"Let's Make A Deal"

There's 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 game show host) what door you want to open, he stops and says, "Okay, you've made your choice. 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?

Logically speaking, this seems obvious to anyone: you can switch if you want to, but it makes no difference. Monty has just eliminated one of your choices. Now you're down to two. You didn't know what was behind the doors before, and by opening one of them, you still don't know what's behind the other two. Your odds are 50/50 no matter which door you choose. So, switch or don't, it makes no difference.

Perfectly sensible, right? Absolutely. But it's dead wrong.

The surprising answer is that you should switch doors.

You are probably reading this and nodding--and then you're thinking to yourself, "No, that's not right. There really is no reason to switch. You can if you want to, or if you think there's cheating going on, but in a pure game, it's 50/50! You've gotten rid of one of the doors is all!"

But if you try it empirically, you'll find that if you stay with your original guess, you'll lose two times out of three. If you switch, you'll win two times out of three. By showing you an empty door after your first choice, Monty's given you information. Your original choice had only a 1 in 3 chance of being right. Odds were 2 to 1 that the money was behind one of the other two doors--and he just showed you which of the other two doors was empty.