The Monty Hall Problem

This is probably one of the most notorious problems in puzzledom. It is loosely based on the game show Let's Make a Deal, hosted by Monty Hall. The problem is typically 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 and the host, who knows what's behind the doors, opens another door which always has a goat. He then says to you, "Do you want to stay with your original choice or switch to the remaining door?" Is it to your advantage to switch your choice?