[jncfhnb]

If he only opens a door when you picked the car, then you will lose 100% of the time by switching when he shows a goat.

I instead love when people get uppity about others being wrong about Monty hall while still being wrong about Monty hall!

[bemusedthrow75]

It doesn't matter what he only does, or always does.

It matters what he did in the problem as it is described. Because that is what you're solving.

[jncfhnb]

The problem as described does not specify his behavior. Only what happened. If all you know is that he opened a goat door you have learned nothing.

It could be that he always opens a goat door meaning you should switch.

It could be that he only opens a goat door if you picked the car and you should not.

It could be he only opens a goat door if you picked a goat and you should.

Probability cannot determine which of these is any more likely than the other. You have learned nothing.

[bemusedthrow75]

"If all you know is that he opened a goat door you have learned nothing."

This is precisely where you are wrong.

[jncfhnb]

Ok let’s play!

We play this game 1000 times.

33% of the time you pick the car and I show you a goat. You switch and lose 100% of the time.

66% of the time you pick a goat and I don’t show you a goat. Assuming you don’t switch because you believe this means you have an equal odd on your current door, you lose 100% of the time.

Congrats. You have lost every single round.

[bemusedthrow75]

You've restated the problem (incorrectly) -- changed it.

There's still a goat you could show me. And it is a fact that you show me a goat. Nowhere in the problem does it suggest there is a chance you show me a goat.

I do honestly admire your dogged commitment, and I think the way you are committed shows up an important point about the article and the history of the problem.

Which is that one can quite clearly fairly argue the point, as you are doing, without resorting to misogynistic or patronising rudeness as so many did at the time!

But you're still wrong. :-)

And I'm going to leave it here.

[rspeele]

I am curious how you feel about the "Monty Fall"/"Monty Crawl" problems linked elsewhere in the thread.[0]

For my part I am somewhat sympathetic to jncfhnb's point. The exact phrasing that Vos Savant was asked did not specify the rules that the host was required to open a door, nor that it always must be a goat door. It simply says that the host has knowledge of what's behind the doors, and in this particular iteration of the game, he showed you a goat and asked you about switching.

That does not exclude a scenario where the host is a manipulative fellow, who chose to show you the goat only because he knows you are about to win, trying to convince you to lose. A contestant on the real show would surely worry about this possibility.

Of course, the people who wrote to disagree with Vos Savant almost never said "the problem is not fully stated", they said "it's 50/50 you fool", which isn't right. Additionally, since it wouldn't be a math problem at all if we let the host have agency, it is reasonable to assume the unspoken rule that he does not, leading to Marilyn's correct 2/3rds answer.

[0]https://web.archive.org/web/20230706235720/https://probabili...

[jncfhnb]

I have not restated the problem.

You’ve incorrectly assumed that I would be showing you a goat in every case. But that is not included in the prompt. All you know is that you were shown a goat on one play. This is possible in the setup I’ve listed. You landed in that 33% case where you chose a car.

There is nothing in the prompt that says we are not playing my variant of the game.

Being told that you SAW a goat does not mean you would ALWAYS SEE a goat if the previous conditions had gone differently. And that is why you’re wrong :)

Again > all you know is that he opened a goat door