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. That, "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)

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)

(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)