Talk:Monty Hall problem: Difference between revisions

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== External link ==
== Archived earlier talk ==


Richard, CZ does not allow self-promotion. Therefore I removed the link to [http://www.math.leidenuniv.nl/~gill/essential_MHP.html your paper].
In order to regain focus I created a first talk archive of talk up to this point: [[Talk:Monty_Hall_problem/Archive_1]]. [[User:Richard D. Gill|Richard D. Gill]] 15:21, 2 February 2011 (UTC)
I hope you understand this and agree with it.
After some progress has been made with the main article, we may put it (together with other references) on the Bibliography subpage.
--[[User:Peter Schmitt|Peter Schmitt]] 23:18, 15 January 2011 (UTC)


:I agree that we ought to see how the article progresses, but I wouldn't see a problem with adding that particular paper to the Bibliography if the math editors think it is appropriate since it is specifically about the subject and he is not selling anything.  It also helps to let someone else place the link for you.  [[User:D. Matt Innis|D. Matt Innis]] 00:28, 16 January 2011 (UTC)
== General remarks ==


::Agree, at a quick glance the paper does not appear to be specific self-promotion, but it should not be on the main page.
This talk page has quickly become very long with a difficult to follow structure. I'd like to make a few general remarks:
::Go ahead and add it to the Bibliography page and let's encourage a maths editor to review it for appropriateness.
* Let us avoid to repeat and continue the endless (and mostly useless) discussion of this problem.
::I'd like to see this article expanded fairly quickly; at present it doesn't tell us exactly what the Monty Hall problem is. Putting the definition as the introduction would probably be enough of a start.
* The MHP is not a "paradox". Its solution may be surprising, but it is not paradoxical.
::[[User:Aleta Curry|Aleta Curry]] 00:43, 16 January 2011 (UTC)
* There are not two (or more) "solutions".
:* Once the question has been unambiguously posed there is only one solution -- the correct solution.
:* There may be (essentially) different arguments leading to this correct solution.
:* There may be several (didactically) different ways to present the same argument.
What should an article on the MHP contain (with the reader searching information in mind)? My answer:
* It should state the problem and present its solution as brief and as clear (and in an as informal language) as possible.
* It should summarize the history of the problem and the disputes it has caused.
* It should not contain a large amount of historical details, different approaches, discussion of subtleties, etc. that the ordinary reader will not want, and that would probably be confusing for him.
Supplementary material can be presented on subpages or separate pages:
* A page on the detailed history of the problem.
* A page on the discussion caused by the problem.
* A (Catalog) subpage containing various ways to present the solution(s). It may help a reader to find an explanation he likes.
--[[User:Peter Schmitt|Peter Schmitt]] 13:42, 1 February 2011 (UTC)


::: That's about what I thought. I was, however, uncertain if current policy classifies such a link as self-promotion. Was I saying it too unfriendly? But, in any case, the link cannot replace an unwritten article. --[[User:Peter Schmitt|Peter Schmitt]] 00:52, 16 January 2011 (UTC)
: I agree with everything you say here Peter, except for one thing. MHP is defined (IMHO) by the definitely ambiguous words of Marilyn Vos Savant quoted in the article. Both before her popularization of the problem, and later, different authorities have translated or transformed her problem into definitely different mathematically unambiguous problems. And I'm only referring to problems to which the solution is "switch"! That is part of the reason why there is, I think, not a unique "correct solution"  - there are as many correct solutions as there are decent unambiguous formulations.  


:::::Oh, I don't know about unfriendly, but Richard is new here so I didn't want him to think his efforts were unappreciated. [[User:Aleta Curry|Aleta Curry]] 01:09, 16 January 2011 (UTC)
: I think there are two particularly common solutions: one focusing on the overall probability of winning by switching, and the other focussing on the conditional probability of winning by switching given the specific doors chosen and opened. The present draft intro contains elements of both and even at attempt at synthesis. [[User:Richard D. Gill|Richard D. Gill]] 23:26, 1 February 2011 (UTC)
::::While we may indeed need to clarify the policy with an EC motion, I've never thought that a link to a clearly noncommercial page is self-promotion. This is even more the case when pointing to one's own peer-reviewed publications and presentations, which I have done -- I might not have written them if I thought there were better references. [[User:Howard C. Berkowitz|Howard C. Berkowitz]] 01:05, 16 January 2011 (UTC)


:::::I had some discussion about this with Larry years ago, with respect to using one's own articles if one were a) an authority in the field or b) the only person writing in the field!  His answer, basically, was 'use common sense and ask an(other) Editor to review/confirm'.
: By the way, the meaning I am used to of the word "paradox" is an <i>apparent</i> contradiction. And there certainly is an apparent contradiction between ordinary people's immediate and instinctive solution "50-50, so don't switch", and the "right" solution: "switching gives the car with probability 2/3". [[User:Richard D. Gill|Richard D. Gill]] 15:39, 2 February 2011 (UTC)
:::::[[User:Aleta Curry|Aleta Curry]] 01:09, 16 January 2011 (UTC)


Thanks everyone. The point of the reference was just to be a resource for anyone interested in joining in. Over on Wikipedia a fight has been going on for two years, basically between laypersons who find a short intuitive solution of MHP completely satsfying, and mathematicians who dogmatically insist on a tricky solution using Bayes theorem. My own modest contribution (he said modestly) was merely to present the mathematical facts of the matter and go some way to resolving the Wikipedia conflict. Partly, by creating a "reliable source" (wikipeda terminology) for both sides of that battle. Partly by showing that the "full conditional solution" can be obtained by making one small step from the "popular simple solution" by the use of symmetry - a neat trick which I learnt from our friend Boris in this context! However the paper is too mathematical for most laypersons.
:: But where is the ''apparent contradiction''? That intuition and correct reasoning lead to different results is not an apparent contradiction, I would say. (But this is only a question of language, not really important here.) --[[User:Peter Schmitt|Peter Schmitt]] 00:54, 6 February 2011 (UTC)


I think it's challenging to get across to laypersons what the difference is between the simple solutions and the conditional solutions, as they are often referred to. More below.
== Which MHP? ==


==Problem variant as a cartoon==
In [[Talk:Monty_Hall_problem/Archive_1]], Wietze Nijdam started discussion of which of the following two problems is "the MHP". That discussion has been raging on wikipedia unabated for over two years, producing only polarization. Wikipedia editors interested in sensible compromise have left in bemusement, disgust or frustration.
I must share a memorable cartoon idea based on this problem, from ''Playboy'' (I read it for the cartoons). The problem is reduced to two doors, and the contestant is faced with legends of "damned if you do" and "damned if you don't". Monty Hall, the game show host, is in devil garb, prodding the contestant with a pitchfork. [[User:Howard C. Berkowitz|Howard C. Berkowitz]] 00:12, 16 January 2011 (UTC)
::As an aside, is Monty Hall his real name, or was it a joke referring to 3 card Monte? [[User:Aleta Curry|Aleta Curry]] 01:09, 16 January 2011 (UTC)


:::I looked it up, Monte Hall was apparently the stage name for Monte Halperin of the TV game show, "Let's make a Deal", so it looks like his mother was the one pulling the 3 card Monte :)[[User:D. Matt Innis|D. Matt Innis]] 01:28, 16 January 2011 (UTC)
Returning to Wietze's text, consider the following two statements ( [[User:Richard D. Gill|Richard D. Gill]] 10:12, 3 February 2011 (UTC) ) :


:::: There is question related to it: Should this be "Monty Hall Problem" or "Monty Hall problem". I tend to the latter, but this is a question for language experts. --[[User:Peter Schmitt|Peter Schmitt]] 01:33, 16 January 2011 (UTC)
*F0: (Conditional formulation) If the contestant is offered to switch after the host has opened the goat door, the decision has to be based on the conditional probability given the initial choice and the opened door.


:::::I remember that!  I was...ahem...two years old, of course, but....[[User:Aleta Curry|Aleta Curry]] 01:34, 16 January 2011 (UTC)
*F1: (Unconditional formulation) If we are asked whether the contestant should switch, even before he has made his initial choice, and we are not allowed to give a solution for every possible combination of initial chosen door and opened goat door, the decision will have to be based on the (unconditional) probability of getting the car by switching.


:::::: It would be nice, @Howard, to find that cartoon on internet! @Peter, Regarding the P for problem: as a rusty native English speaker who was never all that good at spelling or grammar, I'd say that in plain text, whether you write Monty Hall Problem or Monty Hall problem depends on context. If you are writing about many different problems, then the Monty Hall problem is just one of those many problems. However when you are writing about The Monty Hall Problem capitalization of the P is appropriate. The word is part of the common name of one individual problem. [[User:Richard D. Gill|Richard D. Gill]] 13:07, 26 January 2011 (UTC)
The discussion point is: which formulation is more natural to be the MHP and should be presented as such.


:::::::Hmmm...I think ''Playboy'' rather than ''New Yorker''. When I read Playboy, it ''is'' for the cartoons. [[User:Howard C. Berkowitz|Howard C. Berkowitz]] 21:25, 26 January 2011 (UTC)
My opinion: F0. [[User:Wietze Nijdam|Wietze Nijdam]] 22:45, 2 February 2011 (UTC)


==Proposal for main content ==
: Both are of interest; but
:* F1 is of interest to nearly everyone;
:* F0 is of interest for those already understanding F1 and wishing to widen and deepen their understanding.
:In this sense, F1 is the basic MHP while F0 is the advanced MHP.
: --[[User:Boris Tsirelson|Boris Tsirelson]] 07:06, 3 February 2011 (UTC)


Apart from history, sources, variants, and so on, the main content of the article should obviously be the presentation of a solution to MHP. The challenge is to simultaneously satisfy mathematical pedants and be intelligible to ordinary lay-persons. This requires an almost purely verbal solution, using only plain words of everyday English, which does however, sentence by sentence, cover every single logical step, including explicit use of all necessary assumptions. That's what I plan to write first. Now. [[User:Richard D. Gill|Richard D. Gill]] 11:22, 26 January 2011 (UTC)
:: @Wietze, you cannot say "the decision <b>has</b> to be based on the conditional probability". You could say that it would be wise to base your decision on the conditional probability. Moreover, if you want to reach the general public, you had better explain why this would be the wise thing to do. Please draft some material on this in the article on [[conditional probability]].


: First try done. Please edit or comment. [[User:Richard D. Gill|Richard D. Gill]] 13:08, 26 January 2011 (UTC)
:: If indeed you want to reach the general public, it would also be wise to note that given that the player has chosen Door 1, whether the host opens Door 2 or Door 3 has no relevance at all to whether or not the car is behind Door 1 (under the probabilistic assumptions which many people find natural). So the general reader can be informed of the truth <i>and the whole truth</i> of the standard MHP - F0 and F1 combined - using plain non-technical English and without needing to follow a course in probability theory first.


:: Better than WP, which is however not a compliment: on WP it is too bad. Well, this is just good, I think so.
<i>PLEASE</i> let us not repeat this endless discussion here. Draft appropriate subpages to MHP as Peter Schmitt indicated is the next step which ought to be made. Get to work on the articles on [[probability]], [[probability theory]], [[conditional probability]], [[Bayes Theorem]]. [[User:Richard D. Gill|Richard D. Gill]] 09:55, 3 February 2011 (UTC)
:: A remark: you mention Bayesian probability (and I understand why), but just above that you use frequentist probability (you count the winning ratio in the long run!). Some readers may be confused. --[[User:Boris Tsirelson|Boris Tsirelson]] 21:03, 26 January 2011 (UTC)


::: You're right. I should remark explicitly on the "paradigm shift". It was kind of deliberate. I think that the "standard" uniformity assumptions of MHP are only well-justified within a subjectivist notion of probability. For a frequentist, it is harder to come up with any probability model or at all, and even if it does make some sense, the probabilities cannot be considered as known in advace. But the frequentist picture is also valuable. Personally, I understand the arithmetic of relative frequencies much better than the "logic" of subjective probability. Fortunately, whatever your personal choice, subjectivist and frequentist probability satisfy the same rules, so I can always give a frequentist story about a subjectivist probability inspired model.
::: That's not up to me. [[User:Wietze Nijdam|Wietze Nijdam]] 10:22, 3 February 2011 (UTC)
::: One of the wikipedia MHP editors wrote "no one who thinks seriously about MHP cannot avoid pondering on the meaning of probability". My own opinion is that the infinitely many repetitions of the frequentist are equally imaginary to the "parallel worlds" of the subjectivist. Both are equally meta-physical, thus the choice is a matter of opinion, of religion, of choice of meta-phor. The important thing is that we use probability in scientific discourse hence we need some kind of inter-subjectivity. You have to make the repetitions, of whichever kind, appealing and understandable to the people you want to communicate with. [[User:Richard D. Gill|Richard D. Gill]] 23:56, 26 January 2011 (UTC)


:::: I would find a truth table of some sort particularly illustrative in the explaination in the first paragraph or two, showing the 3 possible car locations, the host's choice(s) of doors to open, and the contestants win/loss result of switch/no switch. [[User:David E. Volk|David E. Volk]] 19:36, 27 January 2011 (UTC)
:::: It is up to you. You are an author, a citizen of citizendium? See [[CZ:Myths_and_Facts]]. [User:Richard D. Gill|Richard D. Gill]] 10:25, 3 February 2011 (UTC)


