Talk:Monty Hall problem

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 that this door or one of those doors as the case may be. 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)