Talk:Monty Hall problem/Archive 27

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  (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 .  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)

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)


 * Martin - you're not only putting "simple first" (which, btw, /draft1 does as well so please stop trying to imply I'm suggesting not putting simple solutions first) but putting simple solutions first in a section called "Simple solutions" with no prior mention of any other solution approaches (not even a brief introductory paragraph with forward references!) and all other solution approaches in a section called "Other solutions". By doing this we are not only saying the simple solutions are correct but that they are the undisputed primary approach to the problem (considering all sources). IMO this does not represent (per WP:NPOV) "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources".  -- Rick Block (talk) 14:55, 5 August 2011 (UTC)
 * Rick, putting the simple solutions first says nothing about them except that they are simple. Why do you think this format asserts that the simple solutions are 'correct' or that they are 'the undisputed primary approach to the problem '?  In any case, we are going to give a full explanation of the status of simple solutions, according to sources, later in the article.


 * There is an argument that if the article just had the simple solutions and much later on the 'conditional' solutions this might be construed by some as indicating that the simple solutions were more important but I am not proposing this. I am suggesting that the later text explains the relative strengths and weaknesses of all the solutions.  This surely overrides any vague impression given by the order the solutions are presented in. Martin Hogbin (talk) 17:47, 5 August 2011 (UTC)


 * I repeat, stop trying to imply I don't want the simple solutions first. I'm starting to wonder if you're saying such things to be deliberately misleading.  They ARE first in /draft1 (you have read this, haven't you?).  The major difference between us is NOT that you're suggesting presenting the simple solutions first, but that you want ALL other approaches to be presented much later and in a way indicating that they are less important (the headings "Simple solutions" and "Other solutions" explicitly indicate this).  Later text explaining differences is fine, but can't and won't override the structural POV induced by your outline.  -- Rick Block (talk) 18:41, 5 August 2011 (UTC)


 * Your response shows why I have withdrawn my support for the drafts. This method clearly will not work as even you have misunderstood what I am getting at.  The section titles are irrelevant, these can be decided by cooperative editing once we have the basic structure agreed.  All I am proposing is that the treatment of the MHP as a simple mathematical puzzle is presented first, without mention of the potential mathematical complexities because the simple solutions are:


 * Simple.
 * Most well known.
 * Have by far the most supporting sources.
 * Will be of most interest to the vast majority of our readers.


 * There is nothing in WP:V,WP:NPOV, or WP:WEIGHT which tells us we should not do this. Martin Hogbin (talk) 19:07, 5 August 2011 (UTC)


 * Martin - Have you read /draft1 or not? What about it conflicts with your stated goal of discussing the "simple mathematical puzzle first"?  Are you suggesting the conditional solution in this draft addresses some different problem? It reflects a very widespread interpretation of vos Savant's statement of the problem without editorially taking any stance about which interpretation is more correct.  By omitting any conditional solution from this section of the article you are making the article take the POV that the "correct" interpretation (which, not coincidentally, is your POV as well, right?) is to completely ignore the door numbers and (as well) ignore the example situation described in the problem statement (where the player is asked to switch after initially selecting door #1 and then seeing the host open door #3).  Once again, this does not represent (per WP:NPOV) "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources".  You keep accusing me of bias.  However, look at our stances here.  I'm arguing simple AND conditional solutions be included early in the article (with a forward reference to other approaches as well), with NO "health warnings", with "simple" solutions presented FIRST, with both sorts of solutions presented as EQUALLY correct.  You're arguing anything other than "simple" solutions must be deferred until much later in the article.  Can you honestly not see who's biased?  -- Rick Block (talk) 06:29, 6 August 2011 (UTC)
 * Rick I have looked at /draft1 and it is exactly what I and many other editors do not want. It pushes a POV, contrary to the vast majority of sources, that the partially complete 'conditional' solutions have some special significance that requires them alone to be included near the start of article.
 * The simple solution do have a special status, for the reasons I have listed above. I not you have not challenged these. Martin Hogbin (talk) 08:28, 6 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)


 * I oppose bringing this mess to dispute resolution. Yes, there is an ongoing dispute about what the article should say. There have been multiple attempts to resolve that dispute. There's a perceived impasse. A standard approach to dispute resolution is getting the sides the agree what the dispute is about. After years of effort by some very smart people, I don't think that there is a clear statement of anything. We have somebody saying X believes this, and then X comes back and says, "No I don't".
 * Sitting on sidelines, I don't get a sense of a reasonable dispute. If there's no clear dispute, then what gets resolved?
 * There seem to be some broad agreements among various sets of editors, but those seeming agreements are confounded by disagreements on other issues.
 * Another tactic is to divide the dispute up into smaller pieces and attack them individually. There seems to be an almost studious attempt to avoid resolving smaller pieces. There are discussions, for example, of health warnings, but the health warning is not isolated as a small piece, but rather sucks in all the other disputes.
 * There's also a strange civility going on here. Editors are so polite and so circumspect that they avoid focused disputes.
 * There seems to be an inordinate focus on mathematics here. The debate seems to be much more about mathematics than the actual MHP. There's more discussion about the mathematical clarity of MvS's description than the contestant's confusion. For all the talk about the weakness of MvS description, the math expositions just cover the simple case. What is the conditional probability debate really about if it's only being applied to the simple case? Is Bayes is too difficult for mere mortals?
 * Part of the debate is about terminology: aids to understanding, simple solutions, and other solutions. It's a petty dispute, and I think it is best addressed by avoiding it. Too many people are investing too much in the terms. A demonstation with simulation may not rise to some desired level of mathematical clarity, but it was simulation that convinced MH and lots of school children. And the absence of simulation still has a lot of people claiming that staying or switching are 50:50. Simple solutions: are they right or wrong? Should other solutions be given dominance? I have a lot of trouble with those issues.
 * I was opposed to forking the drafts some time ago, and I'm still opposed now. It's difficult to manage, in discourages edits in the main article, and it suggests ownership.
 * I'm also opposed to this current path of guided defeatism.
 * Glrx (talk) 20:34, 5 August 2011 (UTC)


 * Interesting take. I am going to have to think about the above. On small added point; unlike many disputes, we will be going into this one with the knowledge that the efforts of multiple mediators and editors have failed to get the sides to agree, and with the expectation that a wider circle of editors will arrive at a consensus - which implies that somebody is going to have to resign themselves and say "I still think I am right, but the consensus went against me." Guy Macon (talk)  22:02, 5 August 2011 (UTC)


 * Right, exactly. Trying to reach agreement failed for years because of "I still think I'm right", and mediation has failed twice, because of "I still think I'm right". Then arbcom found it was time to end ownership by clear restrictions. Still seeing that "I still think I'm right" continues, it is time to reveal "but the consensus is against me". Gerhardvalentin (talk) 00:35, 6 August 2011 (UTC)


 * Arbcom banned one editor and restricted another for page ownership yet the views of the original page owners still dominate this article. Martin Hogbin (talk) 08:40, 6 August 2011 (UTC)


 * Trying to reach agreement failed for years for two reasons. The first is that the editors did not have enough insight, which on satisfactory levels appeared only recently, thanks Richard's efforts. The second reason is even simpler: ambitions. Generally speaking ambitions is a Good Thing, which actually brings people to this discussion. Who is more biased? I think Martin. "Simple soltion" is worth three lines, and it explains that one constant-action policy wins in one case, another in two. The cases are disjoint. There is nothing more in it, you can have years of fruitless meditation, talk of simulation and all this nonsense. You can talk of media furore building up on the same story... Martin never confirmed that he understands what I was talking about, positively reacting when I confirmed that simple should go first (I re-confirm this view). Another point is understanding what is optimal and in which sense. Here, Rick is showing some progress and flexibility, although the draft calls for many improvements. Yes, I think we need more time.RocksAndStones (talk) 08:22, 6 August 2011 (UTC)
 * I agree, I do not know what you are talking about, can you explain further please. The simple solutions together with the associated media furore and the fact that most people get the answer wrong do have a special status.  The vast majority of sources on this subject are concerned with those issues not alternative approaches to the problem. Martin Hogbin (talk) 08:40, 6 August 2011 (UTC)


 * Martin, what I am talking about is called Gezond verstand, Common sense, Gesunder Menschenverstand, El sentido común, Здравый смысл, Senso comune. This has been explained to everybody in this forum by Richard and myself at least half a dozen times. You compare strategy A="choose D1 always stick" and B="choose D2 always switch". Whenever A wins, B wins too. That is, A is dominated and must be discarded. In the same way every policy which allows for nonswitching must be discarded. After this simple fundamental observation, the decision-maker is left to meditate about the question which door is the worst (in any sense you like), because the optimal strategy is O="choose worst door then aways switch". Once you see that, your simple solution, conditional arguments etc all belong to the dust bin. If a crowd of academics failed to observe this in 35 years, it is their problem. Gerhardvalentin, what you say makes no sense to me. The conditional probability for switching depends on the distribution of the prize and behaviour of host. Varying these the conditional probability (provided host tosses biased coins) varies between 0 and 1. Example: if D1 is 100% winner and if you choose D1, then the conditional probability to win with switching is 0.RocksAndStones (talk) 13:42, 6 August 2011 (UTC)
 * RocksAndStones, the question is about a "one time game show" that, in the given manner, may never have been on stage in real life. So neither the distribution of the price nor the behaviour of the host in choosing which door to open if he should have two goats to show, has ever been known nor will ever be known. Textbooks in maths can and will and do assume various kinds of distribution of the prize and special kinds of host's behaviour for this case. In the standard version, depending on host's choice if he has a choice, giving conditional probability of at least 1/2 (but never less) to 1 as a result, so always varying around 2/3, but forever unable to show that staying could ever be the better decision. All of this is an interesting mathematical aspect, but never needed to give the only correct answer "switch". Regards, Gerhardvalentin (talk) 15:44, 6 August 2011 (UTC)


 * The crowd of academics were statisticians with a prior interest in teaching Bayes' theorem and generally with no knowledge of game theory or decision theory. The guys who could easily have pointed out the domination argument 35 years ago were games theorists and mathematical economists. But unfortunately, in that field, writing pedagogical articles is considered a waste of time and energy - everyone is working hard to win a Nobel prize. But fortunately, we do now have the domination argument to hand. By simple inspection, any strategy which could involve staying with your chosen door in some situation, is soundly beaten by a carefully paired strategy in which whatever happens, you switch. Wherever the car actually is, whatever the host does, the paired "always switching" strategy gets the car every time the sometimes staying strategy does. And there are situations where staying gets a goat but the paired always switching strategy gets the car. Anyone who is wise enough to realise this, need not think about staying at all. There is no need at all to think about conditional probabilities. The only thing worth devoting some thought to, is how to initially choose your car. Since you're going to switch anyway, you obviously have to pick the door least likely to hide the car. And if every door hides the car with probability 1/3, then it doesn't matter how you choose. You'll get the car with probability 2/3. And this cannot be improved. Discussion of conditional probabilities is superfluous. Discussion of host bias is irrelevant. I think editors would benefit by trying to internalize this argument and looking for ways to explain it to plain folk. Rather than fighting about how to present material which ordinary readers will not grasp. Do simple solutions first to get the probabilistic intuition right. Then present the domination argument to show that always switching cannot be beaten. Intuitively, easily. Richard Gill (talk) 15:16, 6 August 2011 (UTC)


 * And the mathematical puzzle that -  given you assume to have additional information on the actual location of the car, that no one will ever have, then, based on such "private assumption" and distinguishing door numbers, you could be "correctly" varying the conditional probability within a very very restricted range, but always closely around 2/3, representing the very correct effect of such private assumptions, is of no relevance for the famous decision asked for, as recent sources clearly show. This mathematical aspect is what it is and forever was, just a very interesting mathematical aspect but without any influence to the famous optimal decision asked for in this said "just one famous game" that the famous question is about. Textbooks teaching maths problems is quite another issue. Gerhardvalentin (talk) 09:28, 6 August 2011 (UTC)
 * Gerhardvalentin, indeed assuming that you have a small insider's advantage, AND assuming almost symmetric host the conditional probability is close to 2/3. Einverstanden?RocksAndStones (talk) 18:45, 6 August 2011 (UTC)
 * Nein, RocksAndStones, not agreed, for that's the dodgy matter of dispute. Maths teachers like sundry "assumptions" as waste conditions and, according to their input forever will get then the adequate results. So that point is moot. For the "standard problem" all of this is not relevant to get the "only correct answer". You are right, assuming to have insider's advantage will result in probability to win by switching for the most extreme case of at least 1/2 (factor 2/3) and at the same time(!) likewise to 1 (factor 1/3). So always varying around 2/3.


 * What can conditional probability and superfluous assumption contribute to getting the only correct solution, the only correct answer, the only correct decision? Nothing but to confirm once more that nothing can be better than to switch here and now, in this one special game the question is about. That's no news at all.


 * Superfluous additional assumptions and their handling in conditional probability theory belong to the realm of probability theory. Solving the paradox of the MHP is easy, as anyone can clearly find out that two doors have double chance. And if you like you can add  "have exactly double chance, because probability to win by switching will be at least 1/2 (factor 2/3) to even 1 (factor 1/3) in this one special game the question is about".  That's all what conditional probability theory can contribute. So no forum here to teach and to learn conditional probability theory and to teach and to learn application of Bayes' theorem. That belongs to the maths' forum. Regards, Gerhardvalentin (talk) 11:14, 7 August 2011 (UTC)


 * Gerhardvalentin, Perhaps, V R speaking about the same thing, so a "scheinbar" disagreement results from wording, instead of writing formulas. What I meant to say was: if Connie is a bit clairvoyant and can guess the worst door with chance say 0.35, her win probability is close to 2/3 no matter what Monte does. If we understand standard assumption as the uniform distribution, then always-switching gives 2/3 (no matter what Monte as door-opener does), although you really need an additional argument why Connie cannot do better (no matter what Monte does). That an argument is needed, can be seen with 4 doors, when Monte can help win with 100% probability by signaling through sequence of 2 doors to reveal. Now, as far as I know, Olle Haggstrom and Richard were first to focus on this Holy Grail argument. Combinatorially, the Holy Grail result means that if Connie's decision policy is not allowed to exploit the actual location of prize (which formally means she "does not know" where it was hidden) then she cannot win in all three cases out of three under any circumstances. I agree, that the site is not a forum for teaching the conditional probability. However, from the viewpoint of Markov decision processes (and other standpoints), the conditional probability is a natural quantification of the advantage of switching. Although the existence of cond. prob. is unnecessary assumption and its computation adds nothing to the right decision, it is a valuable thing. In particular, if the distribution of the prize is not uniform, and the first quess was not done in optimal way, you may decide for sticking on base of conditional probability. I do not object the cond. prob. approach, it only needs to be given a proper place as a tool to quantify the advantage.RocksAndStones (talk) 12:14, 7 August 2011 (UTC)
 * Rocks&Stones, R U interested in my view?  Again, I M not fully einverstanden. Let us stick to the standard version, where the host not just offers to switch if you should have chosen the prize, etc...  Let us stick to the standard version, where unproven "assumptions" of clairvoyance and opaque doors never will influence the decision asked for.


