# More on the Monty Hall Problem

Once again, here's the Monty Hall problem:

Monty Hall offers you a choice of three doors.  Behind two are goats, and behind the other door is a brand new car.  After you make your choice, Monty opens one of the doors you didn’t choose to reveal a goat, and offers you the chance to switch your choice to the one remaining.  What should you do?  (Assume Monty knows where the car is and always opens a door with a goat after the contestant chooses.)

1. Switch
2. Stay
3. It doesn't matter.

Many people think that once Monty opens a door to reveal a goat, that with only two doors remaining, their odds of winning the car are 50-50.  That explains why many people think it doesn't matter if you stay or switch.  Some people think that perhaps Monty is trying to talk them out of sticking with their original door because it contains the car.  These people are likely to think that staying with their first choice gives them better than 50-50 odds.

However, the correct way to think about this problem is that you had a 1/3 chance of guessing right on your first try. After Monty opens a door and reveals a goat, there's still a 1/3 chance that you were right, which means there's still a 2/3 chance that you were wrong. Before Monty opened that door, that 2/3 chance that the car was not behind your door was split between two doors. Now, however, there's a 2/3 chance that the car is behind the remaining door that you didn't choose. So you should switch. On average, you would expect to win a car 2/3 of the time if you switch.

Here's the tree diagram I showed briefly in class. It helps to model this random process in something of a chronological order. First, the car is placed behind one of the three doors randomly, with each door being equally likely.  Then, you select your door, again with each door being equally likely. Then, Monty opens one of the other doors to reveal a goat. (It's important that Monty knows where the goats are here! Otherwise you get a different tree diagram.) When you look at the outcomes, not all are equally likely because Monty's choices are limited in some cases. Of all the outcomes, weighted by their respective probabilities, your choice of door was correct only 1/3 of the time. Thus, switching doors is the better strategy.