::::: I don't think that a "truth table" or, rather, a tree showing the possibilities is useful. It looks impressive but makes it also look more complicated than it is. There are, essentially, only two distinct cases - the door first chosen is the winning one, or it is not ... (For those looking for such a diagram, it could, perhaps, be put on a subpage.) --[[User:Peter Schmitt|Peter Schmitt]] 20:41, 27 January 2011 (UTC)
:::::@Boris: your reaction in the next section seems contradictory to what you've commented here. So, please, make clear what you mean. The formulation F0 is the one in which the contestant is in the end standing in front of two closed doors and one opened, and then asked whether she wants to switch. Yet you write here above: F1 is of interest to nearly everyone; That's puzzling me. [[User:Wietze Nijdam|Wietze Nijdam]] 09:43, 6 February 2011 (UTC)


:::::: Peter, if we were writing for PhD mathematicians who can obviously grasp that point, I would agree, but others get the aha moment by looking at the possible outcomes. [[User:David E. Volk|David E. Volk]] 21:13, 27 January 2011 (UTC)
:::::: OK, you are right; I did not understand you correctly. I believe that the whole MHP story is of interest "to nearly everyone" first of all because of "the conclusion". Thus, the "two closed doors" situation must be emphasized. But for me it already is: ''"Almost everyone, on first hearing the problem, has the immediate and intuitive reaction that the two doors left closed, Door 1 and Door 2, must be equally likely to hide the car".'' This is the first phrase after general introductory words! Thus, for now I believe that F0 is relevant, and presented. And so I still fail to understand your dissatisfaction. --[[User:Boris Tsirelson|Boris Tsirelson]] 17:06, 6 February 2011 (UTC)
(outindenteed) Boris, does it surprise you that I was dissatisfied when you first said F1 was the important formulation? I hope you also understand my dissatisfaction with Richard and seemingly Peter favouring F1 as the important formulation and hence primarily presented to the readers.[[User:Wietze Nijdam|Wietze Nijdam]] 17:19, 6 February 2011 (UTC)


::::::: I very much doubt that a table (as in your sandbox) makes it easier for a "non-mathematician". I rather expect that the non-initiated looks at it and thinks "well, if the mathematicians say that thse are the formulas to use, then I tend to believe it.) The table does not help to understand the crucial point.
: Well, I am sorry for my error (in fact I thought that your "conditional" means also "asymmetric"). But anyway, we discuss the article, not its talk page. Once again: F0 is relevant, ''and presented,'' boldly. Isn't it? --[[User:Boris Tsirelson|Boris Tsirelson]] 19:12, 6 February 2011 (UTC)
::::::: I think it is easier to argue as informally as possible: Your chances are 1 in 3 (or 1 to 2) to choose the winning door first, thus chances are 2 in 3 (or 2 to 1) that you have '''not''' chosen the winning door and switching opens the winning door.
: But let me formulate my position more exactly. I ''am'' dissatisfied with the article as it is now, because "the conclusion" is missing. And "the conclusion", as I see it, compares the answer (1/3, 2/3) with the naive equal probabilities on the two closed doors. Thus, "two closed doors" is a crucial component of it. Which does not conflict ''(as you agree, if I am not mistaken)'' with calculating the conditional probability via the unconditional probability and the symmetry. --[[User:Boris Tsirelson|Boris Tsirelson]] 19:21, 6 February 2011 (UTC)


:::::::: Everyone understands that you have a 1/3 chance at the onset, and 2/3 chance of being wrong at the onset, but almost everyone fails to understand why the opening of one door, thus leaving the car behind only one of two closed doors, does not equal a 50:50 chance. In other words, <b> they fail to see why opening one of the doors <u>does not change your odds</u></b>. A layman thinks, 1 car in 1 of 2 locations = even odds.
::Boris, I'm glad you also consider F0 the MHP (I knew this from our discussion on Wikipedia). But ... you have to read the article more carefully. Tthe article present initially F1 as the standard form of the MHP. And that is precisely Richard's intention. With the simple solution S1 (without any reference to symmetry or conditional probability) as its solution. I'm strongly against this. Many people will not notice that F1 is presented and from their own imagination think it is F0, just like many texts on internet about MHP that present F0, but with S1 as its solution, what is a logical error. [[User:Wietze Nijdam|Wietze Nijdam]] 22:33, 6 February 2011 (UTC)
:::::::: In this particular problem, the general masses heard the answer of the mathematicians and said no ****** way!. They did not believe that the solution was correct, and still don't after much explaination. [[User:David E. Volk|David E. Volk]] 21:54, 27 January 2011 (UTC)


::::::::: I know the history of the problem and the many disputes caused by it. But calculations, the use of formulas, do not help to '''understand''' the problem. They may help you to '''convince''' the "general masses" because they tend to '''believe''' in mathematical formulas (without bothering to understand them), but not help them to better grasp the solution. --[[User:Peter Schmitt|Peter Schmitt]] 22:19, 27 January 2011 (UTC)
::: As for me, such nuances of the emphasize balance are not as important here as (say) in a political communique. If "the conclusion" (in one form or another) satisfying me will appear, it will clearly juxtapose F0 with F1 (thus closing the problem, in my opinion). For now I am waiting for Rick's opinion about "the conclusion". --[[User:Boris Tsirelson|Boris Tsirelson]] 07:11, 7 February 2011 (UTC)


It's clear from the wikipedia MHP wars that different readers need different ways to get their minds around the problem. Some have to see a table of numbers. Some have to see a formal mathematical derivation. Others need the briefest possible verbal argument, anything more overloads their minds. The MHP pages here should, I suppose, also cater for all tastes, while keeping the structure clean and mean.
== Three editor approval ==


@Peter: you say "there are essentially two cases: the door chosen by the player hides a car or it hides a goat". This is spot-on of course, and it's the first message which has to be got across. The fights on Wikipedia were about the word "essentially". The probability purists insist on a formal probability calculation by Bayes theorem to show that the identity of the door opened by the player is irrelevant. They are supported by every standard probability textbook. Yet no single popular writer, nor academic writers from psychology, ethology, or whatever, sees the point of this extra complexity. It's an interesting problem of demarcation, of ownership. Who does MHP belong to? [[User:Richard D. Gill|Richard D. Gill]] 08:21, 31 January 2011 (UTC)
Just as a point of procedure, there are currently three editors on this page; Boris, Peter, and Richard. Should they agree on content and style, it is possible that this article can be approved and locked allowing editors to move on to other important and related articles. With input from the very knowledgeable authors on this page, I do think you've all illustrated your willingness to create a good article here, thanks for your professional efforts. [[User:D. Matt Innis|D. Matt Innis]] 13:14, 3 February 2011 (UTC)


:Please pardon the intrusion. I'm not certain that this statement:
:Yes, I've been very impressed by the work that has been put into this article and the people it has drawn in. I'd definitely like to see it reach approval. --[[User:Joe Quick|Joe Quick]] 19:19, 3 February 2011 (UTC)
::"...they are supported by every standard probability textbook."
:..Is correct. Or maybe the phrasing implies something other than intended. In any case, I don't agree with it. [[User:Garry L. Kanter|Garry L. Kanter]] 08:25, 31 January 2011 (UTC)


:: Pardon my poetic licence! To be precise: I know a *lot* of standard probability and statistics textbooks and all the ones I'm familiar with have MHP as an example or as an exercise, in the early chapter on Bayes theorem for ordinary (discrete) probability. And they all solve it by making the usual uniformity assumptions and by explicit computation of a conditional probability.
:: My own opinion is that a number of sub-pages need to be written, and that when this is done, the introductory page can be shortened and sharpened (some of the side remarks are really reminders to myself or others of things that need to be explored on sub-pages). Also the list of references - at present it is just a comprehensive list stolen "as is" from wikipedia - needs to be replaced with a shorter and annotated list of key references. I don't know enough yet about citizendium procedures to know it makes sense to "lock" an article when a lot of supporting material still needs to be put into place. [[User:Richard D. Gill|Richard D. Gill]] 09:46, 4 February 2011 (UTC)


:: A good example is the [http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html textbook by Grinstead and Snell], freely available on internet, and supported by the American Mathematical Society. And it is a pretty good book, too. Yet they have the same dogmatic approach as Morgan et al. Actually the text has been released under the GPL so it can be rewritten by you and me! But I think it would be wise first to contact the authors. [[User:Richard D. Gill|Richard D. Gill]] 11:17, 31 January 2011 (UTC)
::: Richard, the main article can be locked and still allow further work on the subpages. Also, when an article is locked, a draft is created that is an exact copy of the approved version where work continues. At any point, it can be re-approved and the new version replaces the original.  That way we get incremental improvements (hopefully). Again, it will take three editors to agree on the improvements. [[User:D. Matt Innis|D. Matt Innis]] 13:29, 4 February 2011 (UTC)


Wouldn't it be 'likely' that a textbook on (conditional) probability would uncritically present a formal conditional solution to the MHP? And then, since it's such overkill for the problem, devise reasons & complexities that engage the students (non-symmetric host bias, forgetful Monty, devious Monty), in order to show the capabilities of conditional probability?


I think this is the relevant passage from G & S. They are *not* describing the recognized MHP, nor are they claiming to. They intentionally restate the problem in order to make 'some' point more evident, and less ambiguous:
:::: Thanks, that's clear. Well, I'm ready to approve. @Peter Schmitt, @Boris Tsirelson, how about you?
:"'''We begin by describing a simpler, related question.''' We say that a contestant is using the "stay" strategy if he picks a door, and, if offered a chance to switch to another door, declines to do so (i.e., he stays with his original choice). Similarly, we say that the contestant is using the "switch" strategy if he picks a door, and, if offered a chance to switch to another door, takes the offer. Now suppose that a contestant decides in advance to play the "stay" strategy. His only action in this case is to pick a door (and decline an invitation to switch, if one is offered). What is the probability that he wins a car? The same question can be asked about the "switch" strategy.


:"This very simple analysis, though correct, does not quite solve the problem that Craig posed. Craig asked for the conditional probability that you win if you switch, given that you have chosen door 1 and that Monty has chosen door 3."
::::: As Boris and Richard have clearly stated to be advocates of the unconditional formulation of the MHP, I urge Peter to think thoroughly about this. I have never seen ordinary people, picturing the MHP, and not imaging the player standing in front of two closed and one opened  door, and only then offered the possibility to swap doors. It is in my opinion not only the charm of the puzzle, it is also the crux. Only when actually confronted with the two remaining doors, people are inclined to think the odds are equal. [[User:Wietze Nijdam|Wietze Nijdam]] 09:16, 5 February 2011 (UTC)


:"At this point, the reader may think that the two problems above are the same, since they have the same answers. Recall that we assumed in the original problem if the contestant chooses the door with the car, so that Monty has a choice of two doors, he chooses each of them with probability 1/2. Now suppose instead that in the case that he has a choice, he chooses the door with the larger number with probability 3/4. In the "switch" vs. "stay" problem, the probability of winning with the "switch" strategy is still 2/3."
:::::: Wietze, I completely agree that "Only when actually confronted with the two remaining doors, people are inclined to think the odds are equal", and I do treat this aspect as very important (see "Toward the conclusion" below). I agree that the conditional probability is important in the ''formulation'' of MHP. But the article already treats it, explaining that the conditional probability is equal to the unconditional probability by symmetry. Indeed, the article contains a link to [[conditional probability]]. Do you see anything wrong in choosing this (rather short and intuitive) way to the conditional probability (via the unconditional one)? --[[User:Boris Tsirelson|Boris Tsirelson]] 15:23, 5 February 2011 (UTC)


G & S do *exactly* what I describe above. Which is *not* a criticism of any other solution, just an example of why you *might* need a conditional decision tree or Bayes, '''for some other problem'''. This is why I argue (for 2+ years, now) on Wikipedia that the so-called '5 specific critics' of the simple MHP solutions (of which it is claimed G & S are 1 of the 5) are not necessarily 'critics' at all. [[User:Garry L. Kanter|Garry L. Kanter]] 11:39, 31 January 2011 (UTC)
:::::: Moreover, looking again at the article I see that all reasonable approaches are already sketched. Including the most traditional treatment of conditional probability. Thus I really do not understand what is the problem. Maybe you want a more formal treatment (something similar to my lectures on a course for math students)? Also no objections from me; but better on a subpage. Or maybe you want to cover asymmetric cases? As for me, a paradox should be always stripped down to the simplest possible formulation (which was indeed made for set-theoretic paradoxes a century ago). Generalizations are a more advanced topic of more special interest. --[[User:Boris Tsirelson|Boris Tsirelson]] 17:47, 5 February 2011 (UTC)


