There are other preselected strategies than the two you mention: "choose Door 1 and switch, whatever the host does" and "choose Door 1 and stay, whatever the host does". The strategy "choose Door 1, watch which door is opened by the host, and only then decide to stay or switch according to the conditional probability that the car is behind the other door" is also a preselected strategy: we can imagine either door being opened by the host and we can imagine both computations and both conclusions, in advance. The strategy "choose door 1 and see which door is opened, then toss a coin whether to stay or switch" is also a preselected strategy. When we talk about strategies we do not restrict ourselves to the two rather special and extreme strategies which you think is the subject of the simple solutions. We also include your favourite stategy, and we also include the obvious strategy following the "it doesn't matter" answer to the question whether or not you should switch.

Now, it is child's play to see that any strategy which would in some circumstance lead you to stay is beaten case by case (ie where-ever the car is, and what-ever the host does) by an appropriately coupled strategy of always switching. So one can *in advance* decide on totally rational grounds to only consider the three strategies: choose Door 1 and always switch; choose Door 2 and always switch; choose Door 3 and always switch (and also, randomized choices from these three strategies). You, Rick, forget that the *only* reason for determining your action via conditional probability is because this is a way which gives a guarantee that your strategy can't be improved; ie, its overall, *unconditional*, win chance cannot be improved. My apologies that the writers of introductory text books on probability theory don't often mention this explicitly. Probabilists in general know this so well that they don't bother to explain it to other folk. Please, please, realise that checking conditional probabilities is not necessarily the only way to get this guarantee! Any way to show that your overal win-chance can't be improved above 2/3 is sufficient to prove the optimality of "Choose Door 1 and switch whatever". A little thought in advance shows us than in the case of MHP, we may completely forget about staying, in any circumstances. However the car is hidden, whatever the host does. The only thing one should pay some attention to is, which door to choose at the start. If all doors are initially equally likely to hide the car, then the three always switch strategies and all randomized combinations of them all have overall win chance 2/3. Since there is no point whatsoever in considering any other strategies, this proves that "choose a door and switch whatever" is the best you can do. As a corollary, it follows that in this case all conditional probabilities must support switching. There is no need whatsoever for Bayes! No need whatsoever to compute them! No need whatsover to worry about possible host-bias! A little strategic insight is enough.

Sure, wikipedia has to follow the reliable sources and reliable sources are typically ten years out of date. But remember that reliable sources have their "sell-by" date. The conditional probability approach need only be a foot-note for specialists. It's foolish to highlight it in the article. Richard Gill (talk) 18:56, 31 July 2011 (UTC)

I want to put the simple approach first not because it is 'correct' but because it is simple. That is how most technical subjects are treated. Once you move away from the simple mathematical puzzle aspect of this problem there are many problems, questions, formulations and solutions that arise. Your preferred approach is just one way to tackle the problem. Why should we single your preferred approach out for special treatment? If we are going to say, '"Another approach is to determine the conditional probability of winning by switching given the player has initially selected Door 1 and the host has opened Door 3' at the start of the puzzle, why do we not also say 'and another approach is to use game theory... and another approach is to consider the symmetry of the situation... and another approach is to consider the Bayesian perspective of the player .... and another approach... '. That would be absurd. Let us get the puzzle bit out of the way then have a proper scholarly discussion of the wider aspects of the problem. Just to have one special case is just your POV. Martin Hogbin (talk) 19:00, 31 July 2011 (UTC)

The "simple" approach is arguably no more accessible than the conditional approach (the decrease in complexity is offset by the need to change the mental model of the problem). I have argued for including the conditional approach not because it is my preferred approach, but because it is extremely prevalent in the literature (much more so than game theory, Richard's "switching beats any other strategy" approach, etc.). WP:NPOV demands that the article "fairly represents all significant viewpoints that have been published by reliable sources, in proportion to the prominence of each viewpoint". Based on my reading of many, many of the sources, my opinion (not my POV) is that the conditional approach is at least as prominent among reliable sources as the simple approach. Putting the conditional solution on a par with the simple solutions is simply fairly representing its prevalence. -- Rick Block (talk) 02:26, 1 August 2011 (UTC)

No approach comes close in number of sources to the simple solutions. Martin Hogbin (talk) 09:04, 1 August 2011 (UTC)

