Introducing the Monty Hall Problem

Theoretically, Bayes' Theorem is sufficient to determine one set of conditional probabilities from other probabilities (both conditional and not). In practice, it is a very difficult problem. I will eventually write about using a software tool, PyMC, to help us to solve this difficult problem, but initially, I will demonstrate applying Bayes' Theorem to some problems that we can solve analytically.

The first example I'll work on is somewhat famous: the "Monty Hall problem."

Monty Hall was the host of an American television game show. One of the games involved a single contestant who had the opportunity to select one of three doors. Behind one door was a very valuable prize; behind the other two doors were prizes like a goat or fake fingernails. After the contestant selected a door, Monty Hall then showed one of the two unselected doors and asked, "Would you like to switch?"

It seems like a simple probability problem. When I was selecting from three doors, I had a 1 out of 3 chance of selecting the prize winning door. After eliminating one, door, I now have a 1 out of 2 chance. Right?

Not quite so fast. It actually depends on our model of the behavior of the host, Monty Hall. For example, Monty Hall will never show a door with the prize. However, suppose we believe that if Monty Hall can show either door, he randomly picks one of the two other doors.

Let's label our doors A, B, and C. Further, suppose you pick door A. Before you select a door, the probability that the big prize is behind any of the three doors is 1/3. This value is the prior probability. Suppose you pick door A.

Monty Hall then opens door C and asks, "Would you like to switch?" How do the probabilities change if I switch? Here's a table with the information.

How did I every come up with that? Unfortunately, because I do not want these posts to become too long - and also, to be honest, it provides a "hook" - you'll need to come back tomorrow.