Chasing this tangent a bit -- I have never been happy with the Monty Hall problem as posed.

To me the problem is that it is posed as a one-shot question. If you were in this actual situation, how do you know that Monty is not deliberately trying to make you lose? He could, for example, have just let you open the first door you picked, revealing the goat. But he chose to ask you to switch, then maybe that is a big hint that you picked the right door the first time?

If the game is just "you will pick a door, he will reveal another door, and then you can choose to switch" then clearly the "usual" answer is correct; always switch because the only way you lost is if you guessed correctly the first time (1/3).

But if the game is "try to find the car while the host tries to make you lose" then you should never switch. His ideal behavior is that if you pick the door with the goat then he gives you the goat; if you pick the door with the car then he tries to get you to switch.