: G & S say explicitly that the simple solution does not answer Craig's question; and they say explicitly that Craig is asking for the conditional  probability, which they have previously defined as the probability that the player chooses door 1 and the host opens door 3 and the car is behind door 2, divided by the probability that the player chooses door 1 and the host opens door 3. They also say that just because both questions have the same numerical answer it doesn't mean that they are the same question. [[User:Richard D. Gill|Richard D. Gill]] 12:45, 1 February 2011 (UTC)
:::::::Strange that you do not understand. It really is a mess. We, you and I, had part of this discussion before on Wikipedia. Let me explain once again. In my and a lot others opinion in the MHP the contestant is offered to swap AFTER the host has opened the goat door, and the contestant is confronted with two closed doors, from which many people are inclined to think the odds for the car are even. This problem has (is wisely) to be solved by calculating the conditional (or if you like the posterior) probability given the situation the contestant is in. (If it makes it easier for the average reader we may formulate it different.) The way the conditional probability is calculated is (of course) unimportant, although by using the symmetry (under suitable assumptions) this may be explained in a more understandable(??) way than just by using Bayes' formula. Richard agrees with me on this. But!!! the simple reasoning, the one that says (in short), you hit the car 1/3 of the time, hence by switching you get it 2/3 of the time, is not a solution to this formulation of the MHP, as it does not calculate the conditional probability, i.e. it does not account for the situation the player is in. Some authors, in my opinion in need to make the simple solution work, change the problem formulation, i.e. they say: the player is asked whether she wants to switch, even before she has made her initial choice. Then no conditions have been imposed, and the unconditional probability is sufficient. This is not what I and not just me, consider the MHP. Did I really have to explain this to you? So, what concerns your question, no I have no objection by calculating the CONDITIONAL probabilty, with the use of the symmetry, through the unconditional, as long as the CONDITIONAL (or how we will call it) is indeed calculated. The simple solution definitely does not mention anything like this at all. [[User:Wietze Nijdam|Wietze Nijdam]] 00:57, 6 February 2011 (UTC)


: And it is exactly the articles and text-books which are written in this vein which have generated the situation that ordinarily decent folks on Wikipedia have been stuck in their WW I opposing trenches for more than two years, waving the same rule book at one another! That is a situation which hopefully will not recur on citizendium. [[User:Richard D. Gill|Richard D. Gill]] 12:56, 1 February 2011 (UTC)
:::::: About three-editor approval: yes, in principle I am ready to join; but see "Toward the conclusion" below. --[[User:Boris Tsirelson|Boris Tsirelson]] 15:36, 5 February 2011 (UTC)


== MHP wars on wikipedia in a nutshell ==
::::::: As a matter of fact, currently both Boris and I have not contributed to the article, therefore both of us could single-handedly approve it. Even if I add an introduction (as I probably will), Boris could do it alone. But, of course, teamwork is always possible. --[[User:Peter Schmitt|Peter Schmitt]] 00:58, 6 February 2011 (UTC)


Consider this four step informal/intutive "good" solution to MHP. Do you think the third step - placed in parentheses - is necessary or unnecessary?
:::::::: Wietze, putting replies into the middle of comments makes talk pages difficult to read and makes it difficult to see who wrote what.
:::::::: I also fail to see a problem: As the question is posed the candidate is '''not''' put in front of two closed and an open door. The problem clearly tells how the situation evolved. However, I think it is not useful to '''number''' the doors unless the numbers are used to identify the "door first chosen", the "other door closed", and the "door opened". (The argument using repetitions does not make clear that the door opened will not always be the same.) --[[User:Peter Schmitt|Peter Schmitt]] 01:14, 6 February 2011 (UTC)


* 2/3 of the time the contestant will select a goat
::::::::: Wietze, you write "I have no objection by calculating the CONDITIONAL probabilty, with the use of the symmetry, through the unconditional, as long as the CONDITIONAL (or how we will call it) is indeed calculated." But your requirement is already fulfilled by the article, isn't it? --[[User:Boris Tsirelson|Boris Tsirelson]] 06:32, 6 February 2011 (UTC)


* The host opening a door to reveal a goat doesn't change this.
::::::::::No, Boris, the article does not. It says: ''One could say that when the contestant initially chooses Door 1, the host is offering the contestant a choice between his initial choice Door 1, or Doors 2 and 3 together.'' Let us continue this discussion in the section "Which MHP?" above. There I formulated the versions F0 and F1. And it is about the difference between these two. [[User:Wietze Nijdam|Wietze Nijdam]] 09:37, 6 February 2011 (UTC)


*(The door being opened being door 3 doesn't change this either)
::::::::::@Peter. Sorry you fail to see the problem, because there is one. Unless the contestant is blind(folded), she sees the doors and hence is able to distinguish between them. The problem formulation speaks also of door 1 and door 3. Anyone so it seems may see the doors and the contestant pointing at one door and the host opening one. Look at all the simulations that are constructed. A specific door is chosen and a specific one is opened. Only mathematicians may come to a formulation in which only is spoken of "the chosen door" and "the opened door". It is possible, but then these doors are random variables, and take values in specific situations. Rather difficult to understand for the average reader, don't you think? Please follow and take part in the discussion under "Which MHP?". [[User:Wietze Nijdam|Wietze Nijdam]] 09:56, 6 February 2011 (UTC)
::::::::::: I have no intention to repeat and continue the endless discussions. Wietze, you forget that we all know the problem and the arguments ... unless something new turns up. --[[User:Peter Schmitt|Peter Schmitt]] 11:45, 6 February 2011 (UTC)


* Therefore the contestant should switch
:::::::::::: I don't know what endless discussions you're referring to. Definitely not here on Citizendum. And as far as I know, you were not involved in the discussions on Wikipedia. You may be an excellent mathematician, but I doubt you really understand the problem as you say. Let alone all the arguments, as you show with your remarks about the door numbers. [[User:Wietze Nijdam|Wietze Nijdam]] 14:41, 6 February 2011 (UTC)


Step 1 uses only "car is hidden at random"
::::::::::::: I am glad that we had no endless discussion on CZ (yet?), but one need not be "involved" in the WP discussions (e.g., en. and dt.) in order to have noticed them and to know about them. --[[User:Peter Schmitt|Peter Schmitt]] 00:01, 7 February 2011 (UTC)


Step 3 uses the symmetry (the probability assumptions don't change on renumbering the doors) of adding to the previously used assumption also the assumption "host choice 50-50"
::::::::::::::Well, do you agree with me, or do you contribute to the endless discussion? [[User:Wietze Nijdam|Wietze Nijdam]] 07:27, 7 February 2011 (UTC)


The four-step argument is intuitive and mathematically rigorous at the same time - each step can be converted into formal mathematical language via the use of Bayes' rule.
(unindent) Peter, no, I do not want to approve alone, in presence of three editors. --[[User:Boris Tsirelson|Boris Tsirelson]] 07:27, 6 February 2011 (UTC)


The "simple solution" or "unconditional solution" corresponds to removing Step 3, the "conditional solution" corresponds to keeping it.  [[User:Richard D. Gill|Richard D. Gill]] 08:23, 28 January 2011 (UTC)
== Toward the conclusion ==


We see how the simple solution does not require the full assumption set: the advantages and disadvantages of both solutions are plainly visible.
But I bother: the conclusion is missing. I mean something in the spirit of the following.


:: Removing step 3 is not really a problem. The problem occurs, I think, in believing that Step 2 is true. It needs to be explained in terms of the possible choices available to the host as to which doors can be opened. If the contestant chooses correctly, the host has 2 choices of which door to open, and if the contestant chooses incorrectly the host has only 1 choice of door to open. [[User:David E. Volk|David E. Volk]] 15:18, 28 January 2011 (UTC)
A paradox refutes some naive belief. For example, set-theoretic paradoxes refuted the naive belief in unlimited freedom forming "the set of all x satisfying (whatever)". Another example: the continuous but nowhere differentiable Weierstrass function refuted the naive belief that a continuous function is necessarily differentiable, except some special points.


::: This is subtle, @David! And depends rather crucially on how Step 2 is to be understood! I meant: a door is opened revealing a goat, but the number of that door has not yet been revealed to the contestant. With this understanding, Step 2 is easily seen to be true, since it speaks of "a door", not of a specific door. Whether or not the car is behind Door 1, the host will certainly open a door a reveal a goat. The fact that a door is opened revealing a goat does not (under the conditions of the game) give us any information regarding the question "is the car behind Door 1, yes or no?"
The MHP paradox refutes the naive belief in such an argument:


::: So once we have completed Step 1 and Step 2 we have arrived at what some people over on wikipedia call "the simple solution". Given a door has been opened revealing a goat (but the identity of the opened door is not yet known), the odds are still 2:1 that Door 1 hides a goat. Hence switching to the other door gives the car with probability 2/3.  
: "According to new data, only ''m'' possibilities remain; apriori, ''n'' possibilities were equiprobable; therefore (?) the ''m'' remaining possibilities are equiprobable aposteriori."


::: So far, we ignored the number of the door opened (at least, that was my intention). Let's look at Step 3 - the one in parentheses, which many authors of "popular" solutions ignore. We have fixed that the player chose Door 1. We know in advance that the host is going to open either Door 2 or Door 3, but not which. By the symmetry of the problem with respect to Doors 2 and 3, (including the assumption that the two host's choices are equally likely if he has a choice), the probability the host opens Door 3 given the car is behind Door 1 is 50%, and the probability the host opens Door 3 given the car is not behind Door 1 is 50%. Thus the identity of the Door being opened contains no information about whether or not the car is behind Door 1 - Door 3 has exactly the same chance of being opened under either hypothesis.
The change of probabilities according to new data (so-called [[conditioning (probability)|conditioning]]) is generally more subtle than just exclusion of some possibilities.


::: I learnt this way of solving MHP from wikipedia: Garry Kanter presented the pithy three step solution there i.e. excluding my parenthetical Step 3 !
--[[User:Boris Tsirelson|Boris Tsirelson]] 15:18, 5 February 2011 (UTC)


::: What I like about this approach is that it is at the same time intuitive and mathematically formalizable. Rather than doing mindless computations using the formal definition of conditional probability (which is what formal solutions using Bayes's theorem in its common text-book form do) it uses the very intuitive Bayes' rule: posterior odds equals prior odds times likelihood ratio. And it is crafted so that the likelihood ratio is always 1, that is to say, each new piece of information has the same chance under the two competing hypotheses, hence is actually non-information. From an educationalist point of view, I would like to see ordinary people be able to gain some probabilistic intuition by becoming familiar with and hence being able to internalize Bayes' rule.
: In my view there is no (true) "paradox" -- though some experience it as one.  
: Moreover, I think the page needs an introduction (and --perhaps-- also splitting into some sections as orintation for the reader).
: --[[User:Peter Schmitt|Peter Schmitt]] 00:46, 6 February 2011 (UTC)


::: There is trickiness in the ambiguous wording "a door is opened". Line 2 has to have added to it "the number of the opened door is not yet revealed to the contestant".  
:: It depends on the meaning given to the word "paradox"; probably there is no consensus on it. But I do not insist on the word. Rather, on a conclusion. --[[User:Boris Tsirelson|Boris Tsirelson]] 06:28, 6 February 2011 (UTC)


::: People with a formal training in probability tend to find Step 3 absolutely crucial. They insist that the competitor's choice must be guided by the conditional probability that the car is behind Door 2 given the host has opened Door 3. Those without formal training in probability theory tend to see absolutely no value in it, because of the combination of semantic ambiguity and the mathematical fact that the number of the door opened is indeed irrelevant regarding the decision of whether to switch of stay. (Which can be seen in many ways, and used in a formal proof in many ways - at the outset or at the end or somewhere in the middle).
== The (new) lead ==


::: In mathematical language: given the contestant has chosen Door 1, whether or not the car is behind Door 1 is statistically independent of whether the host opens Door 2 or Door 3.[[User:Richard D. Gill|Richard D. Gill]] 20:39, 28 January 2011 (UTC)
I like it. However, I (and probably Wietze too) think that even before presenting the right solution we should mention the widespread erroneous argument (since otherwise, why bother at all?). --[[User:Boris Tsirelson|Boris Tsirelson]] 11:18, 7 February 2011 (UTC)


:::: Let us put a hypothetical Step 1a between Steps 1 and 2: The host asks you if you want to switch provided you win if the car is behind one of the two doors. Switching clearly doubles the chances. After you switched, the host teases you by opening the (a) losing door first before revealing whether you have won. Since it is already decided whether you have won this does not change the odds. --[[User:Peter Schmitt|Peter Schmitt]] 21:33, 28 January 2011 (UTC)
Now better; but still not a single word, ''why'' is it counterintuitive, and ''what'' is the wrong answer, and the wrong argument. --[[User:Boris Tsirelson|Boris Tsirelson]] 12:26, 7 February 2011 (UTC)


== Another problem ==
And now the lead says it; but in fact now it more or less explains the solution. Then, what are subsequent sections for? Isn't it better to puzzle the reader in the lead and then (in sections) reveal him the truth? --[[User:Boris Tsirelson|Boris Tsirelson]] 13:48, 7 February 2011 (UTC)


I recall when the original vos Savant column came out and the thousands of letters it generated. To me, the biggest problem was believing that the woman with supposedly the highest IQ in the world was actually named "von Savant" (I mistakenly thought it was "von", not "vos".) It seemed to me like a supreme put-on. It wasn't until many years later, when the Internet, and, I suppose, Wikipedia, had come along, that I Googled this improbable name and discovered, to my astonishment, that it was an actual name. As if the *stupidest* person in the world was actually named George vos Dumb.... [[User:Hayford Peirce|Hayford Peirce]] 21:01, 27 January 2011 (UTC)
: In my view, the lead (I prefer: the introduction) should provide a (brief) summary, sufficient for all those who are looking for basic information. The main part of an article is for those who, judging from the introduction, are looking for information in depth, more details, etc, In this case, it would be fine if a reader is satisfied after the introduction, but while indeed not much more (except historical data and the connection to related topics) needs to be added, the well-known reactions call for some more words telling the same story. --[[User:Peter Schmitt|Peter Schmitt]] 15:32, 7 February 2011 (UTC)


