The Monty Hall Problem

The Monty Hall problem appeared on the movie 21, a film from Columbia pictures inspired by the story of the MIT Blackjack Team. During an advanced class, Professor Micky Rosa, played by Kevin Spacey, challenged Ben Campbell, played by Jim Sturgess, with the Monty Hall paradox which Campbell solves successfully. This meeting eventually led to Campbell joining the blackjack team headed by Rosa. Together with four other MIT students, the team went to sweep Vegas' casinos by storm using card counting. Below is a version of the Monty Hall problem as it appears on the book, Introduction to Mathematical Statistics (6th ed.) by Hogg, McKean and Craig.

Suppose that there are three curtains. Behind one of the curtains is a nice prize while behind the other two are worthless prizes. A contestant selects one curtain at random, and then Monty Hall opens one of the other two curtains to reveal a worthless prize. Hall then expresses the willingness to trade the curtain the contestant has chosen for the other curtain that has not been opened. Should the contestant switch curtains or stick with the one she has? What is the probability that he wins the prize if he switches?

To answer the first question, we only need to know the probability that the contestant wins the prize should he change curtains. Using the Law of Total Probability, this is given to be 2/3. This means that the odds in favor of the event, winning after switching curtains, is 2:1. Thus, it is advisable that the contestant switches.