Read the Recreational Mathematics box on page 702 regarding the Monty Hall problem. Do research and write a paper on Bayes’ theorem. Include a discussion of how Bayes’ theorem can be used to solve the Monty Hall problem.

Data From Recreational Mathematics:

The Monty Hall problem is a probability problem based on the television game show Let’s Make a Deal and is named after the original host, Monty Hall. The problem became famous in 1990 in Marilyn vos Savant’s Ask Marilyn column in Parade magazine. The problem can be stated as follows. You are a contestant on Let’s Make a Deal,’ and there are three doors to choose from: A, B, and C. Behind one door is a new car, and behind two doors are goats. You choose door A. Monty then reveals that behind door B is a goat. He then asks if you’d like to stay with your original choice, door A, or change your choice to door C. What should you do, stay with your original choice or change your choice?

The calculations of the probabilities involved in this problem involve a formula called Bayes’ theorem. Although the calculation of these probabilities is beyond our level of discussion, there are several websites and smartphone apps that can simulate the Monty Hall problem. It can be shown, using Bayes’ theorem, that the probability of winning the new car if you stay with your original choice is 1/3. However, the probability of winning the new car if you change your choice is 2/3! While these probabilities may be surprising, long term repeated trials of the problem using computer simulations confirm these results.