>Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
There's actually some ambiguity here (I'm sorry, I'm a really pendantic person). What it doesn't say is that he always picks the door with the goat. What it says is that in a single trial, he picks the door with a goat. I assumed that in half the trials, he'd pick the door without a goat. I see now that was a wrong (and dumb) assumption.
This is why, rather than reading text, I think people should describe these problems with code. Unambiguous code. With test cases. Then everybody can inspect the unambiguous code rather than having to parse human text.
Since there's ambiguity, you should assign prior probabilities to the two types of behavior Monty can have. Then the overall probabilities are the average of the probabilities for each of the possible cases, weighted by this prior. So of you think there's 1/2 chance Monty always reveals a goat (in which case swiching wins 2/3 of the time) and a 1/2 chance Monty picks randomly (in which case switching wins 1/2 the time), then your overall chance of winning if you switch is 1/2×2/3 + 1/2×1/2 = 7/12. So you should still switch.
Indeed you should switch if you think there's any chance at all that Monty always opens doors with goats. (Unless of course you think that there's also a chance that Monty always reveals cars whenever he can, but that wouldn't make for very good TV.)
We are told that the host knows what's behind the other doors, which implies that the door he opens is deliberately chosen and is not random.
Additionally, to open the door with the prize would terminate the game, so to open the door with the goat is the only action that makes sense given what we know.
I agree, those are reasonable interpretations made from somebody who has done a close reading of the text.
Or you know, somebody could just write a computer program that described the rules unambiguously so everybody could inspect them and not have to make reasonable interpretations.
So, in retrospect that's obvious, but it's also not stated. I just assumed half the time he opened a prize door (I had watched the TV show a few times and... they did weird shit).