 * Yes, Richard's papers confirm that indeed there never will be any better decision than to switch here and now in this very special game the question is about, and moreover that there never will be any better decision than to switch in every one of these games, if this special game should ever be on stage in real life. I would like to show the chances for switching  of 2/3  in "odds form" just at the beginning, also.  Just to help convincing the reader.  And later that the chances forever will remain within the very strict range of  (at least!) 1/2 to even 1,  and without given better knowledge forever exactly 2/3. But I M strictly against the totally unproven brazen assertion that the MHP is incomplete without the conditional calculus showing a variable representing the totally unknown "host's special behaviour" in this special game. Such variant should be treated where it belongs, in some maths forum, for it is not needed within the MHP article, and it does not belong to the MHP article. It is an interesting maths aspect, but is without any relevance to the question asked for and to the correct decision to switch. And the fulminant nonsensical history can be shown in the "history" section. Regards, Gerhardvalentin (talk) 13:57, 7 August 2011 (UTC)


 * Gerhardvalentin, sure I am interested in your viewpoint and appreciate it. Let us review your statements word-by-word, to avoid ambiguity. Richard's papers confirm that indeed there never will be any better decision than to switch here I agree, vorausgesetzt uniform distribution, this was shown by Olle Haeggstroem (Lehrbuch Streifzuege der W-Theorie) and Richard by somewhat different arguments. I would like to show the chances for switching of 2/3  in "odds form" just at the beginning, also. If you mean unconditional odds - I agree this is easy and can be done at the very beginning, but if you mean conditional odds you need the assumption of symmetric host for the case when he has a freedom of choice. And later that the chances forever will remain within the very strict range of  (at least!) 1/2 to even 1,  and without given better knowledge forever exactly 2/3.  I agree, under the assumption that host tosses a perhaps biased coin when he has the fredom of choice.But I M strictly against the totally unproven brazen assertion that the MHP is incomplete without the conditional calculus showing a variable representing the totally unknown "host's special behaviour" in this special game.  I agree with you at this point completely.Such variant should be treated where it belongs, in some maths forum, for it is not needed within the MHP article, and it does not belong to the MHP article. It is an interesting maths aspect, but is without any relevance to the question asked for and to the correct decision to switch. And the fulminant nonsensical history can be shown in the "history" section.  Well, the subject belongs to the MHP site, and the "history section" is indeed to where it belongs. Now, I have a straightforward question to you: have you ever taken 15 minutes to understand the dominance argument, explained since March at least half a dozen of times on these discussion pages?RocksAndStones (talk) 18:01, 7 August 2011 (UTC)


 * Thank you for your reply and your questions, I'll try to answer on the arguments page. Arguments page. Regards, Gerhardvalentin (talk) 22:03, 7 August 2011 (UTC)

Let us go to dispute resolution now!

 * We have a dispute - that is the one thing that is undisputed.


 * We have two statements of the dispute, from different sides.


 * No one here is prepared to produce another statement of the dispute.


 * We have a dispute resolution process.


 * It is absurd to suggest that we somehow resolve the dispute before going to dispute resolution.

Let is do it now! Guy, were you suggesting an RfC on the subject? Martin Hogbin (talk) 08:33, 6 August 2011 (UTC)


 * I am going to handle this one step at a time, while seeking input from others here on which way to go at each step.


 * I am choosing as a first step a vote of confidence; am I the right person to shepherd this through Content Dispute Resolution? Have I shown bias or favoritism toward one side?  Anything else I could be doing better?  Public criticism is welcome, but if anyone wishes to criticize me without doing so in public I welcome email, which will be kept confidential.  Note: if you use the Wikipedia email system, certain other people are able to access it (Wikimedia Stewards and ArbCom members, IIRC, but don't quote me), and there will be a permanent record kept at Wikipedia. Email sent directly to cdrcomments [at] guymacon.com will only be seen by me. Guy Macon (talk)  13:33, 6 August 2011 (UTC)


 * I vote in favour of keeping Guy as shepherd. Richard Gill (talk) 14:56, 6 August 2011 (UTC)
 * Guy please go on as the shepherd. Gerhardvalentin (talk) 15:34, 6 August 2011 (UTC)
 * I fear this dispute will drag on for ever with the original page owners maintaining their dominance over the article. Martin Hogbin (talk) 18:31, 6 August 2011 (UTC)
 * I stick with the herd Richard+Gerhardvalentin. Martin: so far Rick Block showed a good flexibility and will. Join the party!RocksAndStones (talk) 18:45, 6 August 2011 (UTC)
 * Let is see how it goes. I predict no useful progress will be made. Martin Hogbin (talk) 22:01, 6 August 2011 (UTC)


 * OK, let us dispute. My statement of the dispute is an exceptionally simple resolution of your dispute. Discard nonswitching strategies on the ground of dominance. Put the dominance argument in the first position, explain how it implies both simple solution and the decisive inequality for the conditional solution. Explain why the dominance is stronger, as neither requiring assumption on the distribution of the prize, nor on the host's behaviour. Leave the current contents along with media furore (that's you think the problem is about) to the section "History of the problem". Proceed with variations.RocksAndStones (talk) 13:42, 6 August 2011 (UTC)
 * Thank you RocksAndStones, imo this is the best possible way to go, and I especially welcome the section "History of the problem". Gerhardvalentin (talk) 15:34, 6 August 2011 (UTC)


 * I do not have any problem with Guy's impartiality, but I have a sinking feeling that the situation is unmanageable. I find myself agreeing with much of what Glrx said in the previous thread: what gets resolved here? Although valiant efforts have been made to formulate questions to put before the house, I don't get a sense that addressing these questions is likely to resolve the multitude of issues that stymie efforts to rescue this formerly featured article from ignominy and shame. I hope that Guy can prove me wrong. ~ Ningauble (talk) 15:49, 7 August 2011 (UTC)


 * I have no problem with Guy attempting to shepherd this through the dispute resolution process. However it now seems like there are at least 4 different proposals on the table (from Martin, Rick, RocksAndStones, Glrx) which I suspect guarantees none will garner an obvious "consensus".  See, for example, what happened previously when there were three "change" proposals discussed .  Other than a highly biased claim that this discussion showed a consensus for "change"  the outcome was completely inconclusive.  Again, it is extremely clear to me that this is a POV issue and must be treated as such - meaning that we should be talking about which sources say what and how much prominence within the article to give the POV expressed by each set of sources.  Instead, what some here are continually arguing about is which POV is "best".  My answer is that each of them, in their own way, is best but this is utterly, completely, and fundamentally the wrong question.  Our task is to write an article that reflects "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources".  This quote is from WP:NPOV.  If anyone here has not read this, please do so (now).  Mathematicians should be good at following the logical implications of a set of axioms.  WP:NPOV and WP:V are Wikipedia axioms, not truth (the intent is of course that "truth" is implied by wp:npov and wp:v).  We should never directly be talking about "truth" here, but rather who (which source) has already said (published) what, and how much prominence (how many other sources say the same thing) their view should have in the article.  -- Rick Block (talk) 19:33, 7 August 2011 (UTC)


 * We are not looking for a willingly entered consensus now. That's the point. We are going to a higher level to *impose* a consensus. I think the outcome will be random and this is just a thoroughly sensible way of resolving an impasse and moving on. Richard Gill (talk) 07:40, 11 August 2011 (UTC)


 * Just so everyone knows, I am carefully pondering all the information above and thinking about how best to proceed. Several people have made some very good points. Guy Macon (talk)  19:57, 7 August 2011 (UTC)


 * I'm opposed to Guy Macon. There is no question that he is well meaning and has devoted a lot of effort toward resolution. Guy has encouraged editors to talk and to state their opinions. He has shown deference to all sides. Although he has adopted the role of the mediator, that mediation isn't progressing. The argument isn't getting focused. Guy has not been identifying issues and suggesting they be addressed independently. In addition, Guy's efforts have unwitting encouraged ownership of the article. There's a huge amount of text on this talk page, but there's little to show for it. The sense is that the purpose here is just to argue, and the underlying goal is to argue in a higher forum. Glrx (talk) 16:39, 8 August 2011 (UTC)


 * Thanks! Exactly the sort of constructive criticism that I was looking for. It is my opinion (backed up by multiple failed mediations) that there is a basic content dispute that cannot be resolved by argument or compromise.  I really think that all concerned, and especially Martin and Rick, have made a good-faith effort to make their arguments understandable and to evaluate each others arguments, and I have seen multiple attempts at compromise. We will soon see if Conflict Dispute Resolution fails as well. Guy Macon (talk)  04:10, 9 August 2011 (UTC)


 * I think that Guy's diligent efforts have shown, independently of earlier attempts, that there will be no consensus. Argument doesn't help and compromise is not accepted. So it should now go to a "higher court" which will simply enthrone one of the "positions" as being the way forward (for the time being). Seems to me eminently reasonable. Life must go on. Richard Gill (talk) 07:49, 11 August 2011 (UTC)


 * I do not think any "Higher Court" needs to be switched on. There is no instance in the world higher than this forum. I contacted some former prominent editors who rejected to go back to the forum, at the same time convincing me that Richard will fully represent the views of professional mathematicians. The latter confirms, to an extent, that nothing is higher. It seems to me arguments require a bit more time to be dijested, especially with account of the general vacation time. Turning to a court will bring enforced consensus, which is certainly not what we want to have, even if some of the positions will win. The court is useless if a decision is taken without study of the controversy and its roots, and all such people genuinely interested are apparently already engaged in the dispute, so nobody is left over to take the function of advocate or executor.RocksAndStones (talk) 19:14, 11 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) 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.

or

2) 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)

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 Requests for comment/Maths, science, and technology to see how other content disputes have turned out and 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 (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, 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, (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)


 * Martin, I think this is very nicely said, and I like your three levels picture. One might even categorize editors according to which of the three levels is where they can best operate. I am not implying any kind of superiority as one goes "up" levels. Someone with skills and knowledge at level 3 could well be a hopeless Wikipedia editor when trying to work at level 1. I think of myself, in this regard. Now readers will also belong primarily in one of the three levels. All have a right to find what they need when they come to Wikipedia MHP; all of us editors working on the article are doing so, because we think that a) our level is not done justice too; b) the other levels could well use a bit of our level's insight. We are a random sample of readers. Richard Gill (talk) 11:59, 12 August 2011 (UTC)


 * "Random sample of readers"? Was there a lottery or other randomized technique used for selection purposes? 76.190.251.93 (talk) 09:39, 13 August 2011 (UTC)
 * You forget the zero level, on which it is explained that the odds are equal. And also the divine level, stating any solution is right as we are all mortal in the end.77.248.232.44 (talk) 14:28, 12 August 2011 (UTC)


 * I like the general idea of this structure, even though I think some of the spin and rationale veer into a non neutral point of view. I fully endorse the proposition that the article should progress from simple, relatively easy treatments of the problem to more advanced, sophisticated, and general analyses for two reasons: (1) This is a direct application of the WP:TECHNICAL style guideline. (2) This is not counter to the WP:NPOV policy: it does not, in the words of that policy, "create an apparent hierarchy of fact" where the initial treatments appear more "true"' and later treatments are deemed "controversial" but, on the contrary, readers will typically understand this arrangement as leading to "deeper" analysis of the subject. If some readers look on the more sophisticated treatments as mere sophistry, all we can do is write them as clearly as possible and bear in mind that some readers will regard the whole article as an elaborate hoax, believing that there is no advantage to switching at all. ~ Ningauble (talk) 13:36, 13 August 2011 (UTC)
 * I was not really proposing a structure for the article, I was just trying to show how the 'health warning' that some editors insist on represents a rather narrow approach to the problem, essentially that of undergraduate probability. Martin Hogbin (talk) 17:13, 13 August 2011 (UTC)
 * If you think I'm insisting on a "health warning", please tell me exactly what words through the "Solution" section of /draft1 constitutes this "health warning". IMO, the prominence given to the conditional approach in /draft1 being more or less equal to the prominence given "simple approaches" proportionately reflects the prominence of these approaches among reliable sources.  The fact is both of these approaches are extremely common among reliable sources.  Presenting the conditional approach as a minority view because there are (say) 800 sources presenting simple solutions but only 400 presenting conditional solutions is absurd - particularly since the 800 includes a very large number of popular sources more or less parroting vos Savant's view while the 400 includes a very large number of independently published academic sources.  -- Rick Block (talk) 18:36, 13 August 2011 (UTC)
 * Prominence given to the conditional approach among reliable sources and "importance"
 * Actual academic sources concerning the MHP and even concerning the total solution of the MHP show that, for solving the MHP and for making the right decision, a conditional solution using door numbers is an irrelevant aspect, not helpful to make "a better decision". Not addressing the MHP, but using the MHP to show conditional probability theorems. Not indispensable to give the only correct answer and to making the only correct decision to switch in that one very game the question is about.
 * A very large number of independently published academic sources showing conditional probability theory just use the MHP as an example to show conditional probability theory. They are sources showing the correct use of conditional probability theory. They do not belong to the MHP, they belong to maths. Quite unnecessary to making the only correct decision asked for in the MHP, in that very one game the question is about. Regarding weight and importance, that must be considered mandatory. Gerhardvalentin (talk) 08:22, 14 August 2011 (UTC)
 * Rick, in your /draft1 you do worse that have a health warning, you have a diagram that is unnecessarily and arbitrarily over-complicated in that it shows, as two distinct cases, the doors that the host might have opened, despite the fact that we know that it makes no difference, and the fact we know the host did, in fact, open door 3. For some unexplained reason the diagram does not show cases where the player initially chose a different door, even though they might have done so under the game rules.


 * This diagram is not exactly wrong but it is in the wrong place. It belongs to the more complicated case,described by Morgan, where we consider that the player initially chooses uniformly, the car is initially placed uniformly but the host does not choose a goat-hiding door uniformly. When addressing the simple, fully symmetrical, case the diagram is unnecessarily complicated and confusing.  It should come later in the article when we discuss academic extensions to the problem. Martin Hogbin (talk) 09:01, 14 August 2011 (UTC)


 * Martin – I am sorry that I misunderstood your characterization of levels of analysis as a proposal to structure the article with the most understandable parts of the article up front. I evidently do not understand what it is that you are proposing, so please disregard my earlier response in this thread. ~ Ningauble (talk) 17:30, 14 August 2011 (UTC)

The Third Draft
May I just start drafting a new version, through which Guy Macon will lead us on the line-by-line basis.
 * MHP site is falling down, falling down, falling down?
 * MHP site is falling down? My fair lady...
 * Build it up with stone so strong, stone so strong, stone so strong.
 * Build it up with stone so strong, my fair lady.
 * Stone so strong will last so long, last so long, last so long,
 * Stone so strong will last so long, my fair lady!

RocksAndStones (talk) 21:09, 7 August 2011 (UTC)


 * A third draft would be welcome, if neither of the current drafts suits you. And it's OK if someone else writes a fourth. It's also OK for someone to decide they like another version better.  If it ends up that nobody prefers a particular version, we will delete it.


 * Here is what I am looking for in these drafts, and what I am asking everyone to do:


 * Show us what you believe is the best MHP article you can create. No need to be concerned about making someone else happy on this one.


 * Try to help others to improve their version in areas where you don't disagree.


 * Whenever you can, copy the wording of another version exactly. I don't want a bunch of trivial differences. I want the differences to be a true reflection of the places where editors have an intractable  good-faith disagreement about content.