Rather than continue this argument (again), I'll simply refer the interested reader to a previous time we've discussed appropriate and inappropriate ways to weigh the prevalence of sources, see [2]. -- Rick Block (talk) 15:15, 2 August 2011 (UTC)

A new chapter in the history of the MHP is the dispute itself

Dear all (Rick Block: my respect). Apparently, the consensus is coming. Martin clearly stated that he does not give any dominant role to the simple solution, rather it should stay in the first lines just explaining why 50:50 is wrong, and so because it is simple. The overhelming majority of the MHP-article readers will visit the site only to convince themselves that there is one more source stating that 50:50 does not work, as somebody already had explained to them. Now, in my personal (but absolutely objective) POV one of the most notable events in the history of the MHP is this long dispute of editors itself. There is a very reliable and objective source to which we can refer: the documented over 1000 pages Discussion on the MHP article. Perhaps our longest-term editors could prepare a small essay on how the dispute developed...RocksAndStones (talk) 19:48, 31 July 2011 (UTC)

You are right, I think some aspects of the problem have been discussed here in more detail than anywhere else. Martin Hogbin (talk) 20:08, 31 July 2011 (UTC)

How the dispute developed was the article originally had the form Martin seems to prefer (see for example the version when it was first promoted as an FA). In this version, the one and only solution presented in the "solution" section was essentially vos Savant's solution. In February 2008 an anonymous editor opened Pandora's box. This user took issue with not featuring a solution based on conditional probability but instead featuring a solution that one of the basic sources (the much maligned on these pages Morgan et al.) calls "a false solution". In the initial discussion with this user [3] (discussion continues into archive 7) I played the part currently being played by Martin (!), i.e. resisting the addition of a conditional solution early in the article on the grounds that the "simple" solution is perfectly fine. After MUCH discussion on these pages, a compromise was reached where both "simple" and conditional solutions were presented - and this compromise form was vetted by the entire community during the FA review initiated in March 2008 [4]. The inclusion of a conditional solution has irked various users (e.g. Martin) ever since. -- Rick Block (talk) 01:50, 1 August 2011 (UTC)

Rick Block, thank you, very interesting. The opinions drifted to a triangle Simplism, Conditionalism and Strategism. Simplists assert, quite convincingly, that the simplistic case-counting gives the easiest explanation to the paradox of odds. Strategism promotes academically superior game theoretic viewpoint, which is likely to enter text-books on games and behavioural sciences in the nearest future. For all practical purposed Strategism is dominant to answer what to do and to conclude on the inequalities supporting the solution, without actual computation. However, Conditionalism is inevitable to quantify the odds, which every mathematician would like to do, to understand (from certain position) "how much better". Although dominated by the Strategism, the Conditionalism is unlikely to disappear from the text-books, because the teachers are more interested in exercising conditional probabilities than in solving MHP in the most economic way. W.r.t. the golden triangle my position drifts to the Barycentrism. A concrete suggestion is to think more of the consensus and the Wiki article as the most reliable source itself, which need not give proportional weight to tonns of the garbage literature reproducing the same sources. Following the proportionality principle, most of the article should focus on the Simplism because no other source beats http://www.youtube.com/watch?v=mhlc7peGlGg with its record over 639000 hits. The second-popular among academics is Conditionalism, so the most advanced (and really simple) Strategism is given miserable share. So far. But this will change and it is our task to find balance of the triangle and drift to the equilibrium. In older times people used Roman numerals, which nobody could multiply, and positional system just appeared. So Wiki of that time would be exceptionally retrograde if it'd mention the advanced system only on margins. Can we come finally to practical steps? Could somebody advise how to mass-copy all discussion history, before somebody destroyed it? Would somebody be interested to have a meeting of the editors or a section in one of the educational conferences?RocksAndStones (talk) 17:58, 1 August 2011 (UTC)

Wikipedia keep a complete record of every revision of every page, including talk pages, backed up in multiple locations. The problem is not material getting deleted, but rather the huge amount of material that needs to be edited down to something useful - a refreshing glass of water rather than drinking out of a fire hose.

