Sunday, April 26, 2009

The “Monty Hall dilemma”

"Marilyn vos Savant’s column gained national notoriety in the early 1990s, thanks to her response to the “Monty Hall dilemma”: the make-or-break decision facing contestants on the game show Let’s Make a Deal that was then hosted by Hall. The question was posed by Craig Whitaker, of Columbia, Marinaland, on September 9 1990. “Dear Marilyn,” wrote Whitaker. “Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you: ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?”

Savant’s answer, that it was better to switch doors, provoked an extraordinary response: thousands of letters of complaint, many of them from science teachers and academics. “There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!” wrote one reader from the University of Florida. “You are the goat!” said another. “You made a mistake, but look at the positive side,” wrote Everett Harman, of the US Army Research Institute. “If all those PhDs were wrong, the country would be in some very serious trouble.”

But Savant had not made a mistake. In the end it took her four columns, hundreds of newspaper stories and a challenge to children to test the options in classroom experiments, to convince her readers that she was right. “Oh, that was so much fun. I just enjoyed these nasty letters I got,” she said. “The audacity of people! I just loved them.”

The key to the solution lies in the role of the host, who will always pick a door which does not have a prize behind it. Statistics from the game show, in which those who switched won about twice as often as those who did not, bear out Savant’s explanation from her third column: “When you first choose door #1 from three, there’s a 1/3 chance that the prize is behind that one and a 2/3 chance that it’s behind one of the others. But then the host steps in and gives you a clue. If the prize is behind #2, the host shows you #3, and if the prize is behind #3, the host shows you #2. So when you switch, you win if the prize is behind #2 or #3. You win either way! But if you don’t switch, you win only if the prize is behind door #1.”

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