Getting the Goat

In her magazine column for September 9, 1990, Marilyn vos Savant answered a well-known brain teaser submitted by one of her readers. You’re on a game show and you’re given the choice of three doors. Behind one is a car, behind the other two are goats. You choose, say, door 1, and the host, who knows where the car is, opens another door, behind which is a goat. He now gives you the choice of sticking with door 1 or switching to the other door. What should you do?

This was the so-called Monty Hall dilemma faced by guests on Monty Hall’s classic TV game show ‘Let’s Make a Deal’, only the consolation prizes weren’t goats. Vos Savant advised her correspondent to switch doors. Sticking with the first choice gives a one-third chance of winning, she said, but switching doubles the odds to two-thirds. To convince her readers, she asked them to imagine a million doors. “You pick door number 1,” she said, “then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens all the rest of them except door number 777777. You’d switch to that door pretty fast, wouldn’t you?

Evidently not. No sooner had her column appeared than she was besieged by mail from readers who disagreed, including many mathematicians. They maintained the odds were only fifty-fifty, not two-thirds, in favor of switching. The thousands of letters she received were running nine to one against her and included rebukes from a statistician at the National Institutes of Health and the deputy director of the Center for Defense Information. The letters had gotten shrill, with suggestions that she was the goat and that women look at mathematical problems differently from men. “You are utterly incorrect about the game-show question,” wrote E Ray Bobo, a PhD at Georgetown, “and I hope this controversy will call some public attention to the serious national crisis in mathematical education. If you can admit your error, you will have contributed constructively toward the solution to a deplorable situation. How many irate mathematicians are needed to get you to change your mind?”

“When reality clashes so violently with intuition,” vos Savant responded in her column, “people are shaken.” This time she tried another tack. Imagine, she said, that just after the host opened the door, revealing a goat, a UFO lands on the game-show stage, and a little green woman emerges. Without knowing what door you originally chose, she is asked to choose one of the two unopened doors. The odds that she’ll randomly choose the car are fifty-fifty. “But that’s because she lacks the advantage the original contestant had - the help of the host. If the prize is behind no. 2, the host shows you no. 3; and if the prize is behind no. 3, the host shows you number 2. So when you switch, you win if the prize is behind no. 2 or no.3. You win either way! But if you don’t switch, you win only if the prize is behind door no. 1.” Vos Savant was completely correct, as mathematicians with egg on their faces ultimately had to admit.

Paul Hoffman, The Man Who Loved Only Numbers