Re: "In older times people used Roman numerals, which nobody could multiply, and positional system just appeared. So Wiki of that time would be exceptionally retrograde if it'd mention the advanced system only on margins.", here is an interesting quote:

"If Wikipedia had been available around the fourth century B.C., it would have reported the view that the Earth is flat as a fact and without qualification. And it would have reported the views of Eratosthenes (who correctly determined the earth's circumference in 240BC) either as controversial, or a fringe view. Similarly if available in Galileo's time, it would have reported the view that the sun goes round the earth as a fact, and Galileo's view would have been rejected as 'original research'. Of course, if there is a popularly held or notable view that the earth is flat, Wikipedia reports this view. But it does not report it as true. It reports only on what its adherents believe, the history of the view, and its notable or prominent adherents. Wikipedia is inherently a non-innovative reference work: it stifles creativity and free-thought. Which is A Good Thing." --WP:FLAT --Guy Macon (talk) 19:35, 1 August 2011 (UTC)

Guy Macon, thanks for the quotation. You might recall that this citation was reproduced just recently to cool down one indentation-unexperienced newcomer to the dispute. Well, the newcomer had dejavu himself as he forgot the password accompanying a former aliasname... The "universal" Wiki principles should be applied with care when it comes to mathematics. While it requires some empirical experimentation to prove or disprove Galileo's views, the mathematical truth is based on axioms and modus ponens. It is nowhere stated, of course, that Wiki editors of mathematical articles are expected to possess necessary qualification. But if you indeed wish to follow the principle "inherently a non-innovative reference work", OK let us collect the garbage about the problem. Be sure, there will be no place for the conditional probability approach on this scale. So, could you advise me how to copy the discussion pages in some automatic way?RocksAndStones (talk) 20:38, 1 August 2011 (UTC)

It's not that I personally wish to follow the principle "inherently a non-innovative reference work", but rather that Wikipedia policy requires it. I am of the opinion that the policy (which was created out of strong consensus with a lot of input from professional mathematicians) is well crafted and that one is well advised to find out why we have a policy before rejecting it out of hand.

Here are the policies that apply to "mathematical truth is based on axioms and modus ponens":

http://en.wikipedia.org/wiki/Wikipedia:These_are_not_original_research#Simple_calculations

http://en.wikipedia.org/wiki/Wikipedia:Scientific_citation_guidelines#Examples.2C_derivations_and_restatements

http://en.wikipedia.org/wiki/Wikipedia:Routine_calculations#Routine_calculations

As for copying the entire talk page archive, here is a good place to ask that question:

http://en.wikipedia.org/wiki/Wikipedia:Village_pump_%28miscellaneous%29 --Guy Macon (talk) 09:25, 2 August 2011 (UTC)

Guy Macon, for every word of mine a reliable source can be provided... Could you help with a more concrete advice on the issue of discussion pages copying?RocksAndStones (talk) 12:56, 2 August 2011 (UTC)

The Reliable Sources policy was invented so that wikipedia would not be hijacked by crackpots to function as a platform for expounding their crackpot theories of gravity, light, quantum theory, or whatever. The recent discussions during the arbitration procedure on the Monty Hall Problem exposed a strong concensus among the mathematicians that the policy as now stated, if taken literally with regards to mathematics and elementary logical reasoning, is far too restrictive. It is clearly written down by people who have no idea what mathematics is about (lawyers?), and if taken literally would make the job of being editor of a mathematical article almost impossible. In general, articles about subjects belong to well defined academic fields are often written by authors from those fields and take for granted, when getting into the nitty-gritty of the topic, that the reader has sufficient grounding in the field to be able to appreciate what is written. That includes the ability, in mathematics, to appreciate elementary logical arguments or elementary mathematical derivations. One can write, on wikipedia, as an example, that $5x^{4}$ is the derivative of $x^{5}$ without being obliged to find a textbook which includes this specific example. (If challenged, the editor (an academic mathematician) could post a detailed derivation on his university web page and that would automatically become a reliable source; and that without creating a Conflict of Interest). Yet, following a standard algorithm to compute the derivative of a polynomial is, according to wikipedia policies taken literally, an example of "Original Research". So please let's remember that Original Research is not anathema by definition, it's an issue when the results are controversial (challenged) or when they serve personal interests of the editor and in short, when they are not in the interests of readers. The problem we have with MHP is that some editors are invoking "NOR" and "reliable sources" in order to keep an article in a state which represents the state of the art, ten years ago. Of course they are free to do so, and if they do so, the rules of wikipedia are such that probably, they will enventually succeed. But why do such editors insist on withholding beautiful insights from wikipedia readers? This is what I don't understand. We want to write a great encyclopedia, right? The rules were set up so as to serve this purpose. The rules are not an aim in themselves. They should, I think, be applied "in spirit", not "in the letter". When we need to invoke the letter of the rules, collaborative editing has already failed, working on wikipedia is not fun any more, and most important of all, the result is not going to serve either the ideals of wikipedia nor the interests of our readers. Richard Gill (talk) 13:23, 2 August 2011 (UTC)

