The Monty Hall Problem

As you may have noticed on the rapidly evolving sidebar to your right, I've been reading, and recently finished, The Curious Incident of the Dog in the Night-Time. In the very creative novel, there were many cool asides, one of which was The Monty Hall Problem.

I was not familiar with said problem. How many of you are familiar with said problem? Raise your hands. Keep 'em up.. I count one. Cool. For the other two readers, here's the dealio:

You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and there are goats behind the other two doors. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind the doors). Then he says that you have one final chance to change your mind before the doors are opened and you get a car or a goat. So he asks you if you want to change your mind and pick the other unopened door instead. What should you do?

(Curious Incident... pp 78-9.)

Well, it doesn't matter right? At that point it's a 50/50 shot that you'll get the car........... right? That's what intuition will tell you.

Marilyn vos Savant, "an American magazine columnist, author, lecturer, and playwright who rose to fame through her listing in the Guinness Book of World Records under 'Highest IQ'" (wikipedia), in response to that question, answered differently. She wrote that you should always change your pick because there is a 2 in 3 chance that the car is behind the other door. Say? She received thousands of letters scolding her, including some harsh words from mathematicians and scientists.

But she was right, and explained it in a follow-up column. There are three doors to choose from, so there are three possibilities:

1. You choose a door with a goat behind it.
2. You choose a door with a goat behind it.
3. You choose a door with a car behind it.

For 1:

• You stay, you get a goat.
• You change, you get a car.

For 2:

• You stay, you get a goat.
• You change, you get a car.

For 3:

• You stay, you get a car.
• You change, you get a goat.

So, changing will get you a car 2 times out of 3. And staying will get you a car 1 time out of 3.

Cool, eh?

In the novel, the narrator concludes this aside by saying, "And this shows that intuition can sometimes get things wrong. And intuition is what people use in life to make decisions. But logic can help you work out the right answer".

For more on The Monty Hall Problem, click here.