## Thursday, March 4, 2010

### Probability: Behind one of these doors is a new car...

Perhaps you've all heard of the "Monty Hall" Problem? I hope this is new to a few of you. I'm writing about it today because I just learned about a twist in the controversy over it I hadn't heard before.

Monty Hall was the host of Let's Make a Deal, and would often play a game with contestants where he would show them 3 doors.

Behind one was a brand new car, and behind the other two were stinky old goats. You pick a door, and he shows you a goat behind one of the other doors. He now allows you to switch. Do you do it?  (We'll assume you prefer new cars to old goats.)

Marilyn Vos Savant wrote about it in Parade Magazine in September, 1990. She got piles of letters in response, mostly from people who disagreed with her analysis. (I'm trying to avoid giving away the answer here, so you can play with the problem yourself.) She was right, but a number of mathematicians told her she was wrong. How is that possible for such a simple little problem?! I think it's because we trust our intuition too much.

I remember reading that column when it came out, and getting the answer 'wrong'. But that's because I made an assumption that she didn't address one way or the other in her statement of the problem. I assumed the game show host would try to mislead you. To do this as a math problem, it's important to add one thing to the statement I gave above. You need to know that the host will always show another door with a goat behind it, and offer you a chance to switch.

And that is what I just read an article about. Monty Hall himself pointed it out in an interview with John Tierney in the New York Times. (It was published way back in 1991, but a discussion of probability problems this morning led me there.) That article will tell you how to solve the problem, so don't click until you're ready to see their answer.

Have fun!

1. I love to have students design an experiment to confirm their theory or to disprove a competing theory. It's such a good problem for the theoretical/experimental difference and for developing probability modeling.

With younger students you could use one of the many java/flash animations of the problem to run their experiment.

2. Here's a good explanation of it, also from the New York Times: http://www.nytimes.com/2008/04/08/science/08monty.html

3. I love this problem because even though I know what the answer is, if I haven't looked at in awhile, I forget why and have trouble figuring it out.

Looking at the link that Rick posted though, the answer may stick with me now. That is a very clear explanation.

4. Thanks for the link, Rick. It's a great presentation of the reasoning.

@Craig: I love it! How many people will say "I love this because I struggle with it"? There's a great goal in teaching math, to get students to love the parts they struggle with.