If by "some editors are invoking 'NOR' and 'reliable sources' in order to keep an article in a state which represents the state of the art, ten years ago" you are referring to me, you are completely and utterly (willfully?) misrepresenting what I'm saying. -- Rick Block (talk) 15:11, 2 August 2011 (UTC)

It's how it comes across to me (and others). Apologies if it is not your intention. But what about the content of what I said? Richard Gill (talk) 06:13, 4 August 2011 (UTC)

Dispute resolution

Guy and others, I understood we were going to use some form of dispute resolution. So far we have had only two suggestions as to how to frame the dispute any there is no sign of anyone writing another. I therefore we suggest we go with both the questions below: Martin Hogbin (talk) 09:58, 2 August 2011 (UTC)

Rick's question

The dispute is whether the article should primarily satisfy

1) Wikipedia:Make technical articles understandable, with an initial, extended section focusing exclusively on "simple solutions" that makes no mention of any other solution approaches, in particular the approach using conditional probability. All other approaches will be relegated to later sections of the article intended for experts only. This structural outline (but not the content aspects) are shown in this version of the article.

oder

2) Wikipedia:Neutral point of view, with initial sections of the article addressing the most common interpretation of the problem using various approaches specifically including both simple and conditional solutions. The version of the article following the May 2008 FAR (this version) was more or less along these lines, although the "Solution" section in this version of the article arguably expresses a bias in favor of the conditional approach.

Martin's question

Should this article treat the MHP principally as an undergraduate exercise in conditional probability or should it treat it as a simple, well-known, probability puzzle that most people get wrong but which was correctly and simply solved by vos Savant and many other sources and also include a full discussion of all other aspects of the problem for the more specialist reader? — Preceding unsigned comment added by Martin Hogbin (talk • contribs) 09:58, 2 August 2011 (CEST) (UTC)

Discussion

Martin asked: Should this article

a) treat the MHP principally as an undergraduate exercise in conditional probability or

b) should it treat it as a simple, well-known, probability puzzle that most people get wrong but which was correctly and simply solved by vos Savant and many other sources and also include a full discussion of all other aspects of the problem for the more specialist reader?

Gerhard says: b) Gerhardvalentin (talk) 10:23, 2 August 2011 (UTC)

Gerhard thanks for your comment. The purpose of this section was to propose a question that we could use as a basis for a dispute resolution process. I imagine we will have an RfC or the like Martin Hogbin (talk) 10:48, 2 August 2011 (UTC)

RocksAndStones says: I stay for Rick's 1), although I do not see what can make up "extended" section. The two "simple" arguments presented now are almost identical. Simulation is not an argument at all, and increasing the number of doors adds a little too. What is important, is the explanation which questions the "simple" solution answers, and why intuition fails. Regarding Martin's itemisation, I stay for c), explain simple solution first then move to strategism and conditionalism, not missing to say which questions these approaching answer, and how they are connected to "simple solution". Then move to symmetrism, variations, etc. Sample articles whose structure could be helpful: Prisoner's dilemma, Poincare conjecture.RocksAndStones (talk) 12:56, 2 August 2011 (UTC)

You chaps do not seem to have understood what this is all about. We are going to engage in some form of dispute resolution but in order to do this we have to tell other people what the dispute is all about. We cannot even agree on how to do that so here are two ways of asking what is essentially the same question. Rick's 1 is essentially the same as what Gerhard has called my b) which is my proposal of simple first without health warnings followed by discussion of more complex solutions. Rick's 2 is the same as my a) which is Rick's suggestion to have some mention of the conditional solutions right from the start.

We hope to get some other people to help us decide which way to go, although those already here will obviously still have a say.

