Tibor's Musings

Paul Erdös and the Game Show Problem

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, "Would you rather like to pick door #2?" Is it to your advantage to switch your choice of doors?

Please ponder this problem for a few minutes. It is the classic "game show" problem whose solution may seem intuitively surprising.

Marilyn vos Savant, "the highest recorded IQ" for many years, wrote about this problem in her "Ask Marilyn" newspaper column, stirring much debate across the US. Many readers went indignant. Marylin received 10,000+ protesting letters, including 1,000+ from PhDs and math professors! She had to repetitively "fight" for the correct answer.

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Even one famous mathematician is said to have been genuinely puzzled by the problem. Paul Erdös -- yes, the one of the "Erdös number" fame! -- reportedly got the problem wrong initially. Leonard Mlodinow, in his book "The Drunkard's Walk: How Randomness Rules Our Lives", writes:

"When told of this, