The Monty Hall Problem

There are several different valid specifications of the Monty Hall problem. All use Bayes' theorem to incorporate the initial choice of the contestant and the information that the host "gives away" to reach the same conclusion.

This link provides a concise explanation of the set-up and the math.

The Wikipedia article for the problem is long, but contains a very good introduction, as well as some interesting history on the problem, and a detailed list of solutions.

Finally, there's a New York Times interview with Monty Hall himself here that describes some of the intricacies of the actual game show.