It looks as though we may have to explain to those who come to help resolve the dispute that these are just two ways of asking the same question. Martin Hogbin (talk) 13:28, 2 August 2011 (UTC)

Before we go anywhere with this, I think we should have actual content to show people so they aren't misled by intentionally pejorative descriptions (i.e. no one is arguing the article should "treat the MHP principally as an undergraduate exercise in conditional probability"). I've created two copies of the current article content, Talk:Monty Hall problem/draft1 and Talk:Monty Hall problem/draft2. I don't care who edits which copy, let's say I and anyone else interested in the approach I'm talking about (I welcome anyone) edit draft1 while Martin and anyone else interested in his approach edit draft2. Since they have identical starting points (the current article), we'll be able to diff these against each other as well as the current article. -- Rick Block (talk) 15:45, 2 August 2011 (UTC)

No problem. My only interest is in keeping the simple solutions simple with no health warnings. The rest can then be discussed later. I will edit draft 1 if you like. Martin Hogbin (talk) 16:42, 2 August 2011 (UTC)

Martin - I suggested above I edit draft1 and you edit draft2. Is your suggestion you edit draft 1 simply a typo? -- Rick Block (talk) 19:28, 2 August 2011 (UTC)

Not exactly a typo but I did not read properly. I will edit draft 2 to how the article was after my major editing. Martin Hogbin (talk) 21:34, 2 August 2011 (UTC)

Excellent! Thanks to both Martin and Rick. I want the best possible arguments to be made for each position, and this goes a long way towards accomplishing that. Of course there will also be ample opportunity to simply argue your case.

Anyone interested should take a look at Wikipedia:Requests for comment/Maths, science, and technology to see how other content disputes have turned out and Wikipedia:Requests_for_comment for a more general description of the process. I will also be posting invitations to participate at Wikipedia talk:WikiProject Statistics, Wikipedia talk:WikiProject Mathematics and Wikipedia talk:WikiProject Probability. --Guy Macon (talk) 17:18, 2 August 2011 (UTC)

I have now changed draft 2 to show my compromise proposal. I should stress that it is not the detail that I want to show but the basic structure, where the simple solutions are shown and discussed first, with no health warnings. Everything else, much as it is now, can come later. Martin Hogbin (talk) 21:47, 2 August 2011 (UTC)

Martin - you say this draft does not show your preferred content yet, but it does include various content changes [5] (I'm guessing you overwrote the draft I created with something from some time ago). Are the changes relative to the current text part of your proposal or not? Please edit the text to your liking (without this I think it is not clear what you're suggesting). If you'd like, I could make a stab at what I think you're after (which you could revert if you don't agree). I've edited /draft1 through the "Solution" section (diff here) and have left the remainder of the article essentially untouched (although parts of it definitely need work). -- Rick Block (talk) 15:32, 3 August 2011 (UTC)

I have a couple observations on the current drafts:

• The current revision of Draft1,[6] in characterizing the first solution (vos Savant) as "an intuitive explanation," could be taken to indicate that this involves lies-to-children, i.e., heuristic aids to understanding that should not be taken as valid demonstrations. In the interest of taking a neutral point of view, I think we should avoid language that could be interpreted as a disclaimer.
• The current revision of Draft2,[7] (which includes only the initial sections, with a placeholder for the rest) is confusing in the sentence beginning "Although not explicitly stated in this version..." because most of this is stated in the antecedent version (K&W). All that is missing is that "random" is taken to be a uniform distribution. This is very redundant with the previous paragraph: did you mean to refer to the previous version (Whitaker)?

In both versions I heartily endorse moving simulation from "Solution(s)" to "Aids to understanding." It is neither a solution nor an explanation. (I know that some people find this persuasive but, personally, I think that using stochastic modeling for a discrete problem space that can be fully enumerated on a 3x5 card is a bit ridiculous.)

Ok, that was three observations, not a couple. I will have more at a later time. ~ Ningauble (talk) 16:43, 3 August 2011 (UTC)

The "intuitive explanation" wording in draft 1 is directly from the source (Carlton) - it pertains to that one sentence, not vos Savant's solution, and was the result of an extended discussion during mediation. Perhaps this should be made more clear (I'll make a stab at this). Per Martin's comments below he has made no attempt to make the content of draft 2 reflect his intent (other than the outline). I've made some edits that I think reflect Martin's intent. -- Rick Block (talk) 04:59, 4 August 2011 (UTC)

Better, thanks. I confess to quibbling, but since the validity of solutions has been the subject of heated dispute I am inclined to strive for the utmost neutrality. ~ Ningauble (talk) 15:23, 4 August 2011 (UTC)

I withdraw my support for Draft 2

I am not trying to be awkward and I was willing to give Rick's suggestion a try but it seems that even the regulars here are misunderstanding the purpose of the two drafts and quibbling about the details. Newcomers are even more likely to do that.