 * Thanks, everyone, for working so hard on this. Guy Macon (talk) 05:12, 8 August 2011 (UTC)


 * Note: I have been sick for the last few days, so there will be a short delay until I feel better. Guy Macon (talk) 04:27, 9 August 2011 (UTC)


 * I thought that was the third draft. ~ Ningauble (talk) 16:06, 14 August 2011 (UTC)
 * Ningauble, that indeed was a sketch of the third draft. Patience! To be continued.RocksAndStones (talk) 19:59, 14 August 2011 (UTC)

Deal Or No Deal
I realise that you're in the middle of a dispute about the page but I was wondering what people's thought were regarding including info about Deal Or No Deal. I know that it's not the same but I've seen it brought up quite often when the Monty Hall problem is mentioned. Maybe some sort of comparison as to why the situations are different and how much the knowledge of the host affects the problem. Sorry if this has been brought up before, I'm still new and not sure how everything works. AlbionBT (talk) 19:17, 11 August 2011 (UTC)


 * Never mind, I've just noticed this has already been mentioned in the archives. Just ignore me. AlbionBT (talk) 19:19, 11 August 2011 (UTC)


 * I think it is an interesting point which should be mentioned in the article. As you say the difference is that in Deal or No Deal nobody knows which prize is where, but in the Monty Hall problem the host does know. The fact that this makes a difference surprises many people. Martin Hogbin (talk) 20:10, 11 August 2011 (UTC)


 * This goes to the point I remarked earlier about the host providing selective evidence. I don't like the idea of bringing Deal or No Deal into the article because there are too many differences between the problems (not least being the issue of a bird in the hand vs. two in the bush); but it would be very good to find sources to cite in the "Sources of confusion" section that explicitly discuss the issue of selective evidence in MHP because I rather suspect this is the aspect of the problem most overlooked by PhDs and other knowledgeable people who famously got it wrong. ~ Ningauble (talk) 12:07, 13 August 2011 (UTC)
 * Your point is one of the most important points in the MHP. Many people find it surprising that the host's knowledge makes a difference.  A more convincing way of putting it might be to say that the host can never open the door hiding the car.  After that point has been made, Deal or No Deal might be mentioned as an example where the host does not know where the prizes are, although I agree that that game is significantly different from the MHP.  Martin Hogbin (talk) 13:21, 13 August 2011 (UTC)

Academic extensions
Following on from a thread above, I wonder how many here agree with this analysis of the situation here. I am sure all of you will agree with some of the points above but I am interested to know where the disagreement starts.


 * 1) The most well known version of the problem was published in a column in a popular, general interest, magazine.
 * 2) In the 'standard' MHP the producer initially places the car uniformly, the player chooses uniformly, and the host chooses a goat-hiding door uniformly. Many other very natural simplifying assumptions, appropriate to a simple mathematical puzzle,  such as that the host does not try to give a clue to where the car is by the language he uses, are also made.
 * 3) The above formulation is completely symmetrical with respect to door number. That is to say, the door numbers make no difference whatever to the outcomes.
 * 4) As the door numbers are not important we can simplify the problem and the solutions by not considering every door number that the player might initially choose, because we know it can make do difference to the outcome.
 * 5) As the door numbers are not important we can simplify the problem and the solutions by not considering every door that the host can legally open, because we know it can make do difference to the outcome.
 * 6) Everything else is essentially an academic extension to the problem. Interesting to many, but not necessary to solve the simple mathematical brain teaser.

Who agrees with all 6 steps and where do the others disagree? Martin Hogbin (talk) 09:20, 14 August 2011 (UTC)
 * Agree 1-6, and what about that simple but clear table based on door #1 (or alternatively door "A") chosen, showing all possible variants of actual car location, and the respective outcome, no matter which door the host has opened, like there on citizendium.org? You can call the doors "door A, door B and door C, if you like). Most important imho is your item 6. Gerhardvalentin (talk) 11:19, 14 August 2011 (UTC)


 * Comment
 * In #2 there is a standard assumption about the placement of the car, but no such assumption about the player's choice - and you're missing a very important 2.5:


 * 2.5. In the standard MHP the player's decision to switch or stay is made after seeing which door the host has opened. Moreover, the reader is explicitly encouraged to think about the specific case where the player has picked door #1 and the host has opened door #3 in which situation the probability the car is behind door #3 is obviously 0 and the car is clearly behind one of only two doors.


 * Your point #3 is a true statement, but it is shown only by a conditional analysis that you are so desperately trying to cast as an "academic extension". Your points 4 and 5 describe what some, but certainly not all, sources do.  Your point 6 is simply a statement of your POV - it is specifically not a published POV widely held by reliable sources.


 * Starting with #3, sticking with a source-based list of points, we could say


 * 3. Many sources choose to ignore the specific case (player picks door 1 and host opens door 3) described in the problem statement showing instead that a strategy of always switching wins 2/3 of the time while a strategy of always staying wins 1/3 of the time. Many other sources show the probabilities in the specific case mentioned in the problem description are 1/3:2/3:0 using elementary conditional probability.


 * 4. Some sources analyze variations of the problem using assumptions other than the standard ones in #2 - typically using conditional probability or game theory.


 * My point is that this list is simply yet another attempt to justify your bias. Your opinion is that using conditional probability to address the fully symmetric case is an "academic extension" or "unnecessarily and arbitrarily over-complicated" - but this is your opinion, not a statement reflected by a preponderance of reliable sources.  -- Rick Block (talk) 17:35, 14 August 2011 (UTC)


 * Rick, I disagree with your assertion that point #3 (door numbers make no difference) "is a true statement, but it is shown only by a conditional analysis." Conditional analysis is not strictly necessary for the particular variety of standard problem formulations that use an "equal goat" or equivalent stipulation. Consider Morgan's rejoinder to Seymann's comment on Morgan's paper (American Statistician 45 NB: all three were published together with an apparent editorial interest in neutrality. ) "Certainly the condition p = q = 1/2 should have been put on [...] It could also have been mentioned that this means that which of the unchosen doors is shown is irrelevant, which is the basis for solving the unconditional problem as a response to the conditional one." [emphasis added] Perhaps Morgan is not the most reliable source but, although I am aware of multiple sources demonstrating that simple unconditional analysis is inadequate when p != q, I am not aware of any sources that convincingly demonstrate conditional analysis is essential when there is an a priori stipulation that p = q. What are the preponderant sources that show this? ~ Ningauble (talk) 19:39, 14 August 2011 (UTC)


 * What I mean is that the fundamental meaning of Martin's sentence "The above formulation is completely symmetrical with respect to door number." is that the conditional answers are all the same, and that showing this requires (in some form) talking about the conditional cases. Morgan et al. assert that stipulating p=q=1/2 provides a basis for solving the unconditional problem as a response to the conditional one, i.e. that this forces all conditional answers to be the same as each other as well as the same as the unconditional answer (which is indeed what it means for the problem to be symmetrical).  I'm not claiming (and /draft1 certainly doesn't say) that a conditional analysis is essential when there is an a priori stipulation that p = q.  What I am saying is that many, many sources present a conditional analysis even in this case and that considering such an analysis to be an "academic extension" of the basic problem is merely Martin's POV - not the POV expressed by a preponderance of sources.  I am not (as Martin seems to think) saying simple solutions are wrong, or that only a conditional solution is correct.  I'm saying both are commonly presented, so the article should not favor one over the other but rather should present both as equally valid alternatives.  -- Rick Block (talk)


 * Gentlemen, you are in the devil's circle, repeating five yrs long the same simple facts. Under the assumption that it makes sense, the conditional solution implies the simple. The simple solution plus a small additional argument about its optimality implies the conditional.
 * The simple solution is mathematically stronger as it does not require host tossing (perhaps, biased) coin. The conditional solution quantifies the conditional risk, an aspect which the simple solution ignores. What else can be conceptually added to that? I do not see.RocksAndStones (talk) 21:35, 14 August 2011 (UTC)
 * Me neither. The same simple facts are repeated for five years, some other simple facts remain ignored. The simple facts which remain ignored are the ones which allow for the different approaches to be unified and simplified. Why is this possibility ignored by wikipedia editors? It's a fact that people who resolve a paradox are thereafter extremely fond of their own resolution and critical of all others. After experiencing one paradigm shift we cannot accept another. Wikipedia editors have to be aware of this in their own work. In a history section one can give due weight (and due criticism) to the vast literature. Not only were there influential papers by Selvin and by Morgan et al., there were also a large discussion and many responses and criticisms. Standard textbooks duplicated a particular approach suitable in the context of teaching Bayes theorem in Probability 101 courses, for making various didactic points to the students. That approach can be reported in an appropriate section. Richard Gill (talk) 13:54, 16 August 2011 (UTC)


 * My statement, "The above formulation is completely symmetrical with respect to door number" does not require any analysis. It is self evident from the fact that all distributions relating to door number are uniform.  This point is confirmed by Falk and several mathematicians here. Martin Hogbin (talk) 21:47, 14 August 2011 (UTC)


 * Rick, also if there is not an obvious symmetry with respect to door number then what is your justification for not having a diagram showing all the doors which the player might have chosen. The door initially chosen by the player is as much a condition of the problem as the door chosen by the host. Martin Hogbin (talk) 21:51, 14 August 2011 (UTC)


 * My justification is that it is what many, many sources do. What is your justification for treating the many, many sources that present this argument as if they hold a minority viewpoint?  I want the article to be NEUTRAL and to represent what the many, many, many reliable sources say "fairly, proportionately, and as far as possible without bias".  Since you're arguing so hard about it, I can only conclude that you do not.  -- Rick Block (talk) 02:03, 15 August 2011 (UTC)


 * Rick, so it is still your opinion that it is indispensable and necessary to confuse the readers just at the start of the article in forever repeating meanwhile disproved outdated sayings, disproved by reliable academic sources. Read the latest authoritative tertiary literature. And you can show all of that outdated sayings, together with the colorful historical conflicts, in a subsequent section "History of the problem". And there you can also show that an extremely biased host, the more the better, will reinforce the decision to always switch, especially in this one special game called MHP. And that no conditional approach can ever give better advice than to switch here and now.
 * To get the question correctly, to capture the full extent of the issue and to make the only correct decision the reader should be "defused", and it is better not to confuse the reader just in the beginning. That would never be a "neutral point of view". All of unnecessary burden can be shown in the section "History of the problem". Gerhardvalentin (talk) 22:36, 14 August 2011 (UTC)


 * What is "indispensable and necessary" is that the article follow WP:NPOV. This specifically means NOT picking one approach as most correct, but instead presenting all published approaches "fairly, proportionately, and as far as possible without bias".  -- Rick Block (talk) 02:03, 15 August 2011 (UTC)
 * The vast majority of sources concentrate on trying to give simple solutions that show why the player doubles their chances by switching. We must report what these sources 'without adding our own opinion or the opinion of other sources. Our evaluation of the correctness of sources should be based on what reliable secondary/tertiary sources say about them.   Martin Hogbin (talk) 08:49, 15 August 2011 (UTC)
 * Rick, the prior odds on the car being behind each door are 1:1:1, and the charm of the MHP is that most people believe that, after a goat has been shown behind one of the two host's doors, the odds on the car being behind the two still closed doors still are 1:1. To convince the readers, I am in favor of also showing the (conditional) chances, short in odds form, just at the beginning of the article to be 1:2:0 then. Starting simple, and later showing that no reasonable (unnecessary) assumption can proof that switching can nor will ever hurt. But please stop confusing the reader with unnecessary assumptions just at the beginning of the article. Gerhardvalentin (talk) 07:54, 15 August 2011 (UTC)


 * Rick, the quotation from Morgan et al. does not say that "this forces all conditional answers to be the same;" they say this means that which of the unchosen doors is shown is irrelevant. The former refers to invariance among solution results, but it is the latter which is "what it means for the problem to be symmetrical." [emphasis added] There is no basis for saying that "showing this requires (in some form) talking about the conditional cases," and attempting to do so in the midst of presenting analyses that treat the choice of goat as irrelevant only confuses the issue. I completely agree that it is imperative to treat different approaches fairly, proportionately, and without bias, and not favor one over the other. To wit:
 * Treating them without bias means not disparaging any of the approaches as merely academic diversions, whether expressly or by implication, as, e.g., in the manner of Martin's characterization of conditional solutions during discussion.
 * Treating them without bias also means not commending any of the approaches as more meritorious than others, as, e.g., where Draft1 singles out conditional analysis as "Mathematical formulation" with the unsubtle implication that other formulations are not.
 * Treating them fairly means presenting each in an equally clear and coherent manner without unnecessary digressions, as, e.g. where Draft1 interjects conditional analysis betwixt a symmetric combinatorial answer and answers that apply symmetry to the unchosen doors. (IMO these are all of a piece.)
 * I completely agree that the conditional approach merits substantial coverage, but there is more to NPOV than proportionate weight. ~ Ningauble (talk) 15:17, 15 August 2011 (UTC)


 * Ninguable I actually used the term 'academic extensions'. There is nothing pejorative or biased about this.  The original problem was a puzzle in a popular magazine, which was answered quite correctly as such by vos Savant.  The whole media furore was about vos Savant's solution and why the answer was 2/3 not 1/2.  Some months later a bunch of academics, Morgan et al, decided to extend the problem to a particular case where the producer hid the car uniformly but the host did not choose a goat-hiding door uniformly.  This is a different, more complex, problem that requires conditional probability to solve. Since then many academics have added different assumptions and used different methods to solve various extensions and modifications of the problem.  None of these are relevant to the original puzzle.  Martin Hogbin (talk) 15:55, 15 August 2011 (UTC)
 * I apologize for the hyperbole, and confess that it was intentional. It was an exaggeration to say your discussion regarding what is of primary and secondary importance in the article was disparaging of the latter, and I am not aware that you have ever proposed including disparaging language in the article. ~ Ningauble (talk) 18:40, 16 August 2011 (UTC)
 * 1) Conditional probability is not needed to solve the biased host problem. Since you can't beat 2/3 probability of winning by switching, the conditional probability must also favour switching. No need to calculate it.
 * 2) Some academics have solved the original problem by using symmetry (of the problem description) explicitly in advance to discard the "information" given by the numbers on the doors, e.g. Georgii in a standard textbook, also Richard-Gill-the-real-person in an online peer-reviewed statistics encyclopedia. Richard Gill (talk) 13:41, 16 August 2011 (UTC)


 * Agree 1-3.. Agree w/reservations 4-5. Disagree 6. I'm disappointed that Martin asks a clear question, someone makes a clear reponse, and then the section explodes with text. IIRC, VoS said her description could be better, but the complainants were not misled by the description; they just got it wrong. Another source said that VoS conveyed the quantification in her description ("say, "), that quantification was understood by the contestant, but then the thought processes of the contestant got hung up by the specific door numbers. Too much information. By modifying the description, the hang ups can be reduced. I agree that indistinguishable doors simplify the problem. I'm not sure that means they can be completely jettisoned. Door numbers are needed to explain confusion in VoS. Parts of so-called academic extension are needed to explain MHP. Glrx (talk) 16:30, 15 August 2011 (UTC)


 * Responding to Girx: In my opinion MHP is a mathematical joke. A brainteaser. VoS's specific formulation is brilliant because it reinforces the deception, by focussing the person who is being teased (the reader) on what he sees in front of him at a late stage of the proceedings. (a) it brings in irrelevant information, (b) it destroys the memory of the process which led to that situation. Now if you want ordinary folk to get the joke, you have to let them make the shift of perspective which makes the natural wrong answer stupid, and the right answer obvious. That's what most of the wikipedia page should be about since that is what most readers are after, that's why the problem is so famous (because it is such a good joke), etc.
 * The Morgan et al. conditional approach is about how to get the right answer by plodding calculations without changing the initial point of view. It destroys the joke, and converts it into an exercise in Bayes probability calculations. Very valuable in the right context but not at the heart of a famous brain-teaser about a popular game-show!
 * I would prefer, but I fear we'll have to wait for ten more years of both popular and academic publications, not to present the brainteaser joke and the conditional probability calculation as opposed approaches. I would rather work as follows.
 * On the assumption only that your initial door has probability 1/3 of hiding the car, switching gives the car with probability 2/3.
 * Observing that *however* you choose a door and subsequently decide to stay or switch, there is always a door such that you'll never get a car hidden there, we see that if all doors are equally likely to hide the car initially, you cannot do better than get the car with probability 2/3.
 * For Bayes theorem fans, it follows (from the law of total probability) that if all doors are initially equally likely to hide the car, then the conditional probability that the car is behind door 2 given you chose 1 and the host opened door 3 is at least 1/2, i.e. favours switching, as do all the other five conditional probabilities you might like to think of.
 * I think the article could be about 10 times shorter than the present one if it were inspired by this line of argument. Richard Gill (talk) 13:14, 16 August 2011 (UTC)
 * I too would much prefer not to present these as being in opposition, but I fear you are being overly optimistic about resolving it in ten years. It may never happen because it is inherent in the Wikipedia process that once a controversy arises, as happened when Morgan et al. uncollegially accused vos Savant of ignorance, the application of NPOV almost inevitably results in including "balancing" points of view and giving UNDUE weight to the controversy itself, describing everything remotely connected to it in terms of opposites, never in terms of complementary or tangential information. I hope I am wrong. ~ Ningauble (talk) 18:40, 16 August 2011 (UTC)
 * Wikipedia is like Democracy. It is fundamentally flawed. But unfortunately there is nothing better. Richard Gill (talk) 11:48, 18 August 2011 (UTC)


 * Agree 1-6 w/reservations. Reservations: because there is not need of any opposition between popular and academic if we move a bit with the times. Or do we want to stay quarreling here for ten years till the academic and popular literature has again exponentially grown, and the pre-2010 literature on MHP is an insignificant, historical amusement (or embarrassment)? MHP is a joke. VoS asks for a decision. All experts will tell you that phrasing a real world question as "I want to know the probability that..." is a recipe for disaster. Ordinary folk just don't have available in their minds the needed concepts, let alone the calculus or the theorems, to understand what they are asking, and anyway, a whole load more outside information would be needed before some operationalism of the question could be answered. My 1) to 3) above seems to me to be a summary of the all mathematical truth about MHP which needs to be in the article. Richard Gill (talk) 13:30, 16 August 2011 (UTC)
 * Agree. 71.166.38.174 (talk) 20:21, 20 August 2011 (UTC) — 71.166.38.174 (talk • contribs) has made few or no other edits outside this topic.

