## Intuitive explanation of the Monty Hall problem

Yesterday I went to see 21 where one of the scenes brings up the Monty Hall problem: there are three doors behind which there are a car and two goats. You choose a door with the goal of winning the car. Then the host opens one of the remaining doors which hides a goat. The question is whether it is to your advantage to switch your choice to the other door. The answer is yes (and yes, I thought it does not matter while watching the movie). When asked to explain why it is a good idea to change the door the main character utters some gibberish about all the variables being changed, etc. Afterwards I checked this problem out on Wikipedia (follow the link above) which gave a few strict proofs making it clear that indeed changing the door increases your chances of winning the car by 1/3. While the formal proofs do their jobs just fine I always prefer to have an intuitive feeling of why a seemingly counter-intuitive answer is actually correct. In this case I wanted to understand what changes once the host opens one of the doors, what extra information is added that makes the difference.

The part of the rule which says that the host has to reveal the other goat brings in the extra information. This happens in the case when you initially selected the door with a goat behind it. In this situation the host is forced to eliminate the other goat: he cannot open the door you have selected and he cannot reveal where the car is. In other words we have two possible outcomes:

- if you selected a goat then the remaining door hides the car
- if you selected the car then the remaining door hides a goat

The probability of initially selecting the goat is 2/3 (two doors out of three hide goats) and the car — 1/3. Thus it is more likely that you will first select a goat instead of the car. And in this more likely case the host is forced to single-out the door which hides the car. Thus changing your selection gives you a better chance of winning the car.

Note also that the probability of your initial choice being the car remains 1/3 even after the host opened one of the doors. It is the probability of the other remaining door hiding the car that has changed (from 1/3 to 2/3) due to the rules of the game forcing the host not to reveal the car.

June 17th, 2008 at 4:06 am

The way I think of it is that you are asked to divide the doors up into 2 groups. One of one door, one of two doors. What are the probabilities of the prize being in each group? It’s easy to say 33% and 66%. So now only thing to do is pick the group with the 2 doors.

Due to the mechanics of the game you are initially forced in the the group with one door (your choice). You go back to the 2 door group by switching.

The information about the goat is essentially irrelevant, and there just to make the drama of the game, and to fool you into thinking it’s now 50:50.