All I am trying to get across is that we should first concentrate on the simple puzzle by Whitaker/vos Savant in Parade magazine, the simple solutions, why the answer is not 1/2, and the media furore. Everything else is an academic extension. I therefore feel I can only withdraw my support for Draft 2 and stick with my dispute statement ant the start of this section. Martin Hogbin (talk) 19:11, 3 August 2011 (UTC)

Martin, I think your attempt has limited success because there is too much excitement by the furore. Instead of taking clear academic viewpoint people start telling a story about silly PhD's who got it wrong. It is obvious to me that many disagreed with vos Savant's "solution" because the framework of what later became "standard problem" was not firm at the moment, or people confused designs. It was perhaps not clear if host by chance did not reveal prize or it was the rule of the game. The simple argument tells that always 50:50 is impossible. But it is equally impossible for untrained mind to see that under circumstances it could be sometimes this, sometimes that, and in fact under some mode of Host's behavior 50:50 *can* be possible. All what needs and must be said about the simple approach is just that: picking 1 then always staying wins if prize behind D1; picking 1 then always switching wins if prize behind 2 or 3. Same for picking x in some way. If under all cicumstances the conditional odds were 50:50 then it'd make no difference, contradicting to what was just said. There is really nothing more to say about the simple solution. If somebody tosses a fair coin to solve the dilemma, then odds are 50:50, in this sense there is a way to "create" 50:50. Another fundamental question is the following: in Selvin's letter of 1975 the talk is about probability, with the solution based on symmetric assumptions. In vos Savant's column I do not see the word *probability* at all, look:

"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?"

Maybe the word "advantage" means "probability"? Or maybe the advantage is 2000 buks vs 1000? In my view, one can do no mistake by saying "we assume that..., then we obtain that...". Explainin three sentences why there is one case vs two cases and explain that discarding history is a source of confusion, then move on to deeper issues.RocksAndStones (talk) 20:48, 3 August 2011 (UTC)

Martin - I've made some content changes in /draft2. I believe this is more or less in the direction you want the article to go. Why I'm fairly insistent on showing approximate content rather than just an outline with headings is because without specific content different people will imagine the sections to contain what they would like rather than what I think you're actually suggesting (and I believe these may be considerably different). Regarding sticking with your dispute statement, as both Guy and I have mentioned, your description of the conflict comes across as pejorative. For example, are you seriously claiming the content of /draft1 through the initial "Solution" section (the part I've edited to reflect what I'm actually suggesting) treats the MHP "principally as an undergraduate exercise in conditional probability"? -- Rick Block (talk) 04:51, 4 August 2011 (UTC)

My point

I do not want to or claim to be able to write this article by myself, I like the concept of cooperative editing. My point is one of principle not of detailed content. The MHP may be addressed at three basic levels:

The popular level

This is undoubtedly the level at which this puzzle was aimed, it was published in a popular, general interest, magazine. At this level, the solutions of vos Savant and the other simple solutions are perfectly adequate.

The fact that most people get such a simple puzzle wrong is what the MHP is all about.

The undergraduate level

With a little licence (in allowing the show host to exhibit a personal preference on a TV game show) this puzzle can be turned into an interesting and instructive exercise in conditional probability. This is pretty well what Morgan say in presenting their solution.

Apart from students of conditional probability, this solution will be of little interest to our readers.

The professional level

At this level, as alluded to by Seymann, you cannot even start to answer the problem without a clear understanding of the exact circumstances, the exact question that the questioner would like answered and the basis on which they would like that answer. As Richard will confirm, to attempt to answer questions of this nature on the undergraduate level, without properly attending to the other issues can lead to serious real-life consequences.

My objection, in common with a the vast majority of other editors, is that you want to give undue prominence to the undergraduate level approach to this problem. Specifically that you insist on complicating one of the world's hardest simple puzzles by mentioning in the solution one specific, irrelevant, alternative method of solution that is somewhat narrow in its interest and applicability. Martin Hogbin (talk) 09:57, 4 August 2011 (UTC)

Do we need more time?

