Thursday, May 04, 2006

Let's Make A Deal

So I know I owe you guys posts on other things, but I want to take a minute to talk about the counter-intuitive mathmatical phenomenon that my friends and I have been playing around with in our heads for the past couple of days. You are probably already familiar with the set-up: it was used famously by Monty Hall on the old "Let's Make a Deal" show. For those of you who didn't grow up with Nick at Night, here's the premise: A lucky contestant gets to pick among three closed doors, behind one of which is a car, and behind the other two are goats (or some equally undesirable "prize"). The contestant picks a door. Monty Hall opens one of the un-picked doors to reveal a goat. The audience cheers. Now the contestant is given a choice- do they want to switch doors? Now, most people will reason that nothing is gained by switching doors, after all, they have a 50/50 chance of getting the car. But here's the kicker and the mathmatical head-scratcher: they are actually twice as likely to get the car if they switch doors!

No, seriously. You will not want to believe this, because basic logic tells you if you are down to two doors to pick between, one of which has a car, you have a 50% chance of getting the car no matter what. But you are wrong. And I was wrong. And it took me a half dozen hand-drawn diagrams and articles to figure it out. And even now, I don't like it. But it's true. Here's the skinny:

Originally, you have a 1/3 chance of getting the right door. Which means if the car is placed randomly and you choose randomly, you will only choose correctly 33% of the time. When a non-car door is opened, your odds don't improve- there is still a 66% chance that the door you selected is a goat. So 66% of the time, it makes more sense to switch doors. Only 33% of the time does it make sense to stay with your original choice. Thus, your odds really are twice as good if you switch. Make sense? Don't worry if it doesn't at first, a bunch of us Hah-vahd kids sat around and drew it and did mock trials before we would believe it either. But if you want, try it out on your spouse/co-workers/kids: Do it several times (at least 9). See if it doesn't come out this way.

An important caveat: This only works because Monty Hall knows what door the car is behind, so of course he isn't going to open that door, even though you didn't select it originally. If it was a random opening, and the car was not revealed behind an un-picked door, your odds are 50-50. But because Monty can control which door is opened, you are twice as likely to win if you switch. Trust me on this one. It won't work for you all of the time (after all, you might have selected the right door immediately), but hey, doubling your chances can't hurt you.