When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

In 1991, Erdős found himself befuddled when the Parade magazine columnist Marilyn vos Savant published a probability puzzle called the Monty Hall problem, named after the original emcee of the TV game show Let’s Make a Deal. It goes like this. There are three doors onstage, labeled A, B, and C. Behind one of them is a sports car; behind the other two are goats. You get to choose one of the doors and keep whatever is behind it. Let’s suppose you choose door A. Now, instead of showing you what’s behind it, Monty Hall slyly opens door B and reveals … a goat. He then offers you the option of switching to door C. Should you take it? (Assume, for the sake of argument, that you are indifferent to the charm of goats.) Counterintuitively enough, the answer is that you should switch, because a switch increases your chance of winning from one-third to two-thirds. Why? When you initially chose door A, there was a one-third chance that you would win the car. Monty’s crafty revelation that there’s a goat behind door B furnishes no new information about what’s behind the door you already chose—you already know one of the other two doors has to conceal a goat—so the likelihood that the car is behind door A remains one-third. Which means that with door B eliminated, there is a two-thirds chance that the car is behind door C. But Erdős insisted to his friends it wasn’t so. His intuition told him that switching should make no difference in the odds. And this peerless authority on probability...