12 days ago I wrote that I was going to take this to content dispute resolution with the best description of the conflict available at that time. Since then considerable progress has been made. Would it be useful for me to move the deadline back a week to give everyone more time? Guy Macon (talk) 03:44, 4 August 2011 (UTC)

I think we have two descriptions of the disputed issue. Both editors think the other editor's description is biased and no one seems interested in writing another. I think we might as well go with these two. Martin Hogbin (talk) 14:01, 4 August 2011 (UTC)

Although there is much that I agree with in the ideas of both Martin and Rick, the way that the positions have been put forward would lead me to !vote "no" to both. If I may be so bold as to presume to single out what seems to be the most specific point of dispute regarding what is proposed for the article, without reference to the rationales for what is proposed or the details for structuring what is proposed, it appears to me that:

• Rick proposes to introduce conditional probability early in the article, and Martin proposes to defer it until later in the article.

There are collateral issues because the devil is in the details, and I may be missing some distinctly different points of disagreement that might be brought out more clearly, but this seems to me a better way to frame the question. Pardon me for engaging in a little hyperbole, but it just doesn't seem constructive to frame it as a question of whether or not to follow policies and guidelines, or whether or not to write an undergraduate study guide. Would the parties be agreeable to putting the question thusly for dispute resolution, or does it really need to be framed in more general terms, such as what the subject of the article ought to be? (There are other matters of dispute lurking around this article, but I have confined myself to the positions that have been put forward by these two contributors.) ~ Ningauble (talk) 16:38, 4 August 2011 (UTC)

I think this is perhaps the net effect, although IMO Martin's insistence that the initial sections of the article mention nothing other than so-called simple solutions directly conflicts with NPOV. His counter-argument is that having the vast majority of the article be about nothing except the "simple" puzzle as described in popular sources plus any academic sources that happen to use exclusively "simple" approaches (effectively marginalizing all other sources whether they claim to be about the "simple" puzzle or not, specifically including the significant number of sources which directly criticize the "simple" approaches) appropriately reflects the WP:WEIGHT of the respective sources. That this is fundamentally a POV issue is absolutely clear to me, and I don't see how it can be intelligently discussed outside of Wikipedia policies (and, yeah, there are other issues too - but this is the main one). -- Rick Block (talk) 19:34, 4 August 2011 (UTC)

Rick, I do not understand why putting things in a logical order is POV. I am not suggesting that we say the simple solutions are 'correct' (although there are many editors who do say this, supported by sources), I am not suggesting that we say the 'conditional' solutions are 'wrong' or unnecessary. I am suggesting that we discus all forms of solution in a scholarly manner based on what reliable sources say about them.

'Simple first' is the format used by text books and encyclopedia article. What is POV is to insist that just one specific and rather narrow aspect of the problem must be mentioned at the start of the article when we are still trying to discus the basic mathematical puzzle. Martin Hogbin (talk) 09:21, 5 August 2011 (UTC)

Ninguable, why do you not propose a statement of the dispute. Martin Hogbin (talk) 09:21, 5 August 2011 (UTC)

I did propose a different way of stating the present dispute. It may have been overly specific or overly simplistic, but I did so in order to encourage the disputants to frame their positions first in terms of what they propose to put in the article, and to follow that with reasons for doing so. I will participate in the DR when it gets underway, and my response to the reasons will mostly be "both/and, not either/or." The real question is what to put in the article and where to put it, and I expect that the DR discussion, not unlike the literature on MHP itself, will wander all over the map regardless of how the question is posed.

Why do I not propose a different dispute of my own? Although one idea I floated above was labeled disputatious, I do not consider it so. I offered it in the form of a question, and the responses persuaded me that it was not the best way to achieve what I was driving at. Part of Rick's objection to my idea is subsumed by the present dispute, as I see it, and does not need to be raised as a separate one. The deeper problem with my idea, as I see it, lies in the conundrum that although the "simple" solutions are seen as simple answers to a simple question, they are also seen to arise from less simple considerations of less simple questions. I still believe that this needs to be brought out more clearly, but I do not dispute that my idea was inadequate for clarifying something that is truly tangled. I may offer another approach to this at a later time.

I will also take a crack at improving some other aspects of the article. I will not be doing so with the intent of proposing to dispute, but if it results in disputation then that will come after they are broached, not before. Since the article is now entering into formal dispute resolution, I will probably wait until the dust has settled before raising any new issues (or readdressing old ones if such turns out to be the case). ~ Ningauble (talk) 14:47, 5 August 2011 (UTC)