CMSC-203 Discrete Math: Special Problem-Monte Hall (spring 2001)

In the once popular TV game show Let's Make a Deal, at the end of the show the winning contestant would play the following game. Host Monte Hall would show the contestant three closed curtains on stage. Behind one curtain there is a grand prize of considerable value (e.g. a new sports car); behind each of the other two there is a booby prize of essentially no value (e.g. an old donkey). The contestant wins the prize she selects. Monte Hall knows where the grand prize is; the contestant does not. Let's call the curtains A, B, and C.

First, the contestant selects one curtain (let's say B). Second, Monte Hall reveals one of the unselected curtains (say C) that has a booby prize. Because there are two booby prizes, and because Monte Hall knows where they are, Monte Hall is always able to reveal a booby prize. Third, Monte Hall asks the contestant if she would like to stick with her original selection (B) or switch to the remaining unopened curtain (A). What should the contestant do?

In the TV show, great excitement builds as the audience shouts out advice to the contestant. Also, sometimes Monte Hall might offer cash to the contestant to influenece her decision (e.g. Monte Hall might offer the contestant $1,000 in cash to switch). [Please ignore these cash incentives in your analysis of this problem.]

The main question is whether it is better to stick (with the original selection) or switch. There are three possible answers: (a) it is always better to stick, (b) it is always better to switch, or (c) it doesn't matter because sticking and switching each have an equal 50% probability of success.

1. After first reading this problem, what was your initial intuition as to what the contestant should do? (stick, switch, it doesn't matter)

2. Initially, before the game begins, what is the contestant's a priori chance of correctly selecting the grand prize on her first guess?

3. With cups and a penny, play this game forty times with a friend. Twenty times use the stick strategy; twenty times use the switch strategy. How many times did the stick strategy win? How many times did the switch strategy win? On the basis of your experiments, estimate the success probability for each strategy. Are your results statistically valid? What factors might have adversely influenced your data? Do you still believe in your initial intuition?

4. Begin to analyze the problem mathematically by identifying a sample space. Also, draw a complete tree of possible event outcomes.

5. What is the contestant's best strategy? Justify your answer.

6. Using equations of conditional probability, calculate the probability of success for each of the stick and switch strategies. How well does your experimental data match these theoretical success probabilities?

7. Harry argues: "It doesn't matter whether the contestant sticks or switches. There are two curtains and the grand prize is behind exactly one of them. Therefore, the chance of success is 50-50 whether the contestant sticks or switches." Is Harry's conclusion correct? Is Harry's logic sound? Explain.

8. Suppose Larry turns on his TV late into the show, just after Monte Hall revealed a booby prize. Larry does not know which curtain the contestant initially selected. What is Larry's best strategy? Explain.

9. What have you learned from this problem?