Monty Hall Problem

The Monty Hall Problem is a problem in probabilistic reasoning. It is often stated as follows.

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?
[Whitaker/vos Savant 1990]
You can read more about the Monty Hall Problem in a great Wikipedia article.
Door No. 1
Door No. 2
Door No. 3