## Tuesday, September 17, 2013

### Monty Hall Problem

I discussed this problem a while ago, but as it is one of my favorite probability problems I felt it deserved a revisit.  What makes the problem great, is that nearly everyone (myself included) gets it wrong at first.  Wikipedia even claims: "Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation confirming the predicted result (Vazsonyi 1999)."
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

This is the article that brought the problem to a large audience.  She published many of the responses, which are almost all claiming she is wrong.

http://marilynvossavant.com/game-show-problem/

http://en.wikipedia.org/wiki/Monty_Hall_problem