I found this very entertaining and counterintuitive (the Monty Hall problem):
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?
See the extended entry for the answer plus caveats:
The answer is YES. Here is a good explanation of why. It’s a good tip that it helps to think of more doors with just one car behind.
http://encyclopedia.laborlawtalk.com/Monty_Hall_problem
But note that there are some caveats, which are described here:
The basic point is that only if the gameshow host INTENDS to choose the goats and you know that, does the solution hold. If he is chosing at random from the remaining doors, and he happens to choose the goats, then the answer is wrong.
Some analysis here:
http://www.wiskit.com/marilyn/gameshow.html
http://www.inference.phy.cam.ac.uk/itprnn/book.pdf P.57