The mathematics of probability

Abstract

The following decision problem appeared some years ago in Marilyn vos Savant's Ask Marilyn column in Parade Magazine . You are a contestant in a game show hosted by Monty Hall. You have to choose one of three doors, and you win whatever is behind the door you pick. Behind one door is a brand new car. Behind the other two is a goat. Monty Hall, who knows what is behind the doors, now explains the rules of the game: “First you pick a door without opening it. Then I open one of the other doors. I will always pick a door to reveal a goat. After I have shown you the goat, I will give you the opportunity to make your final decision whether to stick with your initial door, or to switch to the remaining door.” Should you accept Monty Hall's offer to switch? The Monty Hall problem has a simple and undisputable solution, which all decision theorists agree on. If you prefer to win a brand new car rather than a goat, then you ought to switch. It is irrational not to switch. Yet, a considerable number of Marilyn vos Savant's readers argued that it makes no difference whether you switch or not, because the probability of winning the car must be 1/3 no matter how you behave. Although intuitively plausible, this conclusion is false. The point is that Monty Hall's behavior actually reveals some extra information that makes it easier for you to locate the car. To see this, it is helpful to distinguish between two cases. When you make your initial choice, the door you select either has a goat or the car behind it. In the first case, you pick a door with a goat behind it. This leaves two doors: one with a goat and the other with the car. Since only one of the remaining doors has a goat, Monty Hall must open that door.