== New Guy Here... ==
:: I am not quite understanding how do you apply these general rules to this situation. Anyway, if an article discusses a theorem, then probably its (rough or exact) formulation appears in the lead, while a proof does not. If we follow this approach here then the answer (1/3,2/3) appears in the lead but the question "why" is answered later. Or not?
:: About "lead" and "intro" I remember that on WP they are two different units; "intro" (if present) is the first section. Does CZ treat it differently? --[[User:Boris Tsirelson|Boris Tsirelson]] 15:58, 7 February 2011 (UTC)


Hi, I've been very active in the Wikipedia MHP discussions for 2+ years. No real CV to speak of, just a lot of OCD.
::: I don't know if this has been defined somewhere. My impression (combined with my personal preference) is as I said above. Ultimately, the EC will have to make up its mind whether there should be a rule, or only a recommendation, (or nothing) on this. (If I say "intorduction" I mean the part before the first section title, corresponds "Edit intro". The lead -- for me -- would be (at most) the first paragraph.)
::: As for having the proof in the intro: This depends, I would say, on the type of proof. In this case the "proof" is more important than the question, in other cases mentioning the main idea of the proof can be included, in others nothing at all.
::: --[[User:Peter Schmitt|Peter Schmitt]] 16:19, 7 February 2011 (UTC)


I prepared this decision tree to explain the 'simple solutions'. The other Wikipedia editors did not feel it was worthy of inclusion in the article. It is derived from Carlton's simple solution, and from Morgan's (false) F5.
::::Upon reflection, I think that the new lead is correct. The problem itself should be stated. The info about the game show and vos Savant, etc. should be in a following paragraph, as has now been done. [[User:Hayford Peirce|Hayford Peirce]] 16:27, 7 February 2011 (UTC)


{{Image|Tree -3.jpg|center|600px|Simple solution decision tree}}
::::: I like the look of the whole article now. [[User:Richard D. Gill|Richard D. Gill]] 16:43, 7 February 2011 (UTC)


:::::: OK. Not exactly to my taste, but this is natural: usually, nothing can be exactly to several tastes simultaneously. I am ready to join three-editor approval process. Are you? --[[User:Boris Tsirelson|Boris Tsirelson]] 17:00, 7 February 2011 (UTC)


I think it is an informative representation of what has been referred to as the (elusive) 'Peaceful Co-existance' of the simple and the conditional solutions. I welcome any comments as to it's suitability for and benefit to the article. [[User:Garry L. Kanter|Garry L. Kanter]] 15:30, 28 January 2011 (UTC)
::::::: I should have said: it looks good enough to be stabilized for a while, and for work on subpages to get going!. So I too am ready for a three-editor approval process. (Just saw and liked some good further edits by Peter). [[User:Richard D. Gill|Richard D. Gill]] 19:24, 7 February 2011 (UTC)


: I think the words "say Door 2" and "Door 3" should be deleted from the picture, because the host's choice is not part of the probability story told in the picture. You could give the picture without those words, and then remark that by symmetry the number of the door opened by the host doesn't change the likelihood that the car is behind the door chosen by the player. But then I don't really see the added value in having the picture.  
::::::::The new lead still gives the wrong explanation. Even if it is true that the first choice wins the car in 1/3 of the cases, this does not logically lead to winning the car with chance 1/3 by sticking to your choice in the case the host has opened a door. The error is still in the explanation. Do we present erroneous reasoning to our readers? [[User:Wietze Nijdam|Wietze Nijdam]] 23:50, 7 February 2011 (UTC)


: The thoughts of some other citizens, both experts and laymen with regards to probability, would be interesting. [[User:Richard D. Gill|Richard D. Gill]] 15:41, 30 January 2011 (UTC)
::::::::: It does logically lead to winning the car with chance 1/3 in the average over all cases. All cases are equal by symmetry. Therefore, 1/3 in each case separately. The explanation is not wrong, but indeed incomplete, since symmetry is not used explicitly. As for me, the lead is often not rigorous, for not being too boring. (In fact I'd prefer not to ''prove'' anything in the lead, only ''claim''; but Peter does not agree.) But in sections, indeed, it could be emphasized that in ''some'' asymmetric generalizations of this problem the conditional probability differs from 1/3. --[[User:Boris Tsirelson|Boris Tsirelson]] 06:07, 8 February 2011 (UTC)


== Should The Contestant Switch? - A Simple Solution To The Door 3 Has Been Opened Problem - Without The 50/50 Host Bias Premise ==
:::::::::: My point of view: there is not a single right or wrong explanation. There are different ways to interpret Vos Savant's words, there are different understandings of what probability means, what probability assumptions you make or not make is a matter of taste, a matter of discussion. An encyclopaedia article is neutral. Present the facts to the reader and let the intelligent reader make up their own mind. The reader first needs to be interested in the problem because of the paradox, to gain insight why 50-50 could be the wrong answer. Vos Savant's question is "should you switch", not "what is the probability...". Probability is one possible tool to analyse the problem and there are different ways to model it and hence different ways to solve it. Economists and decision theorists use game theory.


=== Preamble ===
:::::::::: Also, mathematics does not tell us how we *must* behave in practical problems. It can tell us how it would be *wise* to behave.  
What you are about to read comes from the mind of a lay person. It is unvetted. But I think it is valid, correct, and complete.


I will leave out some supporting arguments for clarity, but welcome questions, and I think I am prepared for them.
:::::::::: If always switching gives a success chance of 2/3 and there is no conceivable way one could do better than that, consideration of the unconditional probability is irrelevant. The "simple solution" determines the optimal strategy but does not mathematically prove it is optimal. There are hundreds of other ways to prove optimality for those who feel it is important.


This solution relies on Logic/Philosophy rather than Conditional Probability.
:::::::::: The present draft nowhere says that there is one and only one way to approach MHP and I insist that this remains the case. [[User:Richard D. Gill|Richard D. Gill]] 07:51, 8 February 2011 (UTC)


As I am often told that I misuse them, I will do my best to avoid technical terms and jargon.
(outindented) @Boris: even that is not true; we need the extra assumption, that the initial choice of door is independent of the position of the car. This seems obvious, but is logically needed. And then, we are not aiming for the average contestant, but for the specific one. I have no objections to explaining in the lead the solution in a popular way, as long as it is done correctly. [[User:Wietze Nijdam|Wietze Nijdam]] 09:24, 8 February 2011 (UTC)


=== The Paradox ===
@Richard: There are definitely wrong explanations, like the combined doors solution, Devlin first came up with. And of course mathematicians may want to vary the formulation of the problem, even to the point where no one recognizes the MHP. But the readers we aim at will consider the interpretation with the contestant standing in front of two closed doors and having to decide which one to choose. Also then automatically people will talk about probability, as the crux of the problem is the wrong idea of equal odds. [[User:Wietze Nijdam|Wietze Nijdam]] 09:32, 8 February 2011 (UTC)
:* "Why is it 2/3 & 1/3 rather than 1/2 & 1/2?"


=== The Problem Statement Of Interest ===
: (Edit conflict)
:* "Suppose you are on a game show..."
: Richard: I agree.
: Wietze: (a) But note Richard's position above. ''(Well, you did already. :-) )'' (b) How could the initial choice of door depend of the position of the car?? Surely the man has no information about the position of the car. Such scrupulous logical analysis as you want, I know of only two situations where it applies. One is [[proof assistant]] and the like. The other is [[Entanglement (physics)|Bell theorem]] and the like. But these are extremely special. Too scrupulous even for sections, the more so, for the lead. I think so. --[[User:Boris Tsirelson|Boris Tsirelson]] 09:42, 8 February 2011 (UTC)
: Whoever really wants to collect ''all'' logically needed assumptions, here is my modest contribution to the collection:
: * "car" and "goat" are not synonyms;
: * cars never transmogrify to goats, nor goats to cars;
: * neither cars nor goats move from one chamber to another during the show;
: * quantum interference does not manifest itself during the show.
: :-) --[[User:Boris Tsirelson|Boris Tsirelson]] 11:28, 8 February 2011 (UTC)


=== The Critical Premise ===
:: Boris, do you suggest we add this to the article? Anyway, I think you're missing the point I want to make. Pity.[[User:Wietze Nijdam|Wietze Nijdam]] 17:40, 8 February 2011 (UTC)
The only valid host biases (when faced with 2 goats) that can be incorporated into the Monty Hall Problem paradox are
:* 50/50 - as per Selvin's 2nd letter
:* unstated - as per vos Savant, as interpreted by Morgan
:* Any hypothetical host bias, a la Morgan, is for academic purposes only, or to show the greater generalization capabilities of formal conditional solutions


Nice, Boris! Especially since I am a co-author of a paper on the quantum Monty Hall problem, where we do take account of some of these phenomena... ;-) (Maybe on a subpage we can do the quantum version) [[User:Richard D. Gill|Richard D. Gill]] 09:53, 11 February 2011 (UTC)


=== The Differences Between Logic and Probability ===
Wietze: the lead doesn't say that the information given there is "the official solution". I am not going to tell anyone what they have to believe. I think we should treat citizendium readers as being intelligent enough to work things out for themselves, and decide for themselves. There's surely room for improvement on the present page but I think it succeeds in providing the interested reader both simple "solutions" and full (conditional) solutions, it shows the relationship between them, and it does this (or tries to do this) without burdening the general reader with mathematical formalism and without being dogmatic - which is the sure-est way to alienate an intelligent reader.
==== The Result  ====
:* The contestant faces a situation where a *decision* is required, not a *precise numeric calculation*
:* Being on a game show, the contestant has the presumption that the car distribution and the host bias are random
:* If the above statement were not the case, the contestant *will not* be informed of any other method of car distribution or how the host decides between 2 goats


==== Different Standards ====
"Or putting it differently: Assuming the (wrong) intuitive answer of equal chances would mean that, after the host has opened the door, the initial door would suddenly win in half (instead of only one third) of all cases." --- This is ultimately correct, but finer than it may seem. After the host has opened the door, the 1/3 of that door suddenly jump, indeed. The question is, why does it jump to the second door only. The answer (not given in the article for now) is: because the host knows where is the car. An alternative scenario (also very instructive) is: the host does not know; opening a door he takes risk; but (this time) he was lucky not to reveal the car. In this scenario, 1/2 is the right answer! --[[User:Boris Tsirelson|Boris Tsirelson]] 06:39, 11 February 2011 (UTC)
:* From what I can gather from 2+ years of discussions, relying upon 'Symmetry' or 'random and uniform' in probability requires a *certainty* that events are equally likely
:* From what I have experienced in real life, and from solving puzzles, a Logical model only requires the *the absence of evidence that indicates otherwise*


::: (RDG-1) I think you have misunderstood something. You are on the game show and know nothing except for the rules, which are: the three doors hide two goats and one car. You will choose a door. The host will open another door and reveal a goat and offer a switch - which he can always do since he knows the location of the car. For you therefore the host is equally likely to open Door 2 or Door 3. This is sometimes called "logical probability". There ought to be an article on it on wikipedia or citizendium or both. I suppose by being certain that the two doors are equally likely you would mean that a totally reliable informant has told you that the host chooses which door to open, if he has a choice, by using a fair randomizer. That would certainly imply that for you the host is equally likely to open either door. But it is a rather special case, and it doesn't apply to MHP since we are not given any such information. [[User:Richard D. Gill|Richard D. Gill]] 13:17, 30 January 2011 (UTC)
:The added sentence "Or putting it differently ..." gives the wrong impression to the reader. And I think the author has the wrong idea of what he has written. [[User:Wietze Nijdam|Wietze Nijdam]] 09:22, 11 February 2011 (UTC)


Richard, maybe you could directly address the 2 statements I made? I'm simply tying to differentiate between being given premises like these from K & W:
:: The sentence was originally placed in a context where it was more clear what the author had intended. Other edits have somehow "orphaned" it. At [http://statprob.com/encyclopedia/MontyHallProblem2.html StatProb.com] (an encyclopaedia jointly sponsored by Springer and by the leading societies for statistics and probability) I wrote:
:* 'The car and the goats were placed randomly behind the doors before the show.'
:* 'If both remaining doors have goats behind them, he chooses one [uniformly] at random.'


and these, that I devised about a game show, which I think you labeled as 'logical probability':
::: The (wrong) intuitive answer 50-50 is often supported by saying that the host has not provided any new information by opening a door and revealing a goat since the contestant knows in advance that at least one of the other two doors hides a goat, and that the host will open one of such doors. The contestant merely gets to know the identity of one of those two. How can this non-information change the fact the remaining doors are equally likely to hide the car?
:* 'The contestant, having no information as to where the car is, [and in the absence of evidence to the contrary] assumes each door is equally likely.'
:* 'The contestant being unaware of the host's strategy when he has 2 goats, [and in the absence of evidence to the contrary] assumes each door is equally likely to be opened.'


