3 ms·

I'm gonna pile on the hundreds of comments already existing and say that the real key to the Monty Hall problem isn't about pure probability, it's about human i

by knzhou 7y ago

I'm gonna pile on the hundreds of comments already existing and say that the real key to the Monty Hall problem isn't about pure probability, it's about human interaction. The answer can be either 1/2 or 1/3 depending on Monty's motivations, his knowledge, and the way he decides what information to give you. This is true for Bayesian updating on any information, and is a good lesson for life in general.

dunmalg 7y ago
>depending on Monty's motivations, his knowledge, and the way he decides what information to give you. The missing piece of the puzzle that seems to frequently be left out is we KNOW Monty's motivations, knowledge, and how he decides. At the time the problem was formulated, when Let's Make a Deal was still on television, all this was common knowledge: Monty is the host of a game show, he knows where the two goats and the car are, and he decides which door to open by picking one of the unchosen doors he KNOWS has a goat behind it. The problem suffers now, 40-odd years later, by the fact that people often don't know who Monty Hall IS, and the problem is usually worded as if it's still 1963-1976, so this foundational knowledge is missing. There is no QUESTION what Monty Hall knew, or what his motivation on the show was... except people who don't know the game show don't know that. Granted, the actual game show was a bit more varied, but the presumption in the thought experiment is that Monty ISN'T trying to goad you into un-choosing the correct door, which he frequently did on the show because the tragedy of it was more entertaining than just giving away a car.
- knzhou 7y ago
  Sure, if you fix the motivations it's easy, but that's missing the point of the problem -- the only reason it generates so much disagreement between smart people is that people make different implicit assumptions about the motivations. And that's the key to many other probability paradoxes as well.
  - smohare 7y ago
    This subtlety, to me, only comes into play when you cast it in a Bayesian context, because then you are forced to consider the probability that Monty reveals a goat. But the approach fundamentally breaks down because you can’t actually compute this, due to lack of any data. All it suggests to me is that this line of reasoning is insufficient, not that Monty’s motivations and knowledge actually matter.
smohare 7y ago
The probability only changes if Monty can remove a prize door and does not reveal a goat. Monty’s motivations and knowledge are completely irrelevant in the context of the original problem, because his actions are effectively forced (he reveals a goat). If he can remove an unchosen prize door with some unknown probability p and does not reveal the contents of the removed door, then the probability of getting the prize by switching is simply (2/3)(1-p). His motivations and knowledge just amounts to some distribution of p, but that’s a rather different problem. When the expected value of p is 1/2, then switching is equally likely to win you the prize as your original choice, but the probability of winning is still only 1/3 in either case. Only when p=1/4 will the remaining door contain the prize with probability 1/2.
- knzhou 7y ago
  This is exactly demonstrating my point. When smart people disagree with you over the answer, it's not because they're misunderstanding the question -- it's because they are not aware that Monty must reveal a goat, which is almost never properly specified in the problem statement. It's usually stated as "you pick a door and then Monty shows a goat" which does have an ambiguous answer.

by knzhou 7y ago

dunmalg 7y ago

>depending on Monty's motivations, his knowledge, and the way he decides what information to give you. The missing piece of the puzzle that seems to frequently be left out is we KNOW Monty's motivations, knowledge, and how he decides. At the time the problem was formulated, when Let's Make a Deal was still on television, all this was common knowledge: Monty is the host of a game show, he knows where the two goats and the car are, and he decides which door to open by picking one of the unchosen doors he KNOWS has a goat behind it. The problem suffers now, 40-odd years later, by the fact that people often don't know who Monty Hall IS, and the problem is usually worded as if it's still 1963-1976, so this foundational knowledge is missing. There is no QUESTION what Monty Hall knew, or what his motivation on the show was... except people who don't know the game show don't know that. Granted, the actual game show was a bit more varied, but the presumption in the thought experiment is that Monty ISN'T trying to goad you into un-choosing the correct door, which he frequently did on the show because the tragedy of it was more entertaining than just giving away a car.

knzhou 7y ago
Sure, if you fix the motivations it's easy, but that's missing the point of the problem -- the only reason it generates so much disagreement between smart people is that people make different implicit assumptions about the motivations. And that's the key to many other probability paradoxes as well.
- smohare 7y ago
  This subtlety, to me, only comes into play when you cast it in a Bayesian context, because then you are forced to consider the probability that Monty reveals a goat. But the approach fundamentally breaks down because you can’t actually compute this, due to lack of any data. All it suggests to me is that this line of reasoning is insufficient, not that Monty’s motivations and knowledge actually matter.

smohare 7y ago

The probability only changes if Monty can remove a prize door and does not reveal a goat. Monty’s motivations and knowledge are completely irrelevant in the context of the original problem, because his actions are effectively forced (he reveals a goat). If he can remove an unchosen prize door with some unknown probability p and does not reveal the contents of the removed door, then the probability of getting the prize by switching is simply (2/3)(1-p). His motivations and knowledge just amounts to some distribution of p, but that’s a rather different problem. When the expected value of p is 1/2, then switching is equally likely to win you the prize as your original choice, but the probability of winning is still only 1/3 in either case. Only when p=1/4 will the remaining door contain the prize with probability 1/2.

knzhou 7y ago
This is exactly demonstrating my point. When smart people disagree with you over the answer, it's not because they're misunderstanding the question -- it's because they are not aware that Monty must reveal a goat, which is almost never properly specified in the problem statement. It's usually stated as "you pick a door and then Monty shows a goat" which does have an ambiguous answer.