Variable Change

The movie 21 was on cable this morning, and I caught the ‘variable change’ scene. In the scene, the professor poses a scenario to the class, calling it “The Game Show Host Problem,” which goes as follows:

Suppose you’re on a game show. And you are given a chance to chose from three different doors. Behind one of the doors is a new car, behind the other two, goats. Which door would you chose?

A student chooses door #1.

Now, the game show host, who, by the way, knows what’s behind all the doors, decides to open another door. Let’s say he chooses door #3, behind which sits a goat. Now the game show host comes up to you, asking if you want to stay with door #1 or go with door #2.

Is it in your interest to switch your choice? The student says, yeah, going on to explain his answer to switch is based on statistics and variable change.

Most people think this is just movie fiction, but it’s not. The easiest way to understand this is to realize your choice has a two-out-of-three chance of being a goat (loss). When this is the case (you chose the goat), once the other goat is revealed by the host, you know where the car is, so switching gets you the correct door two-thirds of the time.