I'm trying to understand if 'logical probability' can be used for the symmetry portion of the simple solution + symmetry + tlop solutions. Or, what is the difference between 'logical probability' and the 'special case of the fair randomizer' (or K & W's premises)? Or, if using the the logic that all 6 door pairings are equally likely, then an indifferent simple solution solves the specific door 1 and door 3 conditional problem by itself. [[User:Garry L. Kanter|Garry L. Kanter]] 13:37, 30 January 2011 (UTC)
::: However, precisely the same reasoning can be used against this answer: if indeed the host's action does not give away information about what is behind the closed doors, how can his action increase the winning chances for the door first chosen from 1 in 3 to 1 in 2? The paradox is that while initially doors 1 and 2 were equally likely to hide the car, after the player has chosen door 1 and the host has opened door 3, door 2 is twice as likely as door 1 to hide the car. The paradox (apparent, but not actual, contradiction) holds because it is equally true that initially door 1 had chance 1/3 to hide the car, while after the player has chosen door 1 and the host has opened door 3, door 1 still has chance 1/3 to hide the car.


:I wouldn't say that your logical contestant *assumes* each door is equally likely. For your logical contestant each door *is* equally likely.  
:: I tried to compose an <i>unopinionated</i> encyclopaedia article for myself and for [http://statprob.org StatProb], there is a draft at [http://www.math.leidenuniv.nl/~gill/mhp-statprob.pdf] (stealing some of the citizendium text!). [[User:Richard D. Gill|Richard D. Gill]] 09:46, 11 February 2011 (UTC)


: According to Laplace, probabilies are *defined* by splitting up all events into equally likely outcomes. From this definition he goes on to define independence of events, and conditional probability of one event given another, in terms of the probabilities already assigned to various events. So everything for him is defined in terms of just one so-called "primitive notion" of "equally likely". Laplace set a firm tradition - from then on, almost everyone started with some kind of primitive concept of probability, and then conditional probability and probabilistic independence in terms of probabilities of other events.
:: I like Boris' remark about the initial probability of 1/3 on Door 3 jumping to somewhere else. And bringing in the so-called "Monty-fall" variant: Monty accidentally trips and accidentally knocks door 3 open and it happens to reveal a goat. The probability of 1/3 jumps to doors 1 and 2 in equal parts. Bayes' rule is the good way to assimilate the mathematics of Bayes' theorem into intuitive/informal/subconscious reasoning! But maybe we need a subpage with variants. [[User:Richard D. Gill|Richard D. Gill]] 10:02, 11 February 2011 (UTC)


: Now Laplace's "equally likely" outcomes are often determined by looking for the symmetries in a problem. You want to avoid probability calculus altogether, and in particular the notion of conditional probability, but you do have a notion of probabilistic independence, and you want to use symmetry to decide if it can be applied. It seems to me that there is nothing wrong with your logic. However, since a large part of the world already uses standard probability calculus, it has been told by the High Priests that this is the only way to solve probability problems. It is quite important that your logic of probability does not clash with standard probability. And it doesn't. Your simple solution together with the remark that by symmetry the probability of winning by switching can't depend on the specific door numbers involved is equivalent to a solution written in terms of the standard probability calculus. [[User:Richard D. Gill|Richard D. Gill]] 20:45, 30 January 2011 (UTC)
::: Yes... Again, "...not determined by the situation alone but also by what is known about the development that led to this situation", which is for the ''the'' ultimate message of all that. --[[User:Boris Tsirelson|Boris Tsirelson]] 11:18, 11 February 2011 (UTC)


No, Richard. This is an incorrect statement:
:::: The "Or putting it differently..." was taken from the argument
:"You want to avoid probability calculus altogether..."
::::: ''The (wrong) intuitive answer "50-50" is often supported by remarking that the host has not given the contestant any information that they did not have before: the contestant knows in advance that one of the other two doors hides a goat. However, precisely the same argument could be given against this answer: if indeed the host's action does not give the contestant any information about where the car is hidden, how could it be that'' opening another door and revealing a goat makes the chance that the contestant's initial choice is correct change from 1 in 3 to 50-50?
I condition on the 100% likelihood the host will reveal a goat from another door and offer the switch. I am *not* concerned about conditioning on the irrelevant door #s. [[User:Garry L. Kanter|Garry L. Kanter]]
:::: put before the brief (correct) argument. For me this seemed to be confusing because I could not see how "not giving any new information" was meant to support the 50-50 answer while it clearly is an argument for the one third probability.
:::: When rewriting it, I tried to avoid the term "probability" and used "number of cases" instead.
:::: Boris, you say that Monty knowing the right door makes a difference: Isn't this only the case if the candidate '''knows''' that the host has to take chances? And doesn't the (implicit and general) assumption -- that the same happens in every show -- excludes this, anyway?
:::: --[[User:Peter Schmitt|Peter Schmitt]] 15:40, 11 February 2011 (UTC)


: You are not concerned about conditioning on the door #s because you know that they are irrelevant. By saying that they are irrelevant and giving a good reason for this - symmetry - you have taken account of them, hence you have given a complete solution.
::::: Yes, of course, your argument is correct. But maybe some readers will ask themselves such questions; should they seek the talk page for (hints toward) answers? --[[User:Boris Tsirelson|Boris Tsirelson]] 16:19, 12 February 2011 (UTC)


: You want to completely solve MHP using ordinary logic and common language only. That is what I meant by saying that you want to avoid probability calculus. Your solution is complete in the sense that it can be translated into formal mathematics with no effort whatsoever, and when that has been done, your solution is the so-called conditional solution of the probability purists over on wikipedia. [[User:Richard D. Gill|Richard D. Gill]] 09:40, 1 February 2011 (UTC)
:::::: I think there should be subpages about these issues. The unconditional probability is 2/3 that switching will give you the car, if and only if we know that the host will certainly open a door and reveal a goat (because he knows where the car is hidden). How the host chooses the door to open, when he has a choice is irrelevant. That the conditional probability is 2/3 that switching will give you the goat given also which door was opened by the host, only holds when the host is moreover equally likely to open either door when he has a choice. What does "equally likely" mean? If you only use probability in a frequentist sense, it means that you know that the host makes his decision by tossing a fair coin. If you use probability in a subjectivist sense, then equally likely means that for you it is equally likely because you have no reason whatsoever to put more money on one door than on another. It means that your knowledge is invariant under relabelling of the door numbers. Someone on wikipedia said very wisely, no one who thinks deeply about MHP can avoid wondering what probability means. And the truth is that there are is no concensus on that. [[User:Richard D. Gill|Richard D. Gill]] 21:28, 12 February 2011 (UTC)


=== The Critical Reasoning ===
::::::: The question what probability "really" is, is a philosophical question, I would say. If we describe probabilites using numbers then we have to use a mathematical definition, and have to argue in the bounds of the definition chosen. --[[User:Peter Schmitt|Peter Schmitt]] 00:07, 13 February 2011 (UTC)
:* The contestant has no reason to think any of the 6 'door selected and door opened pairings' have different likelihoods than the other 5, each at 2/3 & 1/3
:* Door 1 selected and door 3 opened is one of the 6 pairings described above
:* [extra reasoning] The contestant has no reason to think the door 1 and door 3 pairing is somehow contrary to the 2/3 & 1/3 he calculated, and that his door has a (much) greater than 50% likelihood of being the car


::: (RDG-2) I earlier found it difficult to understand what you meant about the 6 different pairings having different likelihoods. You were talking about the likelihood of the car being behind the chosen door or not. You're saying that there is no reason to think that for any of the six different values of (door chosen, door opened, door remaining) the likelihood that it corresponds to (car, goat, goat) or to (goat, goat, car) is different. This is the "symmetry argument" again, applied to your logical probabilities. Here you use it to show that the numbers on the doors are independent of the relationship between their manifest and their hidden roles, by the symmetry of your knowledge or lack thereof. The initially chosen door has a chance of 2/3 to hide a goat. The numbers written on this door and on the door opened by the host don't change this chance. The contestant's chance of his first door hiding a goat remain 2/3, whatever the pairing. [[User:Richard D. Gill|Richard D. Gill]] 13:26, 30 January 2011 (UTC)
:::::::: There is no mathematical definition of what probability is. Geometry doesn't define what a point and a line is. It just lists some relationships between them (the axioms) and shows what can be concluded from those axioms. If we want to link mathematics to the real world then we have to draw up a list of correspondences between the abstract mathematical objects of the theory, and things or concepts in the real world. So if you want to use probability theory to solve the three door problem - which is a problem about a game show on TV - you have to explain the bridge from the game show to the maths, and from the maths back to the game show. There are a number of common different understandings of what that bridge might be. Frequentists, subjectivists, the Laplacian definition... They all lead to the same abstract mathematical structure, so for a pure mathematician, it's of no import how the "user" wants to understand probability. However for the applied mathematician it can be of crucial import. In order to do mathematics about MHP you have to make some mathematical assumptions, build a mathematical model. How you understand probability could well influence what mathematical assumptions you want to make. It also determines what the real-life meaning is of the mathematical solution which you derive. I published an article entitled "MHP is not a probability problem - it is a challenge in mathematical modelling". Mathematical modelling is an art as well as a science. [[User:Richard D. Gill|Richard D. Gill]] 15:29, 13 February 2011 (UTC)


I'm trying to point out that the contestant *wouldn't* have any reason to think any of the 6 pairings (contestant's door, remaining door) have odds other than 1/3 & 2/3. I'm trying to present a rigorous logical argument to explain why the contestant would switch for the specific door 1 and door 3 pairing, rather than using a simple solution + symmetry + tlop. [[User:Garry L. Kanter|Garry L. Kanter]] 13:44, 30 January 2011 (UTC)
::::::::: Richard, you could contribute to "[[Theory (mathematics)]]"... :-) --[[User:Boris Tsirelson|Boris Tsirelson]] 15:52, 13 February 2011 (UTC)


=== The Simple Solution ===
:::::::::: Our views are not much different, I think. Choosing a "definition" is essentially the same as choosing a "model". And since, as you say, it is not difficult to see that the result is the same in all cases unless you assume some influence by astrology, precognition, etc. (or fraud). --[[User:Peter Schmitt|Peter Schmitt]] 23:38, 13 February 2011 (UTC)
:* The contestant knows that he would select a goat 2/3 of the time
:* The contestant receives no information as to the location of the car when the host opens door 3 to reveal a goat
:::The original distribution of the goats is 2/3, 2/3, 2/3
:::The contestant selecting a door does not change the above
:* The contestant has no reason to think the door 1 and door 3 pairing has a likelihood other than 2/3 & 1/3
:* The contestant doubles his likelihood of winning the car by switching


=== The Conclusion - Another Paradox ===
== Underlinked ==
:* I've shown that absent the 50/50 host bias premise, the simple solutions return the result 2/3 & 1/3 for the problem where the host revealed a goat behind door 3
The MHP page is a Mathematics Internal Article, a Mathematics Underlinked Article, an Underlinked Article. Are there pages on mathematical puzzles and diversions, brainteasers and the like, on citizendium? Is there a workgroup for this kind of thing? [[Recreation]]? There is also a big literature in [[psychology]] and in [[mathematics education]] on MHP.  [[User:Richard D. Gill|Richard D. Gill]] 10:04, 11 February 2011 (UTC)
:* The formal conditional solutions cannot calculate any probabilities without a host bias premise to plug in
:* Morgan is wrong that the simple solutions, without the 50/50 host bias premise do not solve the door 1 selected & door 3 revealed MHP.
:* In fact, simple solutions are the only solutions that are consistent with the paradox 'Why is it 2/3 & 1/3 rather than 1/2 & 1/2 ?" without requiring a 50/50 host bias premise.


Actually, I may not have left out any supporting stuff. I welcome your responses, below. [[User:Garry L. Kanter|Garry L. Kanter]] 01:39, 29 January 2011 (UTC)
: CZ has no good method of subject classification. (It needs to be established, though.) Currently the "Related articles" are a kind of substitute for it. Categories are only used for administrative purposes, and Workgroups are for approval. (This is likely to change.) --[[User:Peter Schmitt|Peter Schmitt]] 15:50, 11 February 2011 (UTC)
 
=== Responses ===
 
=== Response by Richard Gill ===
 
See responses (RDG-1) and (RDG-2) above. It seems to me that your arguments are in essence the same as the simple solution ("switching gives the car with probability 2/3") completed with the symmetry argument ("switching gives the car with conditional probability 2/3, conditional on the door number of the initial player choice and of the door opened by the host").
 
Splendid! What I have always wanted to see is a translation of the conditional solution into ordinary layperson's language, and that is what you have been pushing for too. [[User:Richard D. Gill|Richard D. Gill]] 13:37, 30 January 2011 (UTC)
 
:No, I'm trying to make the case that using logic rather than probability, that the [conditional] simple solutions, on their own, solve the conditional door 3 has been opened problem. That is, they don't need to rely on symmetry to do whatever it is you and Boris say simple + symmetry + lotp accomplish. [[User:Garry L. Kanter|Garry L. Kanter]] 13:50, 30 January 2011 (UTC)
 
:: You are using logic first to argue that a certain event has a likelihood of 2/3 (2/3 chance that your initial choice hides a goat) and then to show that this likelihood can't depend on some further information (the identity of the door opened by the host). I would say "no reason the odds should be different for any of the six pairings" is the same as saying that by symmetry, the conditional probability must be the same as the unconditional.  
 
:: Students of probability might find the hint (law of total probability) useful to show that the calculus of probability does conform to one's logical expectations. The old hands don't need the hint.
 
