[pulplobster]

I think the Monty Hall Problem is also extremely misleading in its formulation.

Take Wikipedia's formulation for example: 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 [but the door is not opened], 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?

What's not being explicitly said here is that the host knows what's behind the doors (this part is said) but that he also always chooses the goat. With that information, it's pretty clear that when the contestant first made the choice of a door, the probability of getting the right one was 1/3. The probability that the right door is among the other two is 2/3. NOW however, the host removes the one of (or the only) wrong option among those two doors. The thing to realize is that the host opening one of those doors does not give us ANY new information that would change the distribution. Therefore, the other door that the contestant didn't pick must have the probability 2/3.

[dclowd9901]

It's true that that's the crux of the Monty Hall problem, but it's also the understood nature of it. You just don't realize how much impact that would have on the outcome unless you actually start delving into the outcomes.