Car Goat Goat

Over the weekend my roommate (blogname SL Cool J) and I watched "21" on TV, the movie about big-brained MIT students who team up under the leadership of sneaky Kevin Spacey to become rich by counting cards at blackjack in Las Vegas. But that wasn't what intrigued me. What had me beating my brain for days was the problem that Professor Spacey put to his class, what is often referred to as the Monty Hall problem.

My reading of the problem: You're on a game show, and the host shows you 3 closed doors. Behind 2 of the doors are goats, and behind the other door is a car. You'll win the car if you pick the door that's hiding it. You pick Door #1. The host, who has a grudge against the producers, decides to help you out by narrowing your choices. He opens Door #3, which he knows has a goat behind it. So do you stick with your original choice of Door #1, or do you switch to Door #2?

In the movie, the main character student says that you should always change and pick the second door. Your chances of getting the car are 1/3 if you stay with your first choice and 2/3 if you switch. SL Cool J and I were baffled. Surely your chances would be 50-50 because you'd be choosing 1 door out of 2. I went online to find other explanations that might make more sense to me. I'd read some of them out loud, and sometimes it would click for a second, and one or the other of us would think we got it, but then the sense of it would disappear and we'd be confused again.

So I figured if I couldn't work through it logically, I'd look at it practically. I pulled out a deck of cards and took out 2 7s and an ace. I shuffled them, put them face down and had SL Cool J pick which one she thought was the ace. Then I'd look at the other 2 cards, turn over one that was a 7 and write down whether she should have stayed with her original choice or switched. She got tired of this after a while, so I did it myself, picking a card before I looked at all of them. Out of 18 tries, the original pick was the ace 6 times. A perfect 1/3. Although 18 isn't really enough to be conclusive, it was close enough to make me abandon the 50-50 position, even if I still didn't understand why. Which I didn't.

After more internet research and brain twisting, I finally see the logic of the solution, which you might think would be enough to convince me, but I still don't know that I understand it. It hasn't sunk in and settled in yet. (A little like Betsy and Tacy telling their geometry teacher that they memorized their geometry propositions without understanding them (Betsy and Joe, by Maud Hart Lovelace). You can get a lot of geometry problems right by following the rules, even if you have no idea why they work.) I have the evidence from the cards and the logic of the solution; for the rest I guess I'll have to go on faith.

This is often the way it is for me. If I don't automatically accept a principle, I'll gather evidence and try to make sense of it, and in the end, that'll be enough for me to put my faith in it.