:: Well, it will be interesting to see what some further citizens think, both those who are expert in probability, and those who are not. [[User:Richard D. Gill|Richard D. Gill]] 15:25, 30 January 2011 (UTC)
 
:::I have no background in maths, statistics etc. and took a while to understand the problem (days, actually). What made it click for me was the part which mentions playing the game many times. I imagined 99 repetitions of the game with the car randomly behind any of the doors each time. Obviously, if the player sticks to door #1 each time, he'll win about 33 times, on average. But if he switches on all 99 plays, he'll lose the car only on the roughly 33 occasions that it really was behind door #1 to start with, i.e. he wins about 66 out of 99 times, which is 2/3 odds. [[User:John Stephenson|John Stephenson]] 15:57, 30 January 2011 (UTC)
 
:::: Nice comment, @John. Initially almost everyone gives the wrong answer to vos Savant's question (I was no exception, and I'm a professor of probability and statistics). People do tend to worry for several days before suddenly getting an understanding of it. The present draft article needs a lot of work, and in particular it should start with a collection of different ways which different people find useful to understand why the answer is *not* "no point in switching, it's 50-50". My mother (now aged 94) is the only person I know who almost immediately gave the right answer, when I told her about the problem a couple of years ago. She had imagined the problem altered to 100 doors. You choose one door. Not very likely that the car is behind it. The host then throws 98 doors open revealing 98 goats and asks if you'ld like to switch to the one door left closed. I think almost everyone would switch immediately. My mother had no formal mathematics education. She was however one of Turing's "computers" at Bletchley Park during WW2 - the computers were the many young ladies who turned the handles on the calculating machines applying Turing's algorithm to break the Eniga code. [[User:Richard D. Gill|Richard D. Gill]] 20:17, 30 January 2011 (UTC)
 
::::: Yes, this was ''the'' intuitive explanation I gave to my first-year math students ("Introduction to probability" course) just after formal analysis. I made a little spectacle pretending to be the host that opens the 98 doors. When skipping one I glanced with meaning to the audience, and the hint was well taken! --[[User:Boris Tsirelson|Boris Tsirelson]] 19:47, 31 January 2011 (UTC)
 
:::::: This needs to be in the article. With a link to a youtube video from your course, Boris. [[User:Richard D. Gill|Richard D. Gill]] 07:34, 1 February 2011 (UTC)
 
== The table at the end of the article ==
 
The first column appears to add to 1 1/3. In the next to last column, I think row 1 should have a value of 0 or n/a. In the last column, I think row 4 should have a value of 0 or n/a.[[User:Garry L. Kanter|Garry L. Kanter]] 12:45, 30 January 2011 (UTC)
 
: The first two cells of the first column are the same. They should be merged into one box. And I think it would be better to write "Probability..." rather than "p=.." everywhere. The "p" in each column is the probability or conditional probability or joint probability of something different in each column. This table is really the same as the original Carlton decision tree. [[User:Richard D. Gill|Richard D. Gill]] 15:31, 30 January 2011 (UTC)
 
You crack me up, Richard. Instead of saying, 'Oh, good point(s), I'll fix those', you explain the table to me as if I've never seen anything like it before in my life. I have, though. Which I *think* you should be aware of.
 
I understand the problem with the 1st columns, and any 'common man' would, like me, find it poor form, and easy to criticize, and then call into doubt the rest of the table. One of Morgan's criticisms of vos Savant's solution is that the 'door 2 opened' outcome cannot happen. The Wikipedia article version of Chun's decision tree highlights these outcomes in red. But it's your table, and I presume there's no single 'right way' to present the info. Make it any way you want.
 
It seems you instinctively respond to my comments as if you think I don't quite grasp the 'complexities' and 'subtleties' of various aspects of the MHP. I hope soon I will have (finally) demonstrated such rudimentary skills. [[User:Garry L. Kanter|Garry L. Kanter]] 12:04, 31 January 2011 (UTC)
 
: It's not my table, Garry. I didn't put it there. And it is not my page. Anyway, anything placed on citizendium is available under a Creative Commons CC-by-sa 3.0 licence. This is called "collaborative editing".  My remarks were intended for anyone who cares to get to work and improve the table, in particular, the fellow citizen who put it there. I am not good at formatting tables. [[User:Richard D. Gill|Richard D. Gill]] 16:46, 31 January 2011 (UTC)
 
:: Clearly I am not great at tables either, otherwise I would have color coordinated the first two rows and the 3rd & 4th to indicate the two situations possible (chose right/wrong initially). No time to work on it now tho, but I see that quite a discussion has started on this page. [[User:David E. Volk|David E. Volk]] 20:46, 31 January 2011 (UTC)
 
::: Thanks David! It is good to have such a table in the article (I moved it to a better place and added some text, OK?). And I hope a table-expert will improve it. [[User:Richard D. Gill|Richard D. Gill]] 07:32, 1 February 2011 (UTC)
 
== Conditional Probability on Citizendium ==
 
I think that a formal probability anaysis of MHP belongs as an illustration on the pages on [[conditional probability]], rather than on MHP itself. That page is pretty awful... (I think). [[User:Richard D. Gill|Richard D. Gill]] 11:48, 31 January 2011 (UTC)
 
: Sure it is ridiculously short for "one of the most important concepts in probability theory". --[[User:Boris Tsirelson|Boris Tsirelson]] 19:51, 31 January 2011 (UTC)
 
== Conditional probability with MHP ==
 
In the article, Richard Gill presents a player who picks door 1 and due to some assumptions, hits the car with probability 1/3. And indeed such players will get the car 2/3 of the times when switching, The point however is, this (logically) does not guarantee the same for the actual player, who not only picked door 1, but also sees door 3 opened with a goat. The difference between players in general and any specific player seems to grow into a big issue in the MHP discussions. Yet in the past I seldom heard of anyone considering this, let us call it unconditional, formulation. It seems however that gradually, in order in my opinion to justify the unconditional probability as a solution, people tend to defend this. [[User:Wietze Nijdam|Wietze Nijdam]] 21:19, 31 January 2011 (UTC)
 
: The way I started off this article, it contains two solutions: the first one is a so-called simple or unconditional solution. The second one is a full conditional solution, but presented in ordinary (non-technical) language. Later another editor has added a probability table, from which one can read off both a simple solution and a conditional solution.
 
: I think that several more sections need to be added to the article. For instance
 
::* History
::* Formal solutions
::* Variants
::* Why do people get it wrong?
::* Game theory and economics
 
:By formal solutions I mean solutions written out using probability calculus, the kinds of solutions that teachers of first probability courses for mathematicians want their students to write out. Just as there are many simple solutions there are also many routes within standard probability calculus to getting good solutions, so this section should also present various approaches. [[User:Richard D. Gill|Richard D. Gill]] 07:09, 1 February 2011 (UTC)
 
::There are two separate styles of 'simple solutions':
:::*  The unconditional, as presented by Selvin and vos Savant with simple tables listing each possible combination of car location, door selection and door revealed.
:::* The conditional, which shows that the selected door choice's 2/3 odds of being a goat don't change when multiplied by 100% (the host has revealed a goat behind one of the other doors).
::Posted by [[User:Garry L. Kanter|Garry L. Kanter]] 07:15, 1 February 2011 (UTC)
 
 
::: Garry, I think your notion of "simple solution" and "conditional solution" differs from the notion which some editors on wikipedia were using. I think most people there meant, by a conditional solution, a solution using the calculus of probability and computing the conditional probability of switching giving the car given the door opened by the host. By a simple solution they meant a solution which in terms of the calculus of probability computes the unconditional probability that switching gives the car. [[User:Richard D. Gill|Richard D. Gill]] 07:29, 1 February 2011 (UTC)
 
::::You are likely correct, Richard. They are incorrect/imprecise/making assumptions in their terminology. Probably purposely. [[User:Garry L. Kanter|Garry L. Kanter]] 07:54, 1 February 2011 (UTC)
 
::::: If you've learnt formal probability then you've been taught that that is the only way to solve probability problems. [[User:Richard D. Gill|Richard D. Gill]] 09:29, 1 February 2011 (UTC)
 
::::::I find that response flippant, contrary to the levels of precision you have been (counter-productively) holding me accountable to, and not relevant to the discussion. Remember that innocent nurse who went to jail because of lousy math assumptions by some detective? Misinterpreting the English language used by reliable sources in order to promote some ideology is not a condition for solving probability problems. Besides, the MHP is as much (more) logic as probability.
::::::Are you actually suggesting there is no difference in the 2 types of solutions I described? Please expand on that viewpoint. [[User:Garry L. Kanter|Garry L. Kanter]] 11:26, 1 February 2011 (UTC)

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 Definition:  (also called the "three-doors problem") A much discussed question concerning the best strategy in a specific game show situation. [d] [e]
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Archived earlier talk

In order to regain focus I created a first talk archive of talk up to this point: Talk:Monty_Hall_problem/Archive_1. Richard D. Gill 15:21, 2 February 2011 (UTC)

General remarks

This talk page has quickly become very long with a difficult to follow structure. I'd like to make a few general remarks:

  • Let us avoid to repeat and continue the endless (and mostly useless) discussion of this problem.
  • The MHP is not a "paradox". Its solution may be surprising, but it is not paradoxical.
  • There are not two (or more) "solutions".
  • Once the question has been unambiguously posed there is only one solution -- the correct solution.
  • There may be (essentially) different arguments leading to this correct solution.
  • There may be several (didactically) different ways to present the same argument.

What should an article on the MHP contain (with the reader searching information in mind)? My answer:

  • It should state the problem and present its solution as brief and as clear (and in an as informal language) as possible.
  • It should summarize the history of the problem and the disputes it has caused.
  • It should not contain a large amount of historical details, different approaches, discussion of subtleties, etc. that the ordinary reader will not want, and that would probably be confusing for him.

Supplementary material can be presented on subpages or separate pages:

  • A page on the detailed history of the problem.
  • A page on the discussion caused by the problem.
  • A (Catalog) subpage containing various ways to present the solution(s). It may help a reader to find an explanation he likes.

--Peter Schmitt 13:42, 1 February 2011 (UTC)

I agree with everything you say here Peter, except for one thing. MHP is defined (IMHO) by the definitely ambiguous words of Marilyn Vos Savant quoted in the article. Both before her popularization of the problem, and later, different authorities have translated or transformed her problem into definitely different mathematically unambiguous problems. And I'm only referring to problems to which the solution is "switch"! That is part of the reason why there is, I think, not a unique "correct solution" - there are as many correct solutions as there are decent unambiguous formulations.
I think there are two particularly common solutions: one focusing on the overall probability of winning by switching, and the other focussing on the conditional probability of winning by switching given the specific doors chosen and opened. The present draft intro contains elements of both and even at attempt at synthesis. Richard D. Gill 23:26, 1 February 2011 (UTC)
By the way, the meaning I am used to of the word "paradox" is an apparent contradiction. And there certainly is an apparent contradiction between ordinary people's immediate and instinctive solution "50-50, so don't switch", and the "right" solution: "switching gives the car with probability 2/3". Richard D. Gill 15:39, 2 February 2011 (UTC)
But where is the apparent contradiction? That intuition and correct reasoning lead to different results is not an apparent contradiction, I would say. (But this is only a question of language, not really important here.) --Peter Schmitt 00:54, 6 February 2011 (UTC)

Which MHP?

In Talk:Monty_Hall_problem/Archive_1, Wietze Nijdam started discussion of which of the following two problems is "the MHP". That discussion has been raging on wikipedia unabated for over two years, producing only polarization. Wikipedia editors interested in sensible compromise have left in bemusement, disgust or frustration.

Returning to Wietze's text, consider the following two statements ( Richard D. Gill 10:12, 3 February 2011 (UTC) ) :

  • F0: (Conditional formulation) If the contestant is offered to switch after the host has opened the goat door, the decision has to be based on the conditional probability given the initial choice and the opened door.
  • F1: (Unconditional formulation) If we are asked whether the contestant should switch, even before he has made his initial choice, and we are not allowed to give a solution for every possible combination of initial chosen door and opened goat door, the decision will have to be based on the (unconditional) probability of getting the car by switching.

The discussion point is: which formulation is more natural to be the MHP and should be presented as such.

My opinion: F0. Wietze Nijdam 22:45, 2 February 2011 (UTC)

Both are of interest; but
  • F1 is of interest to nearly everyone;
  • F0 is of interest for those already understanding F1 and wishing to widen and deepen their understanding.
In this sense, F1 is the basic MHP while F0 is the advanced MHP.
--Boris Tsirelson 07:06, 3 February 2011 (UTC)
@Wietze, you cannot say "the decision has to be based on the conditional probability". You could say that it would be wise to base your decision on the conditional probability. Moreover, if you want to reach the general public, you had better explain why this would be the wise thing to do. Please draft some material on this in the article on conditional probability.
If indeed you want to reach the general public, it would also be wise to note that given that the player has chosen Door 1, whether the host opens Door 2 or Door 3 has no relevance at all to whether or not the car is behind Door 1 (under the probabilistic assumptions which many people find natural). So the general reader can be informed of the truth and the whole truth of the standard MHP - F0 and F1 combined - using plain non-technical English and without needing to follow a course in probability theory first.

