The Monty Hall Problem

Suppose you're on a game show, and you are given the choice of three doors: behind one door is a car; behind the other two are goats. Naturally your goal is to win the car, which is equally likely to be behind each door.

You pick a door (e.g. #1), and the host (who knows what's behind each door) opens another (e.g. #3) which has a goat. This means there are two doors left, one with a car and one with a goat. You are now given the opportunity to switch your choice (from #1 to #2). Is it to your advantage to do so?

Since you can't know which of the two remaining doors has the car, and since your initial pick had a chance of one-third, you might think that it does not matter. The chance is still ¹/₃ and you might as well stay with your original choice, right? Wrong! Switching actually doubles your chances to ²/₃.