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 probability of winning by switching, and the other focussing on the conditional probability of switching given the specific doors chosen and opened. The present draft intro contains both. Richard D. Gill 23:26, 1 February 2011 (UTC)

Edit

I changed the following sentences from the intro:

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.

The previous solution used a frequentist picture: probability refers to relative frequency in many repetitions. Also, it didn't address the issue of whether the specific door opened by the host is relevant. Could it be that the decision to switch should depend on whether the host opens Door 2 or Door 3?

into:

The previous solution used a frequentist picture: probability refers to relative frequency in many repetitions. Also, it didn't address the issue of whether the specific door opened by the host is relevant. Could it be that the decision to switch should depend on whether the host opens Door 2 or Door 3?

One could say that in general a contestant, who initially chooses Door 1, is offered a choice between his initial choice Door 1, or Doors 2 and 3 together. However the contestant in a specific issue of the game show, who initially chooses Door 1, also sees an opened door, in the problem as an example this is Door 3.

This was reverted by Garry. Any opinion of other editors? Wietze Nijdam 21:49, 1 February 2011 (UTC)

My original draft was shorter and, I think, more neutral. First an executive summary of the preceding (one door versus two). Then an intro to a more detailed analysis, which makes a further assumption - neutrality of expectations w.r.t. the host's choice - and which argues that the earlier result, probability of winning by switching is 2/3, doesn't depend on *which* door was opened by the host. Somewhere else in the article both arguments can be written out in formal probability language for the benefit of students of probability. I think that such readers are the only ones who need to bother about the nicety of whether we should be determining an unconditional or a conditional probability. So in the intro we shouldn't make heavy weather of it. Let the reader who's able and interested to appreciate the subtlties make their own mind up, what they think about them.

The introductory sections need to be accessible to all, and need to concentrate on the "paradox" (a paradox is an apparent contradiction which vanishes on closer inspection) that the result is "switch" not "you might as well stay".

I did my best though to include in such an introductory section a purely verbal/logical version of the "conditional" result! I think that's a major (collective) achievement, the result of years' discussions on Wikipedia.

Also I deliberately drew attention to the possibility of there being different ways to think of probability. Hopefully not in an obtrusive way, but just enough to show that this can also be a matter of debate. And in order to accommodate readers of different persuasions. And to hint at the issue that your probabilistic assumptions will be tied to your interpretation thereof. Richard D. Gill 22:25, 1 February 2011 (UTC)

If the problem is formulated in such a way that the contestant is offered to switch after the host has opened the goat door, the simple solution, the one you present first, is not adequate. And the sentence: 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. has no bearing. The problem with such presentation is that some readers might get the wrong idea about the problem and its solution. That's why I want to make this clear from the start. Wietze Nijdam 10:19, 2 February 2011 (UTC)

Sorry Wietze, but in my opinion what you say is "just" your opinion, not a universal truth. We two disagree, right? And both of us have thought a long time about it. Please try contributing to the many other sections which need to be written. And please let's make it fun, let's make it rewarding, not confrontational, for other authors, to join in too. Already this talk page has lost all structure and focus. As @Peter Schmitt wrote, there is a whole load of serious work to be done. Let's reserve the Wikipedia talk pages for the Never Ending Discussion (which reminded one "mediator" of an elderly couple bickering because they have got so addicted to it).

You could also consider writing some good material on probability, Bayes, conditioning. That's the clever way to support your point of view. Just repeating a dogma is the worse way to convince other people.Richard D. Gill 10:35, 2 February 2011 (UTC)

Well Richard, you said yourself on Wikipedia that the simple solution does not solve the conditional formulation. So it is not "just" my opinion, it is yours as well, don't you remember? Let's see what other authors think about it. Wietze Nijdam 11:00, 2 February 2011 (UTC)

I also wrote extensively that I don't think that the conditional formulation is the only legitimate formulation. Richard D. Gill 15:07, 2 February 2011 (UTC)