PLEASE let us not repeat this endless discussion here. Draft appropriate subpages to MHP as Peter Schmitt indicated is the next step which ought to be made. Get to work on the articles on probability, probability theory, conditional probability, Bayes Theorem. Richard D. Gill 09:55, 3 February 2011 (UTC)

That's not up to me. Wietze Nijdam 10:22, 3 February 2011 (UTC)
It is up to you. You are an author, a citizen of citizendium? See CZ:Myths_and_Facts. [User:Richard D. Gill|Richard D. Gill]] 10:25, 3 February 2011 (UTC)
@Boris: your reaction in the next section seems contradictory to what you've commented here. So, please, make clear what you mean. The formulation F0 is the one in which the contestant is in the end standing in front of two closed doors and one opened, and then asked whether she wants to switch. Yet you write here above: F1 is of interest to nearly everyone; That's puzzling me. Wietze Nijdam 09:43, 6 February 2011 (UTC)
OK, you are right; I did not understand you correctly. I believe that the whole MHP story is of interest "to nearly everyone" first of all because of "the conclusion". Thus, the "two closed doors" situation must be emphasized. But for me it already is: "Almost everyone, on first hearing the problem, has the immediate and intuitive reaction that the two doors left closed, Door 1 and Door 2, must be equally likely to hide the car". This is the first phrase after general introductory words! Thus, for now I believe that F0 is relevant, and presented. And so I still fail to understand your dissatisfaction. --Boris Tsirelson 17:06, 6 February 2011 (UTC)

(outindenteed) Boris, does it surprise you that I was dissatisfied when you first said F1 was the important formulation? I hope you also understand my dissatisfaction with Richard and seemingly Peter favouring F1 as the important formulation and hence primarily presented to the readers.Wietze Nijdam 17:19, 6 February 2011 (UTC)

Well, I am sorry for my error (in fact I thought that your "conditional" means also "asymmetric"). But anyway, we discuss the article, not its talk page. Once again: F0 is relevant, and presented, boldly. Isn't it? --Boris Tsirelson 19:12, 6 February 2011 (UTC)
But let me formulate my position more exactly. I am dissatisfied with the article as it is now, because "the conclusion" is missing. And "the conclusion", as I see it, compares the answer (1/3, 2/3) with the naive equal probabilities on the two closed doors. Thus, "two closed doors" is a crucial component of it. Which does not conflict (as you agree, if I am not mistaken) with calculating the conditional probability via the unconditional probability and the symmetry. --Boris Tsirelson 19:21, 6 February 2011 (UTC)
Boris, I'm glad you also consider F0 the MHP (I knew this from our discussion on Wikipedia). But ... you have to read the article more carefully. Tthe article present initially F1 as the standard form of the MHP. And that is precisely Richard's intention. With the simple solution S1 (without any reference to symmetry or conditional probability) as its solution. I'm strongly against this. Many people will not notice that F1 is presented and from their own imagination think it is F0, just like many texts on internet about MHP that present F0, but with S1 as its solution, what is a logical error. Wietze Nijdam 22:33, 6 February 2011 (UTC)
As for me, such nuances of the emphasize balance are not as important here as (say) in a political communique. If "the conclusion" (in one form or another) satisfying me will appear, it will clearly juxtapose F0 with F1 (thus closing the problem, in my opinion). For now I am waiting for Rick's opinion about "the conclusion". --Boris Tsirelson 07:11, 7 February 2011 (UTC)

Three editor approval

Just as a point of procedure, there are currently three editors on this page; Boris, Peter, and Richard. Should they agree on content and style, it is possible that this article can be approved and locked allowing editors to move on to other important and related articles. With input from the very knowledgeable authors on this page, I do think you've all illustrated your willingness to create a good article here, thanks for your professional efforts. D. Matt Innis 13:14, 3 February 2011 (UTC)

Yes, I've been very impressed by the work that has been put into this article and the people it has drawn in. I'd definitely like to see it reach approval. --Joe Quick 19:19, 3 February 2011 (UTC)
My own opinion is that a number of sub-pages need to be written, and that when this is done, the introductory page can be shortened and sharpened (some of the side remarks are really reminders to myself or others of things that need to be explored on sub-pages). Also the list of references - at present it is just a comprehensive list stolen "as is" from wikipedia - needs to be replaced with a shorter and annotated list of key references. I don't know enough yet about citizendium procedures to know it makes sense to "lock" an article when a lot of supporting material still needs to be put into place. Richard D. Gill 09:46, 4 February 2011 (UTC)
Richard, the main article can be locked and still allow further work on the subpages. Also, when an article is locked, a draft is created that is an exact copy of the approved version where work continues. At any point, it can be re-approved and the new version replaces the original. That way we get incremental improvements (hopefully). Again, it will take three editors to agree on the improvements. D. Matt Innis 13:29, 4 February 2011 (UTC)


Thanks, that's clear. Well, I'm ready to approve. @Peter Schmitt, @Boris Tsirelson, how about you?
As Boris and Richard have clearly stated to be advocates of the unconditional formulation of the MHP, I urge Peter to think thoroughly about this. I have never seen ordinary people, picturing the MHP, and not imaging the player standing in front of two closed and one opened door, and only then offered the possibility to swap doors. It is in my opinion not only the charm of the puzzle, it is also the crux. Only when actually confronted with the two remaining doors, people are inclined to think the odds are equal. Wietze Nijdam 09:16, 5 February 2011 (UTC)
Wietze, I completely agree that "Only when actually confronted with the two remaining doors, people are inclined to think the odds are equal", and I do treat this aspect as very important (see "Toward the conclusion" below). I agree that the conditional probability is important in the formulation of MHP. But the article already treats it, explaining that the conditional probability is equal to the unconditional probability by symmetry. Indeed, the article contains a link to conditional probability. Do you see anything wrong in choosing this (rather short and intuitive) way to the conditional probability (via the unconditional one)? --Boris Tsirelson 15:23, 5 February 2011 (UTC)
Moreover, looking again at the article I see that all reasonable approaches are already sketched. Including the most traditional treatment of conditional probability. Thus I really do not understand what is the problem. Maybe you want a more formal treatment (something similar to my lectures on a course for math students)? Also no objections from me; but better on a subpage. Or maybe you want to cover asymmetric cases? As for me, a paradox should be always stripped down to the simplest possible formulation (which was indeed made for set-theoretic paradoxes a century ago). Generalizations are a more advanced topic of more special interest. --Boris Tsirelson 17:47, 5 February 2011 (UTC)
Strange that you do not understand. It really is a mess. We, you and I, had part of this discussion before on Wikipedia. Let me explain once again. In my and a lot others opinion in the MHP the contestant is offered to swap AFTER the host has opened the goat door, and the contestant is confronted with two closed doors, from which many people are inclined to think the odds for the car are even. This problem has (is wisely) to be solved by calculating the conditional (or if you like the posterior) probability given the situation the contestant is in. (If it makes it easier for the average reader we may formulate it different.) The way the conditional probability is calculated is (of course) unimportant, although by using the symmetry (under suitable assumptions) this may be explained in a more understandable(??) way than just by using Bayes' formula. Richard agrees with me on this. But!!! the simple reasoning, the one that says (in short), you hit the car 1/3 of the time, hence by switching you get it 2/3 of the time, is not a solution to this formulation of the MHP, as it does not calculate the conditional probability, i.e. it does not account for the situation the player is in. Some authors, in my opinion in need to make the simple solution work, change the problem formulation, i.e. they say: the player is asked whether she wants to switch, even before she has made her initial choice. Then no conditions have been imposed, and the unconditional probability is sufficient. This is not what I and not just me, consider the MHP. Did I really have to explain this to you? So, what concerns your question, no I have no objection by calculating the CONDITIONAL probabilty, with the use of the symmetry, through the unconditional, as long as the CONDITIONAL (or how we will call it) is indeed calculated. The simple solution definitely does not mention anything like this at all. Wietze Nijdam 00:57, 6 February 2011 (UTC)
About three-editor approval: yes, in principle I am ready to join; but see "Toward the conclusion" below. --Boris Tsirelson 15:36, 5 February 2011 (UTC)
As a matter of fact, currently both Boris and I have not contributed to the article, therefore both of us could single-handedly approve it. Even if I add an introduction (as I probably will), Boris could do it alone. But, of course, teamwork is always possible. --Peter Schmitt 00:58, 6 February 2011 (UTC)
Wietze, putting replies into the middle of comments makes talk pages difficult to read and makes it difficult to see who wrote what.
I also fail to see a problem: As the question is posed the candidate is not put in front of two closed and an open door. The problem clearly tells how the situation evolved. However, I think it is not useful to number the doors unless the numbers are used to identify the "door first chosen", the "other door closed", and the "door opened". (The argument using repetitions does not make clear that the door opened will not always be the same.) --Peter Schmitt 01:14, 6 February 2011 (UTC)
Wietze, you write "I have no objection by calculating the CONDITIONAL probabilty, with the use of the symmetry, through the unconditional, as long as the CONDITIONAL (or how we will call it) is indeed calculated." But your requirement is already fulfilled by the article, isn't it? --Boris Tsirelson 06:32, 6 February 2011 (UTC)
No, Boris, the article does not. It says: One could say that when the contestant initially chooses Door 1, the host is offering the contestant a choice between his initial choice Door 1, or Doors 2 and 3 together. Let us continue this discussion in the section "Which MHP?" above. There I formulated the versions F0 and F1. And it is about the difference between these two. Wietze Nijdam 09:37, 6 February 2011 (UTC)
@Peter. Sorry you fail to see the problem, because there is one. Unless the contestant is blind(folded), she sees the doors and hence is able to distinguish between them. The problem formulation speaks also of door 1 and door 3. Anyone so it seems may see the doors and the contestant pointing at one door and the host opening one. Look at all the simulations that are constructed. A specific door is chosen and a specific one is opened. Only mathematicians may come to a formulation in which only is spoken of "the chosen door" and "the opened door". It is possible, but then these doors are random variables, and take values in specific situations. Rather difficult to understand for the average reader, don't you think? Please follow and take part in the discussion under "Which MHP?". Wietze Nijdam 09:56, 6 February 2011 (UTC)
I have no intention to repeat and continue the endless discussions. Wietze, you forget that we all know the problem and the arguments ... unless something new turns up. --Peter Schmitt 11:45, 6 February 2011 (UTC)
I don't know what endless discussions you're referring to. Definitely not here on Citizendum. And as far as I know, you were not involved in the discussions on Wikipedia. You may be an excellent mathematician, but I doubt you really understand the problem as you say. Let alone all the arguments, as you show with your remarks about the door numbers. Wietze Nijdam 14:41, 6 February 2011 (UTC)
I am glad that we had no endless discussion on CZ (yet?), but one need not be "involved" in the WP discussions (e.g., en. and dt.) in order to have noticed them and to know about them. --Peter Schmitt 00:01, 7 February 2011 (UTC)
Well, do you agree with me, or do you contribute to the endless discussion? Wietze Nijdam 07:27, 7 February 2011 (UTC)

(unindent) Peter, no, I do not want to approve alone, in presence of three editors. --Boris Tsirelson 07:27, 6 February 2011 (UTC)

Toward the conclusion

But I bother: the conclusion is missing. I mean something in the spirit of the following.

A paradox refutes some naive belief. For example, set-theoretic paradoxes refuted the naive belief in unlimited freedom forming "the set of all x satisfying (whatever)". Another example: the continuous but nowhere differentiable Weierstrass function refuted the naive belief that a continuous function is necessarily differentiable, except some special points.

The MHP paradox refutes the naive belief in such an argument:

"According to new data, only m possibilities remain; apriori, n possibilities were equiprobable; therefore (?) the m remaining possibilities are equiprobable aposteriori."

The change of probabilities according to new data (so-called conditioning) is generally more subtle than just exclusion of some possibilities.