So let us solve the MHP problem problem
The MHP is a famous paradox, based on human psychology. Unquestioned the Monty Hall Paradox, in incredible ways, is surprising the average citizen, strikingly contradicting our vehement "clear judgment and discernment", and contradicting our perception. Seeing two closed doors, and knowing for sure that one of them hides the desired prize, while the other one must be hiding the second goat, our knee-jerk answer – without any pause for reflection – is  "it makes no difference whether to stay or to switch".  To comprehend that the chances are not 1:1 but 1:2 is easy to grasp when the basics are didactic elucidated and expounded. To give the only correct answer: "switch" here and now, in that actual game. That's the whole secret.

Necessarily (or unnecessarily) far-fetched and boldly supposing that there "could" be some (or even maximal) additional information on the actual location of the prize, anyone can easily grasp also that the chances of staying vs. switching will be 1/3:2/3 on average, but at least 1/2:1/2 for sure, and never less, in two out of three, but will be full 0:1 in one out of three (!), that means always to be around 1/3:2/3, and honestly no-one can in fairness say that he "knows" better. Because there definitely is no additional information, you just can imagine to have "supposed" additional information on the actual location of the car. A very weak and purely hypothetical moot point, to be taken with a pinch of salt, but blaming any ignoring of that ludicrous point for the decision to be made. On the other hand just reinforcing the decision to
 * "switch here and now".

However very suitable for teaching probability theory maths, as umpteen times proven in reality, but forever being without the least effect on the decision to be made. Never able to be helpful or insightful, but very interesting as a mathematical problem in conditional probability theory.

So the article could be short and clear, closing with a revealing section on the colorful history in literature, including pedestrian disputes and numerous textbooks just for teaching purpose.

Let's start. Gerhardvalentin (talk) 21:58, 16 August 2011 (UTC)


 * Gerhardvalentin, OK, let us solve. Discard probs for a while. Connie can guest correctly 2 out of 3 cases A,B,C. Why can't she do better?  Suppose Monte has a known to Connie signal system: shows either ticket L in his left hand or ticket R in the right hand. Connie gets one bit of information, but needs to distinguish among 3 cases A,B,C, hence there is no way to always communicate the location of prize by sending one bit. This may be seen as an instance of pigeonhole/Dirichlet principle: having 3 coins in 2 pockets, in one of the pockets there are at least 2 coins.
 * Put another way, consider transmitting letter from the alphabet {A,B,C} through a noisy channel whose output is L or R. Which way of transmission has minimal distortion? For example A->L,B->L, C->R. When Connie picks, say A, there is Left door (door B) and Right door (door C). A switch offered to B means encoding B->L (Monte shows L ticket), and similarly for C->R (Monte shows R ticket). Indeed, when Connie picks A, and the prize is behind B, then switch is offered to B, which is equivalent to encoding B->L. Similarly for C->R. We see that no strategy wins all three cases. Hence strategy S="pick D1 always-switch" is optimal combinatorially. Assigning equal probabilities to A,B,C, we see that  S is average-case optimal as well.RocksAndStones (talk) 14:18, 18 August 2011 (UTC)
 * Exactly:  to suppose that some message could be given from host to Connie in a "restricted" or say "legal" form, then he is unable to communicate, even in the most extreme case, much more than either "switch and win" or "stay or switch, it will be the same". – He impossibly can give any closer info. Knowing that, and knowing that staying never can be any better than to switch, because switching will never hurt, shows that Connie will be wise to switch in any case, completely regardless of all those signals. The host is never able to signal other recommendation, and so Connie knows already before that hypothetical imaginary game show that she will switch in any case, and – after the host has given all his available signals – she knows just as well, maybe even better than before, that to switch is and forever will be the only correct decision. Btw, did you see Richards remark below on Gnedin's solution, trashing "probability"?  Regards, Gerhardvalentin (talk) 15:16, 18 August 2011 (UTC)
 * Gerhardvalentin, if the host could in some way signal, "don't switch", then the contestant has received useful information that takes her beyond the 2/3 winning %. But this has nothing to do with the MHP of Selvin, vos Savant, Whitaker, Morgan, or any other reliable source. 76.190.236.207 (talk) 16:45, 18 August 2011 (UTC)


 * Answer: You are correct. Under the given rules of the standard MHP (no illegal "sayings")  the host can only signal a "minimum level" to win by switching of  "1/2",  but never ever less, what means equivalent full  "1"  likewise, and purportless any possible grade between  "1/2 to 1"  as per fictive assumptions. He forever is out of position to signal "don't switch".


 * Chances of   "door selected : door offered to switch to : door opened"   originally were "1/3 : 1/3 : 1/3".  And after a door has been opened showing a goat, all we forever will "really know" is that they are  "1/3 : 2/3 : 0"  now. We never can nor will have better "knowledge".  But we are free to "assume" just what we like (though observing some "rules"), as I said on the "Arguments" page:


 * The host could be "assumed" to be (extremely) biased to open only the door with the brightest color e.g., whenever he can, i.e. if a goat is behind his preferred bright door. He can open his preferred door in two out of three cases:
 * if in 1/3 he has got two goats to show and switching will LOOSE the car     ("1 : 0 : 0")  and
 * if in 1/3 he has got the car and one goat, and the goat being behind his preferred bright door, and switching will WIN the car     ("0 : 1 : 0").
 * So, whenever he opens his preferred door in 2 out of 3 cases, you know that the chance to win by switching is 1/2 (and never ever less):  "1/2 : 1/2 : 0".


 * but if, in the last 1/3, he has got one goat and the car, but the car being behind his preferred bright door, he has to open his avoided door of darker color e.g., and in this 1 out of 3 -case switching is very likely to win for sure:  "0 : 1 : 0".


 * So we definitely know, just from the start, that in any case the posterior chances definitely must be within the fixed range of  "1/2 : 1/2 : 0"  to  "0 : 1 : 0",  and all we really "can know" is that they are  "1/3 : 2/3 : 0"  now.  And it's finally clear: the more biased the host, the better. And the only correct answer, the only correct solution must be to "switch here and now", whereas to stay forever is out of the question. Without travail of effectless conditional probability theory-training.
 * But advisable to show all of that effectlessness in the History-section of the article.  Gerhardvalentin (talk) 17:52, 18 August 2011 (UTC)


 * Well, my command of German is not perfect. Ziege must be goat, and gehfaul perhaps lazy to walk, so reading will take some time. I wish to add to the above that with four doors the situation is as you just described, provided two useless doors are revealed at once. Then the signal is the door-to-switch. However, if two doors are revealed in sequence, it is easy to design signaling scheme which communicates the location of prize with certainty. I wonder how this informational aspect of the problem could be quantified under biased allocation of the prize.RocksAndStones (talk) 15:41, 18 August 2011 (UTC)
 * RocksAndStones, try to read it there. Gerhardvalentin (talk) 21:46, 18 August 2011 (UTC)


 * Thanks for the translation. The article makes a clear point, which one can put in a slightly different way. Every strategy is evaluated by a win-loss vector whose components are labeled by the pure counter-strategies. The set of Pareto-optimal strategies consists of three always-switching strategies. All other strategies are irrelevant.RocksAndStones (talk) 00:19, 21 August 2011 (UTC)
 * Yes, that's the point. Any kinds of irrelevant pretended "solutions", never being solutions but just unnecessary theorems using "conditional" probability theory, pretending to be indispensable for making the only correct decision "in a given game" – with "fixed door numbers as the base footing for decision making", based on doubtful and highly questionable fictive assumptions that never ever may be given nor will ever be able to influence the correct decision, not in the slightest, should clearly be shown in the article as what they forever have been and as what they really are: Just only welcome examples to teach and to learn conditional probability theory, without any relevance to the famous paradox, and completely irrelevant for the decision asked for. So 15 - 20 more years to go? Gerhardvalentin (talk) 10:18, 21 August 2011 (UTC)

Ignorant Monty
What is the difference between the “Ignorant Monty” and the standard problem? It sounds like the same, Monty is choosing a random goat. --Chricho ∀ (talk) 12:19, 21 August 2011 (UTC)


 * "Ignorant Monty" in 1 out of 3 will show the car (i.e. out of 3 eliminating a full 1/3-chance to "win"), and only in the subset of 2 out of 3 will show a goat, what means that the probability to win by switching is reduced from 2/3 to 1/2.  Chances no more "1/3 : 2:3 : 0", but within the subset of the actual case now "1/2 : 1/2 : 0".   Gerhardvalentin (talk) 12:38, 21 August 2011 (UTC)


 * And if he shows the car the guest can only lose? --Chricho ∀ (talk) 20:26, 21 August 2011 (UTC)


 * Yes, correctly, the guest has lost. In the hypothetic "Ignorant Monty"  the host, in presenting the car behind an unchosen door, has shown that the guest's choice was wrong, and game over. Regards, Gerhardvalentin (talk) 21:24, 21 August 2011 (UTC)


 * Unless of course, he finds one goat to be especially attractive... :) -Guy Macon (talk) 00:12, 22 August 2011 (UTC)


 * Actually, the usual "ignorant Monty" variant considers only the case in which the host fortuitously happens to reveal a goat (leaving it completely unspecified what would happen if Monty were to reveal the car, because the car was not revealed), and asks the question "in this case (in which Monty HAS revealed a goat) what is the chance of winning by switching". This question is easier to visualize in the context of a specific example, i.e. assume the player has picked door 1 and Monty has (without knowing what's behind it) opened door 3 revealing a goat.  This creates the same apparent condition as the usual problem (player initially picked door 1, host opened door 3 showing a goat, and the player is now standing in front of a closed door 1 and closed door 2 looking at a goat behind the now open door 3)- but the probabilities are different.  In the ignorant Monty case the probabilities the car is behind each door are 1/2:1/2:0, while in the usual problem the probabilities are 1/3:2/3:0.  IMO, it is very difficult to discuss the difference between these two problems without talking about conditional probability.  In particular, if you assume the player loses if the host reveals the car and "simplify" this variant in the same way the usual problem is often simplified by talking about the overall chances of winning by switching or staying (as opposed to the conditional probabilities in a specific case where you're now looking at an open door showing a goat), you find the somewhat disturbing result that you win 1/3 of the time and lose 2/3 of the time whether you switch or not. -- Rick Block (talk) 04:50, 22 August 2011 (UTC)

Breaking News
Gnedin's "dominance" approach is featured in today's "Die Zeit", Science section. Those who can read German will enjoy it. Here are screen-shots from my ZEIT ONLINE iPad app:. And here an English translation. . Richard Gill (talk) 14:04, 18 August 2011 (UTC)


 * With the entry http://de.wikipedia.org/wiki/Ziegenproblem#Strategische_L.C3.B6sung_mit_Dominanz the German Wiki goes ahead.LoveMe2XBaby (talk) 18:20, 23 August 2011 (UTC)

Rough Draft of RfC: Please Comment
I have a rough draft of the RfC here: Talk:Monty Hall problem/RfC

Please help me to improve it before I submit it.

Comments placed in the RfC may be edited. Comments placed here will not. --Guy Macon (talk) 11:04, 20 August 2011 (UTC)


 * There are no links to the two versions. Which are they and where can I find them? Martin Hogbin (talk) 11:08, 20 August 2011 (UTC)


 * In the first sentence, which reads...


 * Which approach to the Monty Hall problem should we choose; VERSION ONE or VERSION TWO?


 * ...does clicking on "VERSION ONE" and "VERSION TWO" lead you to the two versions? Guy Macon (talk) 12:15, 20 August 2011 (UTC)


 * I am sorry Guy, but I do not think the RfC is posing the right question. To choose "which approach", the RfC needs to present some clear statements of the of the approaches these drafts purport to represent. Presenting this as a choice between drafts of article content (even if one of the drafts had not already been repudiated by its drafter!) is not going to settle the question because article content is going to continue changing and there will be future disagreement as to what aspects or characteristics of the drafts have been decided upon. Maybe it's just me, but I am also uncomfortable with the either/or framework in the present formulation of the RfC. It feels like a loaded question because I am not "neutral" about these two drafts and I do not "support" either draft. (This is only partly due to the problems inherent in trying to define approaches by example.) In the words of Neutral point of view/FAQ, part of what is needed is to "step back and ask ourselves, 'How can this dispute be fairly characterized?' This has to be asked repeatedly as each new controversial point is stated." This can be very difficult for disputants to do for themselves. It ain't easy to mediate the question of what the question is, but I think it needs more clarity before the question is put to the larger community. ~ Ningauble (talk) 13:47, 20 August 2011 (UTC)


 * No need to be sorry. Asking whether this is the right question is exactly the right thing to ask, and I have no problem with asking a better question - and that's why it is still a rough draft. In fact, I have repeatedly asked for a brief description of the dispute suitable for formulating the question. Alas, none has been forthcoming so far. Perhaps someone will come up with a better question now that there is a rough draft RfC to examine.


 * What I am not willing to do is to put up an RfC without offering the reader a clear-cut choice. I have no problem if we end up listing five alternatives, but we already have a bunch of "that's not the real issue, this is the real issue" comments that only the person making the comment completely supports.  There are thousands of words on this very talk page exploring every aspect of this issue, all without resolving the underlying content dispute.  If you or someone else can formulate a better question that most people here like, great.  If not, I will move forward with the best I have so far. Guy Macon (talk)  19:33, 20 August 2011 (UTC)


 * I agree with Ningauble that neither the substance of the dispute or the proposed resolution is clear from anything in the current draft RFC. I have tried several times in the past to describe the dispute in neutral terms, e.g.  .  Martin will (of course) argue that these are highly biased descriptions, although I'll note that his own description,, completely mischaracterizes one of the positions.  If we view the dispute as between polar opposites, where on the one side the article endorses the view that the simple solutions are all that's necessary and on the other side the article endorses the view that conditional solutions are the only correct approach, IMO Martin is arguing one side of this but no one is arguing the other side.  In particular, I'm arguing for a balanced NPOV approach which endorses neither of these views but instead presents both approaches as equally valid.  -- Rick Block (talk) 19:37, 20 August 2011 (UTC)


 * I also agree that neither the substance of the dispute or the proposed resolution is clear from anything in the current draft RFC - and I wrote it! I am completely open to suggestions, but I am not willing to spend another year or two stalled because nobody can state the issue in a way that the other side finds acceptable.  To all: Give me a better description of the issue or accept the fact that I had to go forward with the best description of the issue I could come up with without help. Guy Macon (talk)  19:53, 20 August 2011 (UTC)


 * Alternatively, one might consider the reason for the never ending dispute arises from a breach in policies. Rather than issue a flawed RfC for no real benefit, perhaps it should be considered whether one of the editors continues to exert ownership of the article, or whether, maybe, another editor has claimed ownership by bogging down these discussions 184.81.169.186 (talk) 20:25, 20 August 2011 (UTC). — 184.81.169.186 (talk • contribs) has made few or no other edits outside this topic.


