## Tuesday, July 3, 2012

### Monty Hall

The situation that I presented in the Quickie Problem post last week is popularly known as The Monty Hall Problem, where "Monty Hall" is the name given to the host of the fictional game show in most versions. I'll repost the problem here for those who missed it:

Suppose you're on a game show and you're given the choice of three doors (and will win what is behind the chosen door). Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one at random. After the host opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

The correct answer is that it's to your advantage to switch doors (that is if you prefer a car over a goat), as was stated by a couple of readers in the post, although it's very hard to understand and accept why (even with the "solution" given by the movie "21"). There are more than a few ways to arrive at the correct answer, but from my research this short video clip presents the simplest and most understandable explanation.

Remember that the key to the solution--the detail that makes switching the best move--is that the host knows where the car is and always chooses a door with a goat to open.

The best thing about the videos, though, are the comments, which are indisputable proof that trolls and idiots abound online.