--Boris Tsirelson 15:18, 5 February 2011 (UTC)

In my view there is no (true) "paradox" -- though some experience it as one.
Moreover, I think the page needs an introduction (and --perhaps-- also splitting into some sections as orintation for the reader).
--Peter Schmitt 00:46, 6 February 2011 (UTC)
It depends on the meaning given to the word "paradox"; probably there is no consensus on it. But I do not insist on the word. Rather, on a conclusion. --Boris Tsirelson 06:28, 6 February 2011 (UTC)

The (new) lead

I like it. However, I (and probably Wietze too) think that even before presenting the right solution we should mention the widespread erroneous argument (since otherwise, why bother at all?). --Boris Tsirelson 11:18, 7 February 2011 (UTC)

Now better; but still not a single word, why is it counterintuitive, and what is the wrong answer, and the wrong argument. --Boris Tsirelson 12:26, 7 February 2011 (UTC)

And now the lead says it; but in fact now it more or less explains the solution. Then, what are subsequent sections for? Isn't it better to puzzle the reader in the lead and then (in sections) reveal him the truth? --Boris Tsirelson 13:48, 7 February 2011 (UTC)

In my view, the lead (I prefer: the introduction) should provide a (brief) summary, sufficient for all those who are looking for basic information. The main part of an article is for those who, judging from the introduction, are looking for information in depth, more details, etc, In this case, it would be fine if a reader is satisfied after the introduction, but while indeed not much more (except historical data and the connection to related topics) needs to be added, the well-known reactions call for some more words telling the same story. --Peter Schmitt 15:32, 7 February 2011 (UTC)
I am not quite understanding how do you apply these general rules to this situation. Anyway, if an article discusses a theorem, then probably its (rough or exact) formulation appears in the lead, while a proof does not. If we follow this approach here then the answer (1/3,2/3) appears in the lead but the question "why" is answered later. Or not?
About "lead" and "intro" I remember that on WP they are two different units; "intro" (if present) is the first section. Does CZ treat it differently? --Boris Tsirelson 15:58, 7 February 2011 (UTC)
I don't know if this has been defined somewhere. My impression (combined with my personal preference) is as I said above. Ultimately, the EC will have to make up its mind whether there should be a rule, or only a recommendation, (or nothing) on this. (If I say "intorduction" I mean the part before the first section title, corresponds "Edit intro". The lead -- for me -- would be (at most) the first paragraph.)
As for having the proof in the intro: This depends, I would say, on the type of proof. In this case the "proof" is more important than the question, in other cases mentioning the main idea of the proof can be included, in others nothing at all.
--Peter Schmitt 16:19, 7 February 2011 (UTC)
Upon reflection, I think that the new lead is correct. The problem itself should be stated. The info about the game show and vos Savant, etc. should be in a following paragraph, as has now been done. Hayford Peirce 16:27, 7 February 2011 (UTC)
I like the look of the whole article now. Richard D. Gill 16:43, 7 February 2011 (UTC)
OK. Not exactly to my taste, but this is natural: usually, nothing can be exactly to several tastes simultaneously. I am ready to join three-editor approval process. Are you? --Boris Tsirelson 17:00, 7 February 2011 (UTC)
I should have said: it looks good enough to be stabilized for a while, and for work on subpages to get going!. So I too am ready for a three-editor approval process. (Just saw and liked some good further edits by Peter). Richard D. Gill 19:24, 7 February 2011 (UTC)
The new lead still gives the wrong explanation. Even if it is true that the first choice wins the car in 1/3 of the cases, this does not logically lead to winning the car with chance 1/3 by sticking to your choice in the case the host has opened a door. The error is still in the explanation. Do we present erroneous reasoning to our readers? Wietze Nijdam 23:50, 7 February 2011 (UTC)
It does logically lead to winning the car with chance 1/3 in the average over all cases. All cases are equal by symmetry. Therefore, 1/3 in each case separately. The explanation is not wrong, but indeed incomplete, since symmetry is not used explicitly. As for me, the lead is often not rigorous, for not being too boring. (In fact I'd prefer not to prove anything in the lead, only claim; but Peter does not agree.) But in sections, indeed, it could be emphasized that in some asymmetric generalizations of this problem the conditional probability differs from 1/3. --Boris Tsirelson 06:07, 8 February 2011 (UTC)
My point of view: there is not a single right or wrong explanation. There are different ways to interpret Vos Savant's words, there are different understandings of what probability means, what probability assumptions you make or not make is a matter of taste, a matter of discussion. An encyclopaedia article is neutral. Present the facts to the reader and let the intelligent reader make up their own mind. The reader first needs to be interested in the problem because of the paradox, to gain insight why 50-50 could be the wrong answer. Vos Savant's question is "should you switch", not "what is the probability...". Probability is one possible tool to analyse the problem and there are different ways to model it and hence different ways to solve it. Economists and decision theorists use game theory.
Also, mathematics does not tell us how we *must* behave in practical problems. It can tell us how it would be *wise* to behave.
If always switching gives a success chance of 2/3 and there is no conceivable way one could do better than that, consideration of the unconditional probability is irrelevant. The "simple solution" determines the optimal strategy but does not mathematically prove it is optimal. There are hundreds of other ways to prove optimality for those who feel it is important.
The present draft nowhere says that there is one and only one way to approach MHP and I insist that this remains the case. Richard D. Gill 07:51, 8 February 2011 (UTC)

(outindented) @Boris: even that is not true; we need the extra assumption, that the initial choice of door is independent of the position of the car. This seems obvious, but is logically needed. And then, we are not aiming for the average contestant, but for the specific one. I have no objections to explaining in the lead the solution in a popular way, as long as it is done correctly. Wietze Nijdam 09:24, 8 February 2011 (UTC)

@Richard: There are definitely wrong explanations, like the combined doors solution, Devlin first came up with. And of course mathematicians may want to vary the formulation of the problem, even to the point where no one recognizes the MHP. But the readers we aim at will consider the interpretation with the contestant standing in front of two closed doors and having to decide which one to choose. Also then automatically people will talk about probability, as the crux of the problem is the wrong idea of equal odds. Wietze Nijdam 09:32, 8 February 2011 (UTC)

(Edit conflict)
Richard: I agree.
Wietze: (a) But note Richard's position above. (Well, you did already. :-) ) (b) How could the initial choice of door depend of the position of the car?? Surely the man has no information about the position of the car. Such scrupulous logical analysis as you want, I know of only two situations where it applies. One is proof assistant and the like. The other is Bell theorem and the like. But these are extremely special. Too scrupulous even for sections, the more so, for the lead. I think so. --Boris Tsirelson 09:42, 8 February 2011 (UTC)
Whoever really wants to collect all logically needed assumptions, here is my modest contribution to the collection:
* "car" and "goat" are not synonyms;
* cars never transmogrify to goats, nor goats to cars;
* neither cars nor goats move from one chamber to another during the show;
* quantum interference does not manifest itself during the show.
:-) --Boris Tsirelson 11:28, 8 February 2011 (UTC)
Boris, do you suggest we add this to the article? Anyway, I think you're missing the point I want to make. Pity.Wietze Nijdam 17:40, 8 February 2011 (UTC)

Nice, Boris! Especially since I am a co-author of a paper on the quantum Monty Hall problem, where we do take account of some of these phenomena... ;-) (Maybe on a subpage we can do the quantum version) Richard D. Gill 09:53, 11 February 2011 (UTC)

Wietze: the lead doesn't say that the information given there is "the official solution". I am not going to tell anyone what they have to believe. I think we should treat citizendium readers as being intelligent enough to work things out for themselves, and decide for themselves. There's surely room for improvement on the present page but I think it succeeds in providing the interested reader both simple "solutions" and full (conditional) solutions, it shows the relationship between them, and it does this (or tries to do this) without burdening the general reader with mathematical formalism and without being dogmatic - which is the sure-est way to alienate an intelligent reader.

"Or putting it differently: Assuming the (wrong) intuitive answer of equal chances would mean that, after the host has opened the door, the initial door would suddenly win in half (instead of only one third) of all cases." --- This is ultimately correct, but finer than it may seem. After the host has opened the door, the 1/3 of that door suddenly jump, indeed. The question is, why does it jump to the second door only. The answer (not given in the article for now) is: because the host knows where is the car. An alternative scenario (also very instructive) is: the host does not know; opening a door he takes risk; but (this time) he was lucky not to reveal the car. In this scenario, 1/2 is the right answer! --Boris Tsirelson 06:39, 11 February 2011 (UTC)

The added sentence "Or putting it differently ..." gives the wrong impression to the reader. And I think the author has the wrong idea of what he has written. Wietze Nijdam 09:22, 11 February 2011 (UTC)
The sentence was originally placed in a context where it was more clear what the author had intended. Other edits have somehow "orphaned" it. At StatProb.com (an encyclopaedia jointly sponsored by Springer and by the leading societies for statistics and probability) I wrote:
The (wrong) intuitive answer 50-50 is often supported by saying that the host has not provided any new information by opening a door and revealing a goat since the contestant knows in advance that at least one of the other two doors hides a goat, and that the host will open one of such doors. The contestant merely gets to know the identity of one of those two. How can this non-information change the fact the remaining doors are equally likely to hide the car?
However, precisely the same reasoning can be used against this answer: if indeed the host's action does not give away information about what is behind the closed doors, how can his action increase the winning chances for the door first chosen from 1 in 3 to 1 in 2? The paradox is that while initially doors 1 and 2 were equally likely to hide the car, after the player has chosen door 1 and the host has opened door 3, door 2 is twice as likely as door 1 to hide the car. The paradox (apparent, but not actual, contradiction) holds because it is equally true that initially door 1 had chance 1/3 to hide the car, while after the player has chosen door 1 and the host has opened door 3, door 1 still has chance 1/3 to hide the car.
I tried to compose an unopinionated encyclopaedia article for myself and for StatProb, there is a draft at [1] (stealing some of the citizendium text!). Richard D. Gill 09:46, 11 February 2011 (UTC)
I like Boris' remark about the initial probability of 1/3 on Door 3 jumping to somewhere else. And bringing in the so-called "Monty-fall" variant: Monty accidentally trips and accidentally knocks door 3 open and it happens to reveal a goat. The probability of 1/3 jumps to doors 1 and 2 in equal parts. Bayes' rule is the good way to assimilate the mathematics of Bayes' theorem into intuitive/informal/subconscious reasoning! But maybe we need a subpage with variants. Richard D. Gill 10:02, 11 February 2011 (UTC)
Yes... Again, "...not determined by the situation alone but also by what is known about the development that led to this situation", which is for the the ultimate message of all that. --Boris Tsirelson 11:18, 11 February 2011 (UTC)
The "Or putting it differently..." was taken from the argument
The (wrong) intuitive answer "50-50" is often supported by remarking that the host has not given the contestant any information that they did not have before: the contestant knows in advance that one of the other two doors hides a goat. However, precisely the same argument could be given against this answer: if indeed the host's action does not give the contestant any information about where the car is hidden, how could it be that opening another door and revealing a goat makes the chance that the contestant's initial choice is correct change from 1 in 3 to 50-50?
put before the brief (correct) argument. For me this seemed to be confusing because I could not see how "not giving any new information" was meant to support the 50-50 answer while it clearly is an argument for the one third probability.
When rewriting it, I tried to avoid the term "probability" and used "number of cases" instead.
Boris, you say that Monty knowing the right door makes a difference: Isn't this only the case if the candidate knows that the host has to take chances? And doesn't the (implicit and general) assumption -- that the same happens in every show -- excludes this, anyway?
--Peter Schmitt 15:40, 11 February 2011 (UTC)
Yes, of course, your argument is correct. But maybe some readers will ask themselves such questions; should they seek the talk page for (hints toward) answers? --Boris Tsirelson 16:19, 12 February 2011 (UTC)
I think there should be subpages about these issues. The unconditional probability is 2/3 that switching will give you the car, if and only if we know that the host will certainly open a door and reveal a goat (because he knows where the car is hidden). How the host chooses the door to open, when he has a choice is irrelevant. That the conditional probability is 2/3 that switching will give you the goat given also which door was opened by the host, only holds when the host is moreover equally likely to open either door when he has a choice. What does "equally likely" mean? If you only use probability in a frequentist sense, it means that you know that the host makes his decision by tossing a fair coin. If you use probability in a subjectivist sense, then equally likely means that for you it is equally likely because you have no reason whatsoever to put more money on one door than on another. It means that your knowledge is invariant under relabelling of the door numbers. Someone on wikipedia said very wisely, no one who thinks deeply about MHP can avoid wondering what probability means. And the truth is that there are is no concensus on that. Richard D. Gill 21:28, 12 February 2011 (UTC)
The question what probability "really" is, is a philosophical question, I would say. If we describe probabilites using numbers then we have to use a mathematical definition, and have to argue in the bounds of the definition chosen. --Peter Schmitt 00:07, 13 February 2011 (UTC)
There is no mathematical definition of what probability is. Geometry doesn't define what a point and a line is. It just lists some relationships between them (the axioms) and shows what can be concluded from those axioms. If we want to link mathematics to the real world then we have to draw up a list of correspondences between the abstract mathematical objects of the theory, and things or concepts in the real world. So if you want to use probability theory to solve the three door problem - which is a problem about a game show on TV - you have to explain the bridge from the game show to the maths, and from the maths back to the game show. There are a number of common different understandings of what that bridge might be. Frequentists, subjectivists, the Laplacian definition... They all lead to the same abstract mathematical structure, so for a pure mathematician, it's of no import how the "user" wants to understand probability. However for the applied mathematician it can be of crucial import. In order to do mathematics about MHP you have to make some mathematical assumptions, build a mathematical model. How you understand probability could well influence what mathematical assumptions you want to make. It also determines what the real-life meaning is of the mathematical solution which you derive. I published an article entitled "MHP is not a probability problem - it is a challenge in mathematical modelling". Mathematical modelling is an art as well as a science. Richard D. Gill 15:29, 13 February 2011 (UTC)
Richard, you could contribute to "Theory (mathematics)"... :-) --Boris Tsirelson 15:52, 13 February 2011 (UTC)
Our views are not much different, I think. Choosing a "definition" is essentially the same as choosing a "model". And since, as you say, it is not difficult to see that the result is the same in all cases unless you assume some influence by astrology, precognition, etc. (or fraud). --Peter Schmitt 23:38, 13 February 2011 (UTC)

Underlinked

The MHP page is a Mathematics Internal Article, a Mathematics Underlinked Article, an Underlinked Article. Are there pages on mathematical puzzles and diversions, brainteasers and the like, on citizendium? Is there a workgroup for this kind of thing? Recreation? There is also a big literature in psychology and in mathematics education on MHP. Richard D. Gill 10:04, 11 February 2011 (UTC)

CZ has no good method of subject classification. (It needs to be established, though.) Currently the "Related articles" are a kind of substitute for it. Categories are only used for administrative purposes, and Workgroups are for approval. (This is likely to change.) --Peter Schmitt 15:50, 11 February 2011 (UTC)
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