 * I agree with Ninguable that this is not the way to approach an RfC. Presenting two different versions will only cause newcomers to argue about irrelevant details.   You asked for a description of the dispute and got two, one from Rick and one from me.  We might both argue that the other one's version presents a biased view, but that is why we have to versions of the question.   We should simply present them both to the RfC. Martin Hogbin (talk) 22:46, 20 August 2011 (UTC)


 * I have no problem with that. I can simply note on each description that other editorss disagree about whether it is a valid description of the conflict. Does anyone have any objections? Do you wish me to use the exact words you posted before or would you like to rephrase before the RfC is submitted?  You might want to consider whether calling your position "simple" and "correct" while describing the other position as "wanting to treat the MHP principally as an undergraduate exercise in conditional probability" might possibly strike some folks as being biased.  Your choice, of course, but I want each side to put forth the best arguments they can. Guy Macon (talk)  00:35, 21 August 2011 (UTC)


 * Just a thought, but how about I have a go at rewording Rick's statement and he has a go at rewording mine? Martin Hogbin (talk) 08:43, 21 August 2011 (UTC)


 * I'm fine with trying that. -- Rick Block (talk) 15:43, 21 August 2011 (UTC)

Looking at "Martin's modified version of Rick's question" (the only part done at the time I am writing this) it has IMO the advantage of focusing on what the dispute is rather than why one side or the other is preferred. That's important information, but IMO things like what policies apply to one side or the other are best put in the arguments section, simply because someone else is sure to disagree about whether those policies apply. On the other hand, "which will be of greater interest to experts" is, IMO straying from simply describing the dispute and instead gives a reason why one side or the other is preferred. Again, important information, but IMO best put in the arguments section. What interests experts or non-experts is not part of the description of the issue, but rather is a reason for favoring one side over the other. Putting it in an argument section would actually make it a stronger argument.

Rick's original question
The dispute is whether the article should primarily satisfy

1) 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.

or

2) 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 modified version of Rick's question
The dispute is whether the article should primarily:

1) Have an initial section focusing exclusively on "simple solutions" that makes no mention of any other solution or approach (including reference to conditional probability). Other approaches, which will be of greater interest to experts, will be placed in later sections of the article. or

2) Have the initial sections of the article addressing the most common interpretation of the problem using various approaches including both simple and conditional solutions.

Rick's response
Rick, comment here if you think I have seriously misrepresented you. Martin Hogbin (talk) 15:54, 21 August 2011 (UTC)
 * Since my stance is that this boils down to a POV issue, omitting any mention of WP:NPOV seems like a serious oversight. I want the discussion about this dispute focused on how to represent what the plethora of sources say without bias and with appropriate WP:WEIGHT and WP:STRUCTURE, not on what individuals think about the content. This should not be a popularity contest between "simple" solutions and conditional solutions.  Frankly, I'd like to preface any statement of the dispute with a summary of sources (e.g. "many, many sources present simple solutions to the usual problem", "many, many sources (including essentially all introductory probability textbooks) present conditional solutions to the usual problem", "some sources express an explicit preference for conditional solutions, often using variations to show exactly how the simple solutions are deficient", "many variations of the usual problem are presented with solutions using conditional probability or game theory").  IMO this dispute should be focused on what the sources say, and nothing else.  -- Rick Block (talk) 05:20, 22 August 2011 (UTC)
 * I have removed WP:NPOV because I and many others do not think that my proposed structure does promote a specific POV. I have always made clear that all solutions and approaches will be properly presented. We are both supposed to be writing a neutral statement of the dispute, not making a case for our opinion. Martin Hogbin (talk) 08:14, 22 August 2011 (UTC)


 * Without opining on whether it should have been placed or whether it should be removed, please review this list from http://en.wikipedia.org/wiki/Template:POV


 * When to remove:


 * This template is not meant to be a permanent resident on any article.


 * Remove this template whenever:


 * No discussion about neutrality issues was started on this article's talk page.


 * Discussion about neutrality issues is dormant.


 * There is consensus in the discussion that the problems have been resolved.


 * Removing this tag may be tendentious—just like placing it on the article may be tendentious—but it is not an act of vandalism. --Guy Macon (talk) 18:12, 1 September 2011 (UTC)


 * Martin didn't remove the tag on the article, he removed it from the description of the dispute. -- Rick Block (talk) 18:48, 1 September 2011 (UTC)

Making the two versions the same where there is no content dispute
Referring to...

Talk:Monty Hall problem/draft1

Talk:Monty Hall problem/draft2

and

Talk:Monty Hall problem/RfC

...I would like to make the two draft versions so that they only differ where there is an actual content dispute. Is reversing the order of "Simulation" and "Increasing the number of doors" really an integral part of the content dispute, or can we reverse the order in one or the other drafts?

In Draft2, the draft ends with "Discussion of all other variants, formulations, approaches and solutions" Could we cut and paste from draft1 to fill out this section and make them as close to each other as possible?

In those areas where draft1 and draft 2 are identical, could we add a note saying so to the section? Perhaps collapsing or using a different font to show what is different would work. I am looking for a way that the reader can instantly see the differences without doing a side-by-side comparison. Guy Macon (talk) 12:37, 20 August 2011 (UTC)


 * Would it be better to highlight where each differs from the present article? Each proposes to change the article in a different way, and I think the most salient difference for understanding these proposals is the difference vs. status quo. ~ Ningauble (talk) 13:57, 20 August 2011 (UTC)


 * There are pluses and minuses to that approach. One issue is that the statu quo reflects a long history of editors who hold positions in the content dispute temporarily stepping aside while editors holding the other position attempted to reach a compromise. The general feeling is that the attempts at compromise resulted in lower quality without achieving the goal of being acceptable to everybody - certainly the article has degraded since being a featured article.


 * On the other hand, you make an excellent point about salience, so perhaps we should highlight where each differs from the present article. It might be worthwhile to cut and paste sections of the current article into the drafts wherever doing so does not change the drafts being good examples showing the content dispute. Certainly making all three as close to identical as possible while still showing the content dispute is a Good Thing. Guy Macon (talk)  19:19, 20 August 2011 (UTC)


 * As Guy says, the current article was explicitly edited by Martin (and others supporting his view about what the article should contain) as part of a preliminary attempt to create what is now intended to be embodied by /draft2, so I think differences relative to the current article are not really very informative since the starting point was not a "consensus" version. Perhaps these could be diffed relative to the version as of the last successful FAR (this version).  There were some comments about /draft1 from Ningauble a while ago  that I have not responded to, essentially reflecting the fact that this draft is incomplete.  I would be willing to work on it some more and invite anyone else who may be interested to do the same, although if Martin remains unwilling to edit /draft2 to show us what he wants the article to look like the comparisons will be fairly meaningless. -- Rick Block (talk) 15:48, 23 August 2011 (UTC)

Martin's original 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?

Rick's version of Martin's question
Should this article

a) include in the introductory sections sufficient clarification of the problem statement to ensure all solutions have the same answer (i.e. that a player who switches wins with probability 2/3) and an accessible solution based on conditional probability specifically showing a player who picks door 1 and sees the host open door 3 has a 2/3 chance of winning by switching to door 2, or

b) start with more or less a complete "introductory" article (based solely on popular sources such as vos Savant plus any academic sources that happen to use exclusively "simple" approaches) treating the MHP as a simple probability puzzle focused on the strategies of "always switching" as opposed to "always staying" without necessarily clarifying the exact question that is asked and without explicitly addressing the specific "player picks door 1 and host opens door 3" case, and only after these introductory sections then proceed with a discussion of all other aspects of the problem for the more specialist reader.

Martin's response
Martin, comment here if you think I have seriously misrepresented you. -- Rick Block (talk) 04:58, 22 August 2011 (UTC)

Rick, you are meant to be writing something based on my wording. You can remove or change things that you think are misleading or unfair but you need to write something that looks at least vaguely like my words, as I have done with your statement. Martin Hogbin (talk) 08:17, 22 August 2011 (UTC)
 * I honestly struggled with this. Your description of one side of the dispute is so far off the mark that there really wasn't anything to do but rewrite it.  I've simply expanded the description of the other side to be more explicit about what you're actually talking about.  Here's a marked up diff (original in bold, additions in italics):
 * should it treat it start with more or less a complete "introductory" article (based solely on popular sources such as vos Savant plus any academic sources that happen to use exclusively "simple" approaches) treating the MHP 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  focused on the strategies of "always switching" as opposed to "always staying" without necessarily clarifying the exact question that is asked and without explicitly addressing the specific "player picks door 1 and host opens door 3" case, and also include a full  only after these introductory sections then proceed with a discussion of all other aspects of the problem for the more specialist reader?
 * I sincerely believe this accurately captures your stance, and it's definitely based on your words (and edits to the article). You do want the article to start with a more or less complete introductory article based on vos Savant (and other "simple solution" sources) treating the MHP as a simple probability puzzle focused on the strategies of always switching as opposed to always staying without clarifying the exact question and without explicitly addressing the specific case of "player picks door 1 and host opens door 3", don't you?  Your description of your stance is so vague that nearly anyone would agree with it (for example, I agree with it if we consider an approachable [not based on Bayes theorem] analysis of the specific door 1/door 3 case as "simply solved" and include nearly all introductory probability textbooks in the scope of "vos Savant and many other sources").  Please correct me if I'm wrong, but hasn't your stance always been fundamentally based not on what you want IN the article but on what you want OUT of the early sections of the article (i.e. anything other than "simple" solutions based on analyzing "always switching" as opposed to "always staying", and most specifically anything whatsoever that even hints that the question is about the difference between the prior and posterior probabilities of the car being behind door 1 and door 2 before and after the host opens door 3)?  -- Rick Block (talk) 15:51, 22 August 2011 (UTC)
 * The deal was that you rewrite my wording and I rewrite yours. If you are not willing to give that a try there is no point in proceeding.  I though it was worth a try.  I guess we will have to stick with the original statements then. Martin Hogbin (talk) 21:46, 22 August 2011 (UTC)
 * Does that mean you disagree with something I'm saying here? If so, what? -- Rick Block (talk) 00:17, 23 August 2011 (UTC)

Re: "I guess we will have to stick with the original statements then", actually, as the person requesting comments, I need to choose what I think is the best way to request comments (of course anyone else is free to write Their own RfC with whatever question they like). I am not going to present two wildly different descriptions of the content dispute just because you two cannot agree upon what you disagree upon. That does not allow outside opinions to choose between two clearly described positions. Instead, I will go with the basic logical principle of "whatever the heck Person X says the dispute is about, he has to write the page if consensus favors his position"` and present the two drafts (possibly after editing out trivial differences myself if I can't get cooperation in making them differ only where this an actual content dispute). I can understand that the two of you may disagree about what the dispute is, but I do not believe that either of you are incapable of editing the article to reflect your position should you prevail. Guy Macon (talk) 09:25, 23 August 2011 (UTC)
 * That is somewhat at odds with WP principles, which favour cooperative editing. I do not want to, or claim to be able to write the article by myself.  My point is one of principle, that the article should principally represent the subject as a simple mathematical puzzle, solved simple and correctly by vos Savant and others.  It should also deal with the subsequent media furore and the psychological reasons that people get the answer wrong.  The 'conditional solutions' represent the narrow viewpoint of undergraduate probability and should have no special standing in the article but should, of course, be mentioned along with the many other methods of solution. Martin Hogbin (talk) 14:52, 1 September 2011 (UTC)


 * Do you believe this approach represents (per WP:NPOV) "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources", or does this approach explicitly endorse those views with which you personally agree? -- Rick Block (talk) 15:31, 1 September 2011 (UTC)


 * Beware of false conclusions. "Significant views" vs. "insignificant views".


 * No circumvention, please. All you really "know" is that by staying you have a "1 in 3 chance", whereas by switching you have a "2 in 3 chance" to win the car. The illusion that you eventually "could" know better if you just knew better, the illusion that any (in real words) unavailable but just assumed additional information on the actual location of the car ever could recommend to stay and not to switch is not significant for the MHP, but is outmost insignificant for the MHP. So no outwitting, please. Such mental games are an illusion and belong to the realm of teaching and learning to do correct calculi in conditional probability theory, that's where they belong, but never to making the correct decision and to give the correct answer to the famous question. In the aricle it may be mentioned en passant, although it never is nor ever was "necessary" to make the correct decision. Gerhardvalentin (talk) 16:55, 1 September 2011 (UTC)


 * What measure do we use to, "fairly, proportionately, and as far as possible without bias, all significant views that have been published by reliable sources"? On the number of sources alone, simple solutions probably outnumber 'conditional' solutions by 10 to 1 or more.  On the other hand if we consider only undergraduate level textbooks on probability then the 'conditional' solutions probably win.  Should we take into account the source of the puzzle and its intended audience in our calculation?  I think so.  It was clearly intended to be a simple popular puzzle not an exercise in probability for students. What about game theory and other types of solution.  Why should they get less exposure in the article than the 'conditional' solution? Martin Hogbin (talk) 17:48, 1 September 2011 (UTC)


 * What measure? Well, first, we need to agree that this is what we're trying to do.  Then, we need to make some effort to honestly and objectively assess sources, paying particular attention to reliable secondary sources.  Your 10-1 number is absurd - in my reading it's perhaps 2-1 but the 2 is dominated by popular sources which would be considered much less reliable than the predominantly academic sources constituting the 1.  By any measure, both simple and conditional solutions are extremely common (many dozens of sources), and are thus both "significant".


 * The original source of the puzzle was Selvin's letter to the American Statistician (not vos Savant's column in Parade), so I suspect you don't actually want to consider the source of the puzzle but rather the source of the popularization of the puzzle. Since the puzzle itself originated in academia, and there are not an insignificant number of sources explicitly critical of its treatment in popular sources, it seems highly biased for the article to (in your words) "principally represent the subject as a simple mathematical puzzle, solved simple and correctly by vos Savant and others".


 * Sources presenting game theory and other types of solutions are MUCH less common than either simple or conditional solutions, and should be presented in a way reflecting their prominence. Clearly where we disagree is about the relative prominence of sources presenting conditional solutions and sources presenting simple solutions.  I think these two are essentially equal.  You do not, but it seems to me your view about this primarily reflects your own opinion about the topic itself rather than your familiarity with the sources. -- Rick Block (talk) 19:39, 1 September 2011 (UTC)

Moving Forward to RfC despite lack of agreement on what the Issue is
Clearly this content dispute is completely intractable. Having failed to get a brief description of the conflict that all can agree upon, I am going with the two draft versions.

References:

Talk:Monty Hall problem/draft1

Talk:Monty Hall problem/draft2

and

Talk:Monty Hall problem/RfC

I am rejecting any further claims that the plan of basing the RfC on the two versions is not what we should do, based upon two self-evident facts: First, no alternative has been proposed that all agree upon. Second, if an editor cannot translate his position into a version of the article now, he obviously will be unable to do so should the consensus favor him.

