Brain Teaser

You find yourself on a game show called "Let's Make A Deal." The game is very simple. There are three doors: door #1, door #2, and door #3. Behind one door is a million dollars. The other two doors contain worthless joke prizes. All you have to do is pick which door you want to open, and you get whatever is behind it. But you only get to open one door. By simple math, then, you obviously have a 1 in 3 chance of picking the correct door and becoming an instant millionaire.

You pick a door. As soon as you tell Monty (the gameshow host) what door you want to open, he stops and says, "Okay, you've made your choice. Now, I'm going to do what we always do here on this game. I'm going to open one of the other two doors for you that I know has a booby prize." And he does so. Then he asks, "Okay, now, would you like to stay with your original guess, or would you like to switch to the other door that's still closed? You only get one shot, so do you want to stay with your original choice, or switch?"

Here's the question: is there any compelling reason to switch doors?

To be clear, there is no trickery, and Monty is not cheating. Furthermore, the money has not moved, will not be moved, and if you open the right door, you win the cash. Money is either behind the door you first picked, or behind the remaining unopened door.

Should you switch?