[RANDOM] Monty Hall Problem

The "Monty Hall" problem is a famous probability question that stumped me when I heard it. Even though I guessed it was a trick question, I couldn't mathematically prove why either solution was correct, which I would consider as getting the question false (I don't think blind guesses should count for anything).

If you've never heard it, it's a fun one to puzzle through. I'll put the problem description below and a link to the Wikipedia page with the solution afterwards.

"Suppose you're on a game show and you're given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats [unwanted booby prizes]. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?"
-Krauss and Wang 2003:10

http://en.wikipedia.org/wiki/Monty_Hall_problem#Extended_problem_description