In have taken the liberty of making draft1 and draft2 closer in content, hopefully without erasing the differences that highlight the content dispute. Please review the changes and correct any places where I got it wrong, keeping in mind the goal that the two versions should differ only where they have to to show the dispute. Thanks! --Guy Macon (talk) 05:28, 2 September 2011 (UTC)


 * I appreciate your efforts to help, Guy. I suggest you proceed with and RfC any way you like.  Rick and I will be able to comment in the RFC itself to make our positions clear. Martin Hogbin (talk) 09:46, 2 September 2011 (UTC)


 * I understand what Guy is trying to do here, but I must reject the rejection of rejecting a choice between these two alternatives. This is not simply a choice between X and not X, it proposes to force a choice between apples and oranges for replacing a banana. Yes, the banana is overripe, but I believe the choices being offered both involve too many disparate changes and are both too extreme in some respects. To give just one example of a change that ought to be addressed as a separate question, both current draft versions (1&2) eliminate any mention an "equal goat" stipulation from the Problem description section, apart from retaining the interpolation of a single word "[uniformly]" in one of them. (Note that Martin's original draft proposal did not do this.) I reject, as no choice at all, the choice between two ways of making what I consider to be the same mistake because, IMO, this needs to be brought out more clearly than in the current article, not eliminated or obscured. I made a feeble attempt to frame the main issue in contention as a yes-or-no question, but folks wanted to argue the answers rather than the question of what the question is. That may have been the wrong question, but I still believe we need to find a better question than an either/or Morton's fork that is "flawed by the fact that its tines are not true opposites." (Amalya Lyle Kearse, Burroughs v. Metro-Goldwyn-Mayer, Inc., 683 F.2d 610 (1982)) ~ Ningauble (talk) 18:31, 2 September 2011 (UTC)


 * A completely reasonable position, and a valid argument supporting it. What I need is one of the following:


 * [1]: One brief description of what the conflict is that all can agree upon.


 * ...or...


 * [2]: One draft page per person who disagrees, with the minimum amount of differences between the draft versions needed to show what the conflict is.


 * Clearly we are getting nowhere on [1], it looks like we never will, but of course someone may have a breakthrough tomorrow.


 * for [2], I don't need anyone to agree, nor do I care whether anyone chooses to participate. I can move forward with two, three or four draft versions, and I can ignore anyone who disagrees with all of the other draft versions yet has failed to write up an alternative draft version. --Guy Macon (talk)  19:51, 2 September 2011 (UTC)


 * I am, of course open to suggestions as to some other way to move forward. Discussing this for another year without a resolution is not an acceptable option. --Guy Macon (talk)  19:51, 2 September 2011 (UTC)
 * Very well then, since my proposed version of the question [1] has been ignored and since I am not going to undertake a singlehanded rewrite of the article [2], I consider myself duly banned from these proceedings. Good luck with that. ~ Ningauble (talk) 01:15, 3 September 2011 (UTC)


 * Thank you Guymacon for your efforts and thank you Ningauble, I appreciate your view and the view of Glrx. Basically, at present we are where we were years ago. No fresh air for the readers to clearly conceive the diverging aspects and to discern what the "dilemma" of that misty article is all about. Still crusted "secrets" of outworn perception. Let's finally put them into perspective. There are plenty of actual most reliable sources that enable us to make the article "such as what the subject of the article ought to be": No more stale, crusty one-sided Myopia. Let's renew the article as modern actual sources firmly require, not just tamper.  Gerhardvalentin (talk) 21:51, 2 September 2011 (UTC)


 * Looking back at Ningauble's proposed statement of the dispute, I think it actually does capture the essence - i.e. should conditional probability (specifically an accessible solution based on conditional probability) be introduced early in the article, or should any hint of conditional probability be deferred until later in the article (specifically treating as a minority viewpoint the view that the problem concerns the probabilities the car is behind each door in an example case such as the player initially picked door 1 and the host has opened door 3, and that these are by definition conditional probabilities). -- Rick Block (talk) 19:36, 3 September 2011 (UTC)


 * What am I missing? The article needs improvement in many areas, but I read Martin's  section as finding a consensus (aka acceptance) that door numbers do not matter. For other issues, the Bayes explanation is lumped in with all solutions and has been simplified (made more "accessible") and trimmed (no door numbers). Class labels have been attenuated. If anyone wants to seek consensus on those or other specific issues, just start a section on this talk page that raises the issue -- just like Martin did. Glrx (talk) 20:41, 3 September 2011 (UTC)


 * I disagree there was consensus that door numbers do not matter (and have reverted your change deleting the distinguished door number version of Bayes' theorem treatment - I must have missed this at the time). Let's discuss this in a new section. -- Rick Block (talk) 22:01, 3 September 2011 (UTC)

Motion to close RfC discussions
A content dispute is one thing, but an unconstrained dispute about a murky dispute is madness. Move to close the discussion on multi-drafts and RfC. Glrx (talk) 02:11, 3 September 2011 (UTC)


 * Support as proposer. Addressing smaller, specific, topics is a better thing to do. For example, Martin sought agreement on his items 1-6 at ; a result appeared to be a consensus to suppress the significance of door numbers in the article. Take smaller steps. Creating N drafts is pointless. Glrx (talk) 02:11, 3 September 2011 (UTC)


 * Proposer? While nobody has actually nominated an RfC yet, there is a draft RfC (still being refined) that I wrote, based upon draft versions prepared by Martin and Rick. You have not made a single edit to any of the three.   I don't know whether you proposed an RfC sometime in the past or whether you were the first to do so, but I am pretty sure you have not listed one (nothing at Requests for comment/All), and the one I am preparing to nominate was entirely my idea. --Guy Macon (talk)  07:58, 3 September 2011 (UTC)


 * Strong Oppose as nominator. Rfcs are Requests for Comments. Any editor can nominate a RfC without asking for permission, and I fully intend to propose this one no matter what motion you post or what the responses to the motion are.


 * I am glad that you made the motion, though. It is my contention that discussing smaller steps has not only been totally ineffective at resolving the ongoing content dispute, but has indirectly led to a lowering of article quality as evidenced by MHP being delisted as a Featured Article. I am pretty sure that the responses to this motion will show you that there is a general agreement that this is true, and a consensus to settle the issue rather than continuing to discuss it. --Guy Macon (talk) 07:58, 3 September 2011 (UTC)

Something to consider
What is the most important thing about the MHP? I don't want you to respond with what you think it is; I just want you to think about what is most important. There may be a quiz later. Glrx (talk) 02:11, 3 September 2011 (UTC)
 * Not answering - just suggesting some possibilities:


 * 1) The most important thing is the difference between this situation, where we don't know anything about where the car is so the probabilities for each door are obviously 1/3

1      2       3  ^  | picked

and this one, where we know the probability the car is behind door 3 is 0 but don't know the probabilities for doors 1 or 2 (so it seems like they each should be 1/2, but with "normal assumptions" a very elementary conditional probability analysis shows they're 1/3 for door 1 and 2/3 for door 2)

1      2       3  ^  | picked
 * ? |  | ?  |  |goat|
 * ? |  | ?  |  |goat|
 * ? |  | ?  |  |goat|


 * 2) The most important thing is that there is no difference (with respect to the total probability the car is behind door 1) between this situation

1      2       3  ^  | picked

and the combination of these two (since the host MUST open one of door 2 or door 3 if the player picks door 1), so picking door 1 and not switching wins with probability 1/3 while picking door 1 and switching (to whichever door the host does not open, whether it's door 2 or door 3) wins with probability 2/3

1      2       3          1       2       3     ^                          ^                     |                          |                    picked                     picked
 * ? |  | ?  |  |goat|     | ?  |  |goat|  | ?  |
 * ? |  | ?  |  |goat|     | ?  |  |goat|  | ?  |
 * ? |  | ?  |  |goat|     | ?  |  |goat|  | ?  |


 * 3) The most important thing is whatever reliable sources say, and since many say #1 and many say #2 it's important for us to say both of them as well. -- Rick Block (talk) 21:36, 3 September 2011 (UTC)

Bayes' theorem
Glrx deleted the long standing (sourced) Bayes' theorem analysis showing the conditional probability the car is behind door 2 given the player picks door 1 and the host opens door 3 is 2/3, leaving instead an unsourced analysis using Bayes' theorem showing that the probability the car is behind the initially selected door is 1/3 given the (probability=1) "event" that the host opens a door showing a goat with the (at least misleading) conclusion that this means the probability the car is behind the unopened, unpicked door is 2/3. I've reverted this, restoring the sourced analysis and deleting the unsourced one.

Two points:

1) sourced is better than unsourced

2) using Bayes' theorem to show a probability 1 "event" doesn't change the probability the car is behind the player's initially selected door is extreme overkill, and (confusingly) means something completely different than that the probability the car is behind door 1 given the host opens door 3 is 1/3. In particular, the fact that the probability the door picked by the player remains 1/3 after the host opens "a door showing a goat" does not mean the probability the car is behind the remaining unopened door is 2/3.  A probability 1 event doesn't change any of the probabilities, so given this event the probability of each door (not just the door the player picks) remains 1/3 - the two unpicked doors still total 2/3 and switching without specifying which door you're switching to will win with probability 2/3, but this event distinctly does not say there is one door with probability 0 and another door with probability 2/3. Subtle errors like this are one of the primary reasons sourced is better than unsourced. -- Rick Block (talk) 22:52, 3 September 2011 (UTC)

Revert without discussion?
The following revert:

https://secure.wikimedia.org/wikipedia/en/w/index.php?title=Monty_Hall_problem&curid=6026198&diff=448295220&oldid=448289792

Appears to have no discussion associated with it. (It may be that I somehow missed the discussion)

According to WP:BRD and WP:REVEXP (also see WP:ROWN and WP:DRNC), the revert should be followed by a discussion.

Please note that I am NOT taking a position on whether the original edit or the revert was good or bad - I just want to see Wikipedia Policy followed. Thanks! Guy Macon (talk) 02:35, 4 September 2011 (UTC)


 * OK, I see Rick Block discussing this (section directly above). I had searched for a talk page or edit comment from Glrx. No doubt he will respond to Rick, but explaining your reverts as you make them is preferred. Guy Macon (talk) 10:59, 4 September 2011 (UTC)


 * Guy, please self-revert this edit. I have explained my reasons for the edits in the original edit summaries two weeks ago and in some more recent talk page text above; see, for example, talk page edit. Your revert is inconsistent with your many claims of neutrality -- including your recent "I am NOT taking a position" claim above. The revert opposes a consensus view that doors numbers are not important: it makes a content judgment. Your edit summary is also devoid of content; it's about my editing practices (which typically run a day behind my watch list). Rick's section above is to seek an alternate consensus from others; his and my positions are clear. Your revert suggests such consensus is not necessary. Your revert has other, more serious, issues. Glrx (talk) 16:16, 5 September 2011 (UTC)


 * I will be going to revert Guy and to revert Glrx as follows. Is this okay? Gerhardvalentin (talk) 18:09, 5 September 2011 (UTC)


 * On what basis? A revert for revert sake? Maintain some status quo? This is a content issue. Weigh in on the content. Weigh in on consensus. What about your agreement on Martin's points 1-6? Glrx (talk) 18:21, 5 September 2011 (UTC)


 * My proposal is to a certain extent consistent with Martin's items 1-6. Imho it's not good enough, I know. Other suggestions? Gerhardvalentin (talk) 19:44, 5 September 2011 (UTC)
 * Your post is confusing and off the mark. Your link is essentially silent on a main issue of the revert. Furthermore, you have continually lobbied for an odds-form expression of Bayes, but you have not garnered a consensus for the odds-form. Why does the odds-form belong in MHP? Most of the article speaks in terms of probability or chance; why the diversion into odds? I reverted the odds-form for a reason; neither Rick nor Guy restored it. If you want to include it, then open a section and get a consensus. Glrx (talk) 20:22, 5 September 2011 (UTC)


 * What is the source for the "Bayes' theorem" section in this draft? And, more specifically, what is the source for the sentence that says "That also means that switching is advantageous because the probability that the remaining door has the car is 2/3."  Please note that the exact same formula can be used to show the probability the car is behind any of the doors after "the host reveals a goat behind one of the other two doors" is 1/3 (let A be the event "the car is behind one of the unselected doors").  What this formula is actually saying is that switching is advantageous because the probability that the car is behind one of the other two doors is 2/3.  But, we know that at the beginning - without using Bayes' theorem.  This is an exercise in (mis)using complicated machinery to show something that is terribly obvious in a way that makes it seem like it is showing something else (which it doesn't).  Without a source behind it, it's simply WP:OR.  Even with a source, it would be stupid.  It adds nothing, other than a veneer of respectability (oooh, look, there's a complicated formula - this MUST be right). -- Rick Block (talk) 20:09, 5 September 2011 (UTC)


 * In a more positive vein, if we're going to have a section using Bayes' theorem we should have one that isn't dumbed down to the point of insipidness, but rather one that reflects the many, many, many sources that present this style of solution. For many years, the Bayes' theorem approach was effectively presented as an appendix (e.g. this version).  I'd be fine with going back to that approach.  If we want a Bayes' theorem section inline, I'm fine with that as well.  What I'm not fine with is a "simplified" version that's unsourced, misleading, and not reflective of the many sources that present this sort of solution.  And lest anyone misread what I'm saying here, I'm stating my opinion as one editor of this article. -- Rick Block (talk) 20:56, 5 September 2011 (UTC)


 * Just so my position is clear, I am neutral on the actual content, favoring neither Rick Block, Glrx, or Gerhardvalentin. I just want WP:BRD to be followed.


 * I would encourage Glrx and Gerhardvalentin to write up a draft for use in the RfC. We are only going to be able to choose between point of view that have been proven by example to be something we can use to creatte an article. Guy Macon (talk) 21:59, 5 September 2011 (UTC)


 * Guy Macon (talk) 21:59, 5 September 2011 (UTC)


 * I have already stated a clear objection to your N-draft proposals. It would be absurd for me to take part in or to condone such ownership of the article.
 * I have stated positions, and then I have gone off to edit those positions into the article. In large part, those edits have stuck. Rick claims he was asleep at the wheel two weeks ago, but he's the only editor who has made that claim. Or is everybody here ignoring the article? Rick's recent edit was adding content; I reverted him; it is Rick who needs to argue the addition of a dense BR proof; so far he hasn't shouldered that burden; he's arguing against simplicity but not for complexity; he has also acquiesced about the dense proof. Go figure. (BTW, I agree that a BR presentation of Monty's unopened door may be worthwhile.)
 * Your (i.e., Guy's) position is far from clear. If you want to be neutral on content, then you should stay off the article page where content appears. Reverting to Rick's version is not being neutral. Furthermore, you are going against the intent of arbcom in at least two ways. Arguably, Gill has made comments that minimize an explicit-door BR proof. You are not the final arbiter of WP policy.
 * If you are going to assume the role of a policeman, then you need to interpret what happens on the talk pages and comment on content. You continually eschew that role. You are not functioning as an arbitrator or a mediator. I don't think you see what happened.
 * Please self revert.
 * Glrx (talk) 23:57, 5 September 2011 (UTC)


 * I have self-reverted while we discuss this. I reverted you because at the time you had offered no explanation in your edit in the edit comment or on the talk page, and had gone on to edit several other articles without discussion your revert. I believe that my decision was content-neutral, and I would have undone someone reverting you without explanation or discussion.


 * As to me not being the final arbitrator of Wikipedia policy, completely true, but not helpful. I do try to follow the consensus that shaped those policies, and if someone argues that I am getting it wrong (as has happened a couple of times, most recently on whether a Masters Thesis is a reliable source) I go to the appropriate noticeboard and seek a clarification. If you have a specific area wher I am getting it wrong, tell me.


 * Regarding "If you are going to assume the role of a policeman, then you need to interpret what happens on the talk pages and comment on content." I strongly disagree and wish to remain neutral on content, but I am throwing it open for general discussion: Is Glrx right?


 * Regarding rejecting the N-Draft method, do you have any alternative at all other than what we have been doing for the last few years? I really do want to at least consider following your advice, but so far your advice has been, in effect, that taking this particular content dispute to content dispute resolution is impossible.  Help me to understand how to take the long-standing and intractable content dispute to content dispute resolution without you objecting. Guy Macon (talk) 03:54, 6 September 2011 (UTC)


 * Well there is no agreement for removing the detailed 3 step Bayes explanation from the article (despite the misguided of some people to do so for years). The problem regarding the arguments here is that many people simply stop to closely listen to this neverending and mostly futile discussion. Though I don't agree with Rick on his so called "NPOV approach", I agree with him that detailed 3 step Bayes is a useful addition to article. Moreover the current version is even correctly attributed and the formely correct attribution is changed into see for further treatment. And yes it is more technical than the other ones, but that is somehow the point to offer that as well. There are already several other less technical or less detailed approaches and also some to do ignore the chosen door, so it isn't of additional benefit for readers to turn this into an additional one (of what we have already). The articles has enough room for several Bayes treatments and for the detailed 3 step version among.--Kmhkmh (talk) 03:39, 6 September 2011 (UTC)
 * So I suggest this clear and well-sourced version. Gerhardvalentin (talk) 22:32, 5 September 2011 (UTC)
 * You are missing the core objection. Strangely, your second draft seems to support deleting the dense BR proof that Rick restored. Is that your intent? This talk section is not about your desired version of an odds-form BR. Take that up in your own section; it is a separable issue. This section is not for side debates. I'm being curt because I want to keep focus. Glrx (talk) 23:52, 5 September 2011 (UTC)


 * 'Rick's recent edit was adding content; I reverted him; it is Rick who needs to argue the addition of a dense BR proof; so far he hasn't shouldered that burden; he's arguing against simplicity but not for complexity; he has also acquiesced about the dense proof."


 * I restored sourced content that you deleted. You need to argue why this should be deleted in favor of the "simpler" unsourced content.


 * I'm not arguing against simplicity, or for complexity - I'm arguing for accurately representing what the sources say. Among sources presenting a BR analysis, an overwhelming majority (it's at least 90% and perhaps as high as 99%) use BR to compute the probability in a specific case, such as the player has picked door 1 and the host has opened door 3 (with door numbers that are properties of the doors, not substitute labels for the roles of the doors).


 * Where have I acquiesced about the dense proof? -- Rick Block (talk) 02:51, 6 September 2011 (UTC)

Just a reminder: when editors disagree about content, the page is supposed to stay at the last stable version while it is discussed. I just self-reverted while seeking opinions on my neutrality (If I am not being neutral and don't see it, I don't want to edit under that condition) but can we agree to leave the page at the last stable version (editing/reverting as needed to get there)? Guy Macon (talk) 03:54, 6 September 2011 (UTC)
 * Yes, but that last stable version includes the 3 step Bayes. Also I would very much prefer if the involved parties sort out the other longstanding problems first, instead of opening a "new" sideshow regarding "the best" formal Bayes treatment.--Kmhkmh (talk) 04:11, 6 September 2011 (UTC)

Request for Arbitration Enforcement
Glkanter 's latest IP addresses have now been blocked.

https://secure.wikimedia.org/wikipedia/en/wiki/Wikipedia:Arbitration/Requests/Enforcement#Request_concerning_Glkanter

https://secure.wikimedia.org/wikipedia/en/wiki/Wikipedia:Sockpuppet_investigations/Glkanter

If he finds new IP addresses we will explore a range block. Guy Macon (talk) 14:51, 5 September 2011 (UTC)

Final draft of the Rfc
The RfC at Talk:Monty Hall problem/RfC is as good as I can make it. This is your last chance to comment/edit before I pull the trigger. In particular, I had to do some major surgery to the two drafts so that (I hope) they only differ where there is a content dispute. Please review and edit as needed; no doubt I got it wrong in places. Guy Macon (talk) 23:03, 12 September 2011 (UTC)
 * I am not going to edit or comment on what is in the draft RfC, of course, because you have declared that you are ignoring me. However, I feel compelled to comment on what is not in it. I made a good faith effort to formulate a statement of dispute in a manner that that is sufficiently clear and direct to be placed before the wider community for decision. That statement was endorsed by Rick (first tentatively and then more difinitively) who is a principal disputant and is, most significantly, the person who placed a POV tag declaring on the face of the article that it is in dispute. Although it has been noted that there are additional areas of contention, no one has denied that this is indeed one of the principal matters in dispute. I appreciate that you are trying in all good faith to find some resolution that can end the interminable argumentum gratia argumenti, but I cannot comprehend why you insist on ignoring a question that might actually do so for one of the principal disputes by fairly characterizing it and settling it in a manner that would give clear direction for article development. You are entitled to ignore me if you believe my proposed question is unconstructive, but I do not understand why you would think so. Perhaps you feel that tackling the points of disagreement one at a time would take too long, but failing to do so will, in my opinion, take forever. ~ Ningauble (talk) 15:28, 13 September 2011 (UTC) Self-striking my mistake, as explained below. ~ Ningauble (talk) 22:38, 13 September 2011 (UTC)


 * I agree with Ninguable that a simple question would be much better than having two drafts but I think Ninguable's statement of the dispute presupposes that the conditional solutions are in some way 'the right answer'. Many editors, not just myself, have considered that, bearing in mind the context, the simple solutions are 'the right answer' and the conditional solutions apply only to a variant of the problem.  Why not just present Rick's and my statements of the dispute, along with Ninguable's if that helps. Martin Hogbin (talk) 16:47, 13 September 2011 (UTC)
 * No Martin, I do not presuppose that one is "the right answer." Rather, as I have repeatedly stated, I suppose, based on reliable sources, that both are correct answers and that each should be presented clearly and neutrally in the most widely understandable manner possible. Evidently you do not agree with this (though I believe we do agree about the order of presentation), and that suggests a different direct and simple question that might better be addressed first as a more fundamental one: Should the article assert that one or the other of the answers commonly given in citation-worthy sources is a fringe theory that is not quite "right" in some respects? ~ Ningauble (talk) 22:28, 13 September 2011 (UTC)
 * I agree that having the solutions in the order that you support might be a good compromise between the considerable majority here who believe that 'conditional' solutions do not answer the question that was intended but answer an academic extension of that question, and others who believe that the 'conditional' solution is the 'correct' one. Martin Hogbin (talk) 14:13, 14 September 2011 (UTC)
 * On the other hand, I am not going to stand in the way of anything that will bring new editors to this discussion (but please bear in mind that I was originally a new editor who came to an RfC on the same subject). Martin Hogbin (talk) 16:47, 13 September 2011 (UTC)

Ningauble, clearly I have offended you and, looking back at my words, you have reason to be offended. For that I am deeply sorry, and I apologize and retract my previous statement. Let me try to restate what I was getting at in a way that I hope will not cause offense.

I am not ignoring you. I am not giving you veto power over whether I follow the steps of Content Dispute Resolution, but I have put that as an option you can vote for in the RfC. If the consensus is to continue talking indefinitely, then that's what we will do. (And if "we should continue talking indefinitely" does not describe what your position is, you have been repeatedly invited to suggest some other wording. I did read every suggestion you have made, and that's the best I could do to turn it into a simple statement of what you think we should do.)

I am not going to put anyone's statements of what the problem is in the RfC questions, because there is no agreement on what the dispute is. Everyone, including you, is free to place such a statement in your arguments section.

Whether because you are unable or unwilling, the fact is that you have not shown by example what the article will look like should you choose to participate in the RfC and the consensus goes your way. An opinion as to how the article should be edited that can not be used to edit the article is unlikely to gain wide support.

You can, of course, choose to not participate despite my apology (and it was a sincere apology - I really am sorry I offended you), but there is no provision in the dispute resolution policies that exempts those who refuse to participate, and if we get a general consensus resolving the dispute you will have to abide by the consensus just like everyone else.

Needless to say, if you can show me a statement of the dispute that everybody agrees really is a statement of the dispute, then I will go with that. Good luck with that.

Martin, we can have the simple question you describe, but you will have to post it in your arguments, because Rick does not agree that your simple question describes the content dispute, Likewise, Rick (and possibly Ningauble) can pose simple questions, but in their argument sections because you don't believe they are accurate descriptions of the content dispute.

The question is, what if one of those arguments gets an overwhelming consensus? If the "winner" cannot take that consensus and edit the article to conform to it, what good is it? I am already anticipating a lot of problems from whoever "loses" saying that the "winners" edits do not reflect what was voted on. Without an example showing roughly what the page would look like such problems will be a thousand times worse. Only two editors have shown that they can create an article if the consensus goes their way. Guy Macon (talk) 18:15, 13 September 2011 (UTC) Guy Macon (talk)
 * Guy, I disagree. I think two (or three) descriptions of the disagreement including one from each side makes it easier for newcomers to see where the difference lies.  We have already seen how asking people to compare drafts leads to arguments about irrelevant details.  Martin Hogbin (talk) 21:25, 13 September 2011 (UTC)
 * The two sides disagree on the weight of conditional probability theory for "solving" the paradox. Conditional probability theory that forever is out of position to show that staying could ever be of any advantage. See "Conditional probability and the MHP".  Regards, Gerhardvalentin (talk) 22:07, 13 September 2011 (UTC)


 * Game theory should be a topic in the article, as I said in the Arguments page. I will be trying to write a draft including Game Theory. Gerhardvalentin (talk) 09:18, 14 September 2011 (UTC)
 * I agree. Once we move away from the obviously intended simple puzzle, game theory has a greater claim to be the right answer than the 'conditional' solutions do. Martin Hogbin (talk) 14:13, 14 September 2011 (UTC)
 * Guy, I accept your apology because I believe you are acting from good intentions, and I am striking my rather severe response above. I can see that I made two errors:
 * Given the replies of Martin and Gerhard above, I see there is actually a much more extreme position than is indicated in either of the drafts, and the question I proposed is therefore not sufficiently fundamental.
 * Given your hyperbolic claim that I propose we should continue talking indefinitely, which I never said, and your perspective on what does and does not constitute legitimate participation in the editorial process, it is I who should be ignoring you for the time being because it is, after all, your RfC and I should simply wait for you to post your simple and succinct question before responding.
 * Sorry, my mistake. ~ Ningauble (talk) 22:38, 13 September 2011 (UTC)
 * Guy, why do you not try to formulate a simple description of the dispute? You have always maintained a neutral position but you have followed all the arguments so this should not be too hard. Martin Hogbin (talk)
 * Yesterday I spent the better part of an hour trying to do exactly that. Every version I created seemed to me to show a bias towards one position or the other and/or to directly contradict what at least one person has claimed that the conflict is. So I slept on it and tried again this morning, with no better results.


 * So my best effort is to have as my simple and succinct question, "choose between Version A, Version B, Do nothing, or Rewrite and Restart the RfC with more versions or with a yet-to-be-created question that there is broad agreement describes the content dispute." That's the best I can do. Guy Macon (talk) 21:34, 18 September 2011 (UTC)


 * Thanks for your efforts Guy. All we can do now is get some new editors in.  We could then start by having a section where current editors could write a short section explaining their views to the newcomers, if we get any. Martin Hogbin (talk) 23:12, 18 September 2011 (UTC)

It is essential to include Carlton's "intuitive" Solution
I don't know what RfC means and I don't understand the previous sentence. So I am putting my comments here. In my view, it is essential to include Carlton's "intuitive" Solution. This is a very slick and yet transparent way to understand the problem. Until I saw that, my favorite explanation was: If you pick one door, then the car has a 2/3 expectation of being behind one of the other two. Since the car is not behind the door Monte opened, that 2/3 expectation of the car being behind one of two doors collapses into a 2/3 expectation of the car being behind the door Monte did not open.

Another point. Monte's knowledge or state of mind has nothing to do with this problem. The event happens only once and we are discussing expectations related to that single event. It is the principle of indifference that is the decisive factor. --Dagme (talk) 05:18, 19 September 2011 (UTC)


 * An RfC is a Request For Comment asking other editors who don't normally look at a page to comment about one narrow topic where the current editors are having trouble agreeing. It is still unfinished and not something you need to be concerned about right now (there will be an announcement when it goes live and at that ttime your comments will be welcome just like everybody else's). Your comment above is about improving the Monty Hall problem Wikipedia page, and you put it in the right place, the Monty Hall problem talk page. --Guy Macon (talk) 11:45, 19 September 2011 (UTC)
 * On the other hand, Dagme, you are welcome to join in the discussion at any time. Martin Hogbin (talk) 10:38, 25 September 2011 (UTC)


 * As many, many sources say, Monte's knowledge is critical. If Monte opens a door without knowing what's behind it and happens to reveal a goat, then the result is a 50/50 choice (for example, see vos Savant's explanation of this here).  The notion that the 1/3 probability the car is behind the player's originally picked door remains unchanged regardless of what the host does is simply incorrect (as many, many sources say).  Whether or not the article should propagate this misunderstanding is at the root of much of the conflict about this article.  -- Rick Block (talk) 01:01, 30 September 2011 (UTC)
 * It is indeed crucial that Monty selectively reveals a goat behind an unchosen door, and Carlton's paper actually stresses this point. (It is implicit in the "game show" context that the grand prize is not given away before offering the contestant a final choice and, as vos Savant said, "anything else is a different question.") Perhaps this could be made more clear in the article, but I do not believe "regardless of what the host does" is at the root of dispute among regular participants here: we all agree that variants in which Monty does not intentionally reveal a goat behind an unchosen door are distinctly different problems having distinctly different solutions. ~ Ningauble (talk) 14:36, 30 September 2011 (UTC)
 * Yes you are, of course, quite correct Ninguable, none of the dsputants is talking about the case where the host has the option of opening the door hiding the car. That fact has no connection with this dispute, although it does seem that Dagme is unclear about it. Martin Hogbin (talk)
 * IMO, most of the conflict concerns whether the article should primarily focus on solutions presented by authors such as vos Savant (and many others) based on the intuition that the player's original 1/3 chance cannot change upon actually seeing a goat behind the door the host opens - in preference to solutions based on a careful conditional analysis as recommended by authors such as Carlton (and many others). The reason the "simple" argument applies in the standard variant but does not apply in numerous slight variants is not at all clear to most people who have learned this argument, and they then proceed to misapply it to variants where it does not apply (as Dagme does, above - "Monte's knowledge or state of mind has nothing to do with this problem").  See previous discussion about this archived here. -- Rick Block (talk) 18:23, 30 September 2011 (UTC)


 * Mark this section for deletion* Jettatore (talk) 08:08, 4 November 2011 (UTC)

Found what is most likely an error.
Location of error: Section "Solutions", 2nd paragraph, last sentence.

Quote of potential error: "If the card remaining in the host's hand is the Ace of Spades, this is recorded as a round where the player would have won by switching; if the host is holding a red two, the round is recorded as one where staying would have won." as of 10/20/2011 8:54PM UTC.

It would seem to me this was accidentally written backwards and that if the host is holding the Ace of Spades that this would be a round where he would have won by staying, and then the next part if he is holding a red two that this would be recorded as a round where he would have won by switching.

I have not edited the main article as yet without a discussion/investigation first. — Preceding unsigned comment added by Jettatore (talk • contribs) 20:55, 26 October 2011 (UTC)

edit: I found that after re-examining it all, "host's hand" which does make the original article correct, but given my confusion possibly it could be re-worded to be initially more clear to new readers? Anyways, delete all this if it's of no further use Jettatore (talk) 21:30, 26 October 2011 (UTC)


 * The "card remaining in the host's hand" is the one that is offered by the host to the player to switch on. In 2 out of 3 it will be the Ace, and then switching will win. Gerhardvalentin (talk) 13:14, 27 October 2011 (UTC)

Yup. Mark this section for deletion Jettatore (talk) 08:09, 4 November 2011 (UTC)

The strategic solution by dominance
A most fundamental and sourced solution, answering the famous question "is it to your favor to switch or to stay" by using the principle of dominance and avoiding doubts on the validity of the correct answer has been made available by editor RocksAndStones. This fundamental solution had been inserted exactly where it belongs, the correct place for this approach is of course to immediately follow the famous answer of MvS.

Unfortunately Rick Block has moved this important view and has placed it where it does not belong. Because of its importance with regard to the correct answer to the famous question "yes or no", I will be going to revert this improper displacement. Gerhardvalentin (talk) 17:37, 16 November 2011 (UTC)
 * This solution was inserted between a section presenting vos Savant's (simple) solution and another section titled "Other simple solutions". It is FAR from simple, so belongs elsewhere.  Placing it this early in the article also gives it a much greater prominence than is warranted - it is but one of many ways to show switching is never worse than not switching which is the same result shown by all the "simple" solutions and by the much more standard Bayesian treatments as well.  Before simply reverting I would encourage you to wait for others to comment. -- Rick Block (talk) 20:54, 16 November 2011 (UTC)
 * I agree with Rick that this is not simple in the appropriate sense, and support the move. Although I appreciate the sense in which this is logically simple and elegant, this sort of simplicity represents a level of sophistication that is not simple in the sense of being accessible to the widest possible audience. Furthermore, and for the same reason, this section really ought to link to an article that explains what is meant by the concept of "dominance" in formal logic or game theory, whichever more closely matches the argument presented, as is done, e.g., for solutions entailing conditional probability. ~ Ningauble (talk) 17:21, 21 November 2011 (UTC)

Monty Hall Paradox (simple answer)
I really hate that I am just a high I.Q. Ga. boy and am about to make you all feel silly, but this is childs play!!!!!!! Slow your brain down for a second and breath deep, it goes like this: Its not the switching of the doors that makes you win more (so to speak), its "NOT SWITCHING" that "LOCKS" you into the 1/3 odds of winning! You see, the other numbers have nothing to do with it-if you dont switch, its a one time deal. If you had 2 doors to choose from it would be 1/2, if you had 8 doors it would be 1/8, and so on and so on, if you dont switch, then you are stuck with the original set of odds, and this CAN NOT BE CHANGED UNDER ANY CIRCUMSTANCES! TADA!!! You all are welcome, David Micha Cross. — Preceding unsigned comment added by Davecross1 (talk • contribs) 08:25, 24 November 2011 (UTC)

If you were as bright as you think you are, you would have noticed the sentence near the top of this page that states that this is a place for a discussion of the article. Like all Wiki discussion pages, it is not a place for a general discussion of the issues raised in the article.Hellbound Hound (talk) 09:32, 5 December 2011 (UTC)

The trick

 * lies in that the host will always open a door with a goat, If the host opens one of the remaining doors at random and reveals the car (by accident) the whole paradox falls to pieces 184.88.37.235 (talk) 13:38, 9 December 2011 (UTC)
 * Is that not already clear enough in the article? ~ Ningauble (talk) 16:37, 9 December 2011 (UTC)
 * the article seems pretty adamant that the host will always pick a goat door and not that he picks a random door that may or may not contain the car 184.88.37.235 (talk) 20:45, 9 December 2011 (UTC)
 * No, this doesn't only "seem" so, but it is explicitely said so with intent, as this is the basic of the question "Is it to your advantage to switch your choice?"  Gerhardvalentin (talk) 21:15, 9 December 2011 (UTC)
 * That's not "the trick," as you call it, that's the entire point of the problem, and is made clear in every solution presented in the article — Preceding unsigned comment added by 108.45.70.208 (talk) 06:26, 19 January 2012

R code for simulating the problem
Hello All.

I code faster than I read, so decided to get my head around the problem by simulating it in R. This is 50 lines of messy code (apologies, I'm sure it could be improved!). I'm not sure if this is the kind of thing moderators would feel it would be useful to include, but I thought I may was well offer it up! Hope I've posted correctly.

Best,

--BmrE2 (talk) 18:27, 11 December 2011 (UTC)


 * @BmrE2, thanks, the code is fantastic. When you optimise it to three lines, please just post a weblink. Users of this page are mainly involved in the discussion of logical aspects of the paradox and its variations, rather than checking if a particular program works.RocksAndStones (talk) 22:15, 11 December 2011 (UTC)
 * It could be done as a one-liner in APL, but this is not germane to the purpose of this talk page, which is for discussing changes to the Monty Hall problem article itself. ~ Ningauble (talk) 18:21, 12 December 2011 (UTC)
 * Thanks for the feedback, both. I agree that it wouldn't have a useful place in the article, apologies for filling talk pages with more junk! My reaction to this article seems pretty common... --BmrE2 (talk) 01:46, 13 December 2011 (UTC)

Intuition suggestion
The article seems to focus a lot on pitting math and experimentation against a failure of intuition, but there's another tactic to try too: elicit a stronger (and correct) intuition and pit it against the false one. In the example of a million doors, I'm thinking we can ask the reader to, for example, "Just imagine the host marching down the line of a million doors, opening them one by one...and then he skips one! That door seems very suspicious - why did Monty skip that door?  It is very likely that Monty is skipping that door because the car is behind it.  So obviously, the player should switch to that door.  The only way the player would lose is if he'd picked the car already, and the host was skipping any old door - and that's very unlikely."

What do people think of this idea? —AySz88\ ^ - ^ 22:34, 28 December 2011 (UTC)
 * To clarify, this sort of idea is referenced several times (in "million doors" and "N doors"), and the mathematics is described, but it's never presented in a way that appeals to intuition. —AySz88\ ^ - ^ 23:26, 28 December 2011 (UTC)
 * i like it, as so much of this problem is the result people ignoring math and going with their faulty intuition. what you're presenting is similar to expanding the "missing dollar" riddle, so that people would be less susceptible to that faulty intuition — Preceding unsigned comment added by 108.45.70.208 (talk) 06:29, 19 January 2012 (UTC)

I am in favour of more way to help readers to understand this problem, however, they should be based on reliable sources. That does not mean that we must copy exact wording from sources but that we should give explanations like those in sources. The proposed explanation is essentially that given by vos Savant. Martin Hogbin (talk) 09:52, 19 January 2012 (UTC)
 * I'm a bit confused about how your comment applies. Are you saying that my explanation is too similar to something that von Savant wrote?  I actually wrote it myself, from the information currently in the article; I didn't refer to any of von Savant's articles (though I wouldn't be surprised if she took a similar approach at some point).  Could you point me to exactly what you say it's similar to? —AySz88\ ^ - ^  05:31, 26 February 2012 (UTC)
 * The "million door" analogy is from vos Savant's very first column about this problem, see http://www.marilynvossavant.com/articles/gameshow.html (her first column is the initial question plus the response in red text). On the other hand, note Ruma Falk's article which (at the very end) says (of vos Savant's million door analogy) "There can hardly be a more smashing argument.  Only that its truth depends on the understanding that if the prize is behind door no. 1 [the door the player initially picks], the host decides by a fair draw which of the remaining 999 999 doors to leave closed.  If, however, you know that for some reason or other that the host is determined to leave door 777 777 closed [the door the host skips as he opens all the rest], whenever possible, observing that situation will render that door as likely to hide the prize as door no. 1.  You would then have no reason to hurry to switch".  The point here is that how the host decides what door to open, in the event that the player has initially (however improbably) chosen the door hiding the car is critically important.  You're  never worse off switching, and if you decide before seeing which door the host opens that you're going to switch you'll win on average (n-1)/n times (where n is the number of doors) - but given the knowledge of which door the host opens, your chance of winning by switching depends on how committed the host is to leaving the door that he doesn't open closed.  If we call this p (he'll leave this door closed with probability p in the case where you happen to initially pick the car), your chances of winning by switching are 1/(1+p) - regardless of how many doors there are. In the 3-door version p is generally assumed to be 50% meaning if you initially pick the door hiding the car the host opens either remaining door randomly - and your chance of winning by switching is therefore 1/(1+1/2) = 2/3.  In the million door version, assuming the host randomly picks which door to leave closed if you've initially selected the car means p is 1 / 999 999 - so your chances of winning by switching switching are 1/(1+1/999999) = 999999/1000000.  If you don't know what p is, your chances of winning by switching are somewhere between 1/2 and 1 - on average, (n-1)/n where n is the number of doors - but if you don't know what p is the bottom line is you don't know your probability of winning by switching.  It's at least 1/2, might be as high as 1, and averages (n-1)/n where n is the number of doors - but it's actually 1/(1+p). -- Rick Block (talk) 07:47, 26 February 2012 (UTC)
 * Although our explanations should be supported by reliable sources there is nothing is WP policy that says that we must quote sources verbatim, in fact this is not the preferred approach.


 * I therefore suggest that AySz88 should propose some form of wording that he considers the most intuitive for discussion. We already have a reliable source that supports that solution and we are free to try to improve the wording.  Bear in mind that what might seem very intuitive to one person might not seem so to another.  Martin Hogbin (talk) 10:24, 26 February 2012 (UTC)


 * @Rick Block: Although your reasoning:
 * "depending on p, the chance to win by switching must be within the range of at least 1/2 to even 1/1, without exception, and on average it is 2/3"
 * is mathematically sound and completely correct, the literature on MHP shows very clear that this cannot be of any relevance whatsoever for the actual question the MHP asks for, the actual decision to be made,
 * as long as p is unknown. And, for the MHP, p  is  unknown.
 * And as long as p actually is unknown, p can be of effect and of relevance only in mathematics lessons on probability theory, but remains without any effect for the MHP, and can never have any influence on the right decision to be taken: definitely to switch in any case and without exception, based on and tightened to the only definitely "known" solid base, on the chance to win by switching of 2/3. To actually switch to the door offered.


 * The literature on MHP has shown that abundantly, meanwhile. Gerhardvalentin (talk) 10:49, 26 February 2012 (UTC)


 * Martin, I'm still confused about what you're asking. Do you feel that my proposal (in my first post above) is not intuitive enough?  I should note that I make it clear what is happening in strategic terms - the player is taking advantage of the host's knowledge - while MvS does not until later columns.  I also offer it to be improved upon by anyone who so wishes. (continued later; originally AySz88\ ^ - ^  17:07, 29 February 2012 (UTC))
 * I personally did not find your original proposal particularly intuitive but I am just one person. My real point is that I do agree with you that the article could make a better attempt at persuading people of the correct answer by making it seem more intuitive. The problem is that what one person finds intuitive another does not. The other problem is that, once you get the correct answer it is hard to put yourself in the position of someone who does not.  The most convincing simple argument to me is that, if you swap, you always get the opposite of your original choice. Martin Hogbin (talk) 18:27, 29 February 2012 (UTC)
 * I should try to clarify what I mean, I suppose. By eliciting an intuition, I don't mean just an argument where the explanation is intuitive.  I mean something that makes the game intuitive - such as, some sort of narrative or story that brings some intuitive insight into what is happening.
 * A case analysis is much simpler to prove and has better rigor, but it has none of the satisfying power I'm hoping to get at.
 * From my observation, I think that the basic thing that trips people up is that it's not immediately obvious that the host opening the door actually imparts any information (i.e. in a Bayesian sense, if you want to be technical about it). For example, one incorrect reaction to the goat reveal is, "What does that matter?  I already knew that there was a goat in at least one of those two doors."
 * In these cases, what is not obvious is that the host is actually forced into revealing something useful (the location of the car) the majority of the time. This is what I was trying to bring out by my original proposal.
 * Perhaps I should break things down a little bit further:
 * It is important to note that if the player chooses a goat first, the host is forced to reveal the location of the other goat to you, thereby revealing the car by process of elimination. This might not seem too useful at first - how would one intentionally choose a goat first?  But even if the player can't always choose a goat, it can still be useful.  To make it more apparent, imagine that there are a million doors.... (And walk the reader through deriving the winning strategy, like in my first iteration.)
 * Thus, this is the winning approach: "Attempt to choose a goat first, forcing the host to reveal the other goat. Switch to the last remaining door to win the car."  This strategy succeeds if you initially pick a goat (two out of three tries) and fails if you initially pick the car (one out of three).  Success depends entirely on the first pick.
 * Better? Again, feel free to edit or build upon this. —AySz88\ ^ - ^  06:09, 7 March 2012 (UTC)
 * (As an aside, I noticed another cute demonstration in the comments at that MvS site. Show that there's no difference between this and the following: (1) pick a door, (2) declare, before the host opens a door, "I would like to switch to both of the remaining doors", thereby increasing your odds to 2/3, (3) the host declares, "Let's show you which one of your doors had a goat", and shows you that door first.  However, I'm not aware of a good source for this.) —AySz88\ ^ - ^  17:07, 29 February 2012 (UTC)
 * This is more or less shown on the current article. Martin Hogbin (talk) 18:27, 29 February 2012 (UTC)
 * Well, I'm more interested in the narrative used, not what is demonstrated per se. Though actually, upon further reflection, this story still has a pitfall - the reader might still think that, since there are now two remaining doors, it's simply 1/2.  It also doesn't explain why it's not equivalent if the host opens the door first.  —AySz88\ ^ - ^  06:09, 7 March 2012 (UTC)
 * It also doesn't explain why it's not equivalent if the host randomly opens a door, but happens to reveal a goat (the "Monty forgets" variant - Monty forgets where the car is, holds his breath, opens a door and [fortunately for Monty] there's a goat behind the door he opens). In this variant the player's chances of winning by switching (or staying) are 50/50.  I sincerely believe it's very difficult to truly understand this problem without resorting to (high school level) conditional probability.  As Carlton (referenced in the article) says: "... the Prisoner's Paradox and the Monty Hall Problem, as the two examples are known, both illustrate the hazards of not carefully designing a tree diagram or incorrectly assuming certain outcomes to be equally likely."  The carefully designed tree diagram he suggests for the Monty Hall problem is the same as the one in the "Decision tree" section of the article.  Whatever your intuition is, there's some version of the problem for which your intuition will be wrong.  Most people's intuition is that with two doors left they must be equally likely to be hiding the car.  In the standard version this intuition is wrong, while the intuition that the initial chances the car is behind the door you picked (1/3) are not affected by what the host does is right.  In the "Monty forgets" version, the situation is reversed (the "equal probability" intuition is correct, while the intuition that what the host does cannot affect the initial 1/3 probability is wrong). Falk (also referenced in the article) says these are both very strongly held intuitions, although the latter is even stronger than the former. As it turns out, the question of whether or not there's a difference between these versions was the actual point of Whitaker's original letter to vos Savant.  She chose to delete this part of his question and presented an intuitive solution only for what has become the "standard" version.  She had this to say (much later):
 * Back in 1990, everyone was convinced that it didn’t help to switch, whether the host opened a losing door on purpose or not. Assuming a knowledgeable host who would always open a losing door, that was incorrect. (A knowledgeable host who opened a winning door on purpose wouldn’t have much of a show, would he?!)


 * Now everyone is convinced that it always helps to switch, regardless of what the host knows. But this is just as incorrect!


 * Indeed. -- Rick Block (talk) 08:45, 7 March 2012 (UTC)