MONTY HALL PROBLEM - BOY, THIS IS NOT HARD!

This problem is based on a "Let's Make a Deal" game. The contestant is offered 3 doors; behind 1 of the doors is a car, behind the other 2 doors are donkeys. The contestant picks a door, then Monty Hall reveals what is behind one of the other doors (it is always a donkey). The contestant then is given the choice to keep his original door or pick the other door.

This problem was presented to Marilyn vos Savant, who said you should ALWAYS pick the other door. This doubles your chances of winning. She received over 10,000 letters (1000 of them allegedly from PhD's), all claiming she was wrong. She was not.

The Wikipedia page on this problem gives a complex explanation, and there have been formal mathematical proofs, but the reasoning is actually very simple, and does not require a formal proof. How does it work?

First of all, when Monty Hall opens a door to reveal a donkey, he is giving you NO NEW INFORMATION - you know he is going to reveal a donkey. Now, when you start the game and first pick a door, there is a 1/3 chance you have picked correctly. If you stick with that choice, you have a 1/3 chance of winning the car. Period.

Conversely, when you start the game, and first pick a door, you have a 2/3 chance of picking the wrong door (i.e a door with a donkey).

So, 2/3 of the time you have picked the donkey door, and the other donkey door is shown to you, so the remaining door must have the car. So pick the other door, and 2/3 of the time you will win the car.