Talk:Monty Hall problem

Fallacy

Today I find this in the article of which this is the Talk page:

QUOTE: What is the probability of winning the car by always switching? What is the probability of winning the car by switching given the player has picked door 1 and the host has opened door 3? UNQUOTE

But those are not remotely equivalent questions. The Monty Hall Problem is the computation of your TOTAL odds of winning by an "always switch" strategy over EVERY possible permutation of the game (each permutation being weighted in the probabilities by the likelihood that it will come up). That doesn't allow for stipulating things like "it's a given that the player first picked Door 1" and "it's a given that the host then opened Door 3". Where do those "givens" come in. For instance, it's never a given that the Contestant will pick Door 1. The odds of any one selected permutation being a win or a loss have nothing to do with the odds pertaining to the sum of all possible permutations.

The 2nd question above is about as relevant as asking "What is the probability of winning by always switching in the cases where switching causes the Contestant to lose?" Well, yes, they're 0%, but you've limited the test to cases where the Contestant loses by switching, so, why would it NOT be 0%?~2026-20619-07 (talk) 20:48, 3 April 2026 (UTC)Christopher Lawrence Simpson

1. "But those are not remotely equivalent questions" → Indeed; and it fact this is exactly the point of that section: "the distinction between [these questions] seems to confound many" and "the following two questions have different answers".

2. " Where do those "givens" come in." → See e.g. Law of total probability.

Malparti (talk) 01:08, 4 April 2026 (UTC)

Problem statement is not complete

The italic text giving the problem statement is not complete. In order for the MH problem to have the solution "switching wins 2/3 of the time", the rules of the game must be made clear to the contestant prior to the game starts. That is, the contestant must know that the host will always open a door, no matter the outcome of the contestant's initial guess. If the host only reveals a door when the contestant was initially correct, the a switching strategy looses with 100% probability.

This is a frequent mistake of phrasing the MH problem. The "standard assumptions sorts this out, but I think it could be given already in the formulation. — Preceding unsigned comment added by 2A00:1310:202:3013:0:DDDD:1:5 (talk) 07:42, 24 March 2025 (UTC)

Yes, this is a valid point. I have made some edits to address this problem. Specifically, there is a sharp gap between the probability problem (i.e., here is a probability puzzle) and the decision-theory problem (i.e., what should a rational contestant do), because the contestant's state of belief is not captured in the likelihood model of the problem. If the contestant doesn't know the rules, then a whole bunch of possibilities are "rational" depending on the contestant's assumptions/beliefs. Gill's paper actually sets this up in a fascinating way as a Nash equilibrium problem for an adversarial game between the contestant and host. He does ultimately arrive at the 2/3 result, through a rather different set of assumptions. (And I think just personally, his model is unconvincing from a purely decision theory perspective, but still very interesting. E.g., I don't think Gill's assumptions are any more realistic than those of the "standard" problem.) To be clear, those assumptions are different from the standard setup in important ways, and may also be different from those that the contestant has reason to believe. Sławomir Biały (talk) 09:18, 8 April 2026 (UTC)

Interactive version

Play the Monty Hall game

🚪1🚪2🚪3

Behind one door is a car; behind the other two, goats. Pick a door!

You chose door ?. Monty, who knows what is behind the doors, opens door ?, revealing a goat. Do you stay with door ? or switch to door ?? Click a door to decide.

…

Strategy	Wins	Games	Win %	Expected
Stayed	0	0	0	33%
Switched	0	0	0	67%

▶ Auto-play using:  stay  switch

0 Show where the car is, to verify the game is fair

Play again

Reset

0 0 0 0 0 0 12 0 12 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The three equally likely cases (you pick door 1; the host opens a goat door)

Car is behind	If you stay	If you switch
Door 1	Win	Lose
Door 2	Lose	Win
Door 3	Lose	Win

Staying wins in 1 of 3 cases; switching wins in 2 of 3. An interactive version of this demonstration is available with JavaScript enabled.

Hey all. On an wiki-interactivity kick and though to try to use {{calculator}} for a Monty Hall problem simulator. After many iterations and edits (adding the "transparency mode" and auto-plays and whatnot), I present it here. {{Monty Hall problem}}. Any objections to including it in the article somewhere? Recommendations for any other explanatory text to include? Disclosure: I did get Claude Code's help with calculator template syntax. — Rhododendrites ^talk \\ 02:15, 24 July 2026 (UTC)

This is very cool. Sławomir Biały (talk) 13:26, 24 July 2026 (UTC)

Good job. An on-page simulator will be a great benefit to the page. MrOllie (talk) 13:31, 24 July 2026 (UTC)

Typesetting of article

The article would look better and read more smoothly if the simple fractions were set on-line as 1/3 rather than stacked. Is there a compelling reason to leave them stacked? Paul C. Anagnostopoulos (talk) 11:25, 1 August 2026 (UTC)

I agree, that looked awful on my computer because it made the spacing between pretty much every line larger than the spacing between paragraphs.

I've tried to fix it. I assume some people are going to complain and so we'll have to look for a consensus. Malparti (talk) 16:20, 1 August 2026 (UTC)

Wikipedia:Manual_of_Style/Mathematics#Fractions is pretty clear that the {{frac}} template should not be used. It needs to either go back to the prexisting {{sfrac}} version or be fully inline, like 1/2. MrOllie (talk) 16:24, 1 August 2026 (UTC)

Thanks, I didn't know that. I've fixed this. Malparti (talk) 17:54, 1 August 2026 (UTC)