The Monty Hall Paradox

Scott Hardie | October 18, 2002
Here's a game show puzzle for you. Let's say you're on the show "Let's Make a Deal." Host Monty Hall presents you with three doors. One of them has a car behind it, and the other two have goats. Monty knows which one has the car, and he gives you a choice. No matter which one you choose, Monty goes to one of the other two and opens it, showing you a goat of course. He asks, "Last chance: Keep the door you chose, or switch to the other one?" Which choice gives you better odds of winning the car?

The intuitive answer is to say that they have 50/50 odds, but that's why intuition isn't always correct in statistical mathematics. It appears that since Monty is always going to eliminate one wrong answer, the others must be one right and one wrong, thus 50/50 odds.

But remember, at the start of the contest, the car has equal likelihood of being behind any of the three doors. The fact that Monty is revealing one of the wrong doors is irrelevant, since he's always going to show a wrong door. It's a trick! When you choose a door, there's a 33% chance that you've chosen the car. But there's a 66% chance that the car is behind one of the other two doors, right? So your odds are twice as good if you switch to one of the other doors (66%), instead of staying with the one you've got (33%). How do you know which door to switch to? The one that Monty hasn't opened, of course!

Sounds silly, doesn't it? The only way to know for sure is to map out the possible outcomes. Fortunately, since I'm lazy, I found this UCSD page that already did it. You can see from the grid, your odds of winning if you stick with your chosen door are 1 in 3, but your odds of winning if you switch doors are 2 in 3, or twice as much.

I find this paradox interesting because it reminds us that we sometimes think statistical factors are relevant when they are quite the opposite. I first read about this paradox in The Straight Dope, but their explanation was vague, and I didn't really get it until I read the above web page. Neat, huh?