A popular probability problem refers to a once popular game show called "Let's Make a Deal." In this game, the host (named Monty Hall) hands out large prizes to contestants for no reason at all. In one situation, Monty would show the contestant three doors, named door 1, door 2, and door 3. One would hide a new car, one $500 worth of false eyelashes, and the other a goat (deemed worthless by the purveyors of the show). The contestant picks door 1. But instead of showing her the prize, Monty opens door 3 to reveal the goat.
1. If the car is really behind door 1, what happens if she switches?
2. If the car is really behind door 2, what happens if she switches?
3. Should the contestant switch her guess to door 2?
4. If she uses the right strategy, what is her probability of getting the new car?
5. It is later revealed that Monty does not always show what is behind one of the other doors, but does so only when the contestant guessed right in the first place (the so-called "Machiavellian Monty"). How often would a contestant who used the strategy in Exercise 47 get the new car?
6. What is the right strategy to use for dealing with the Machiavellian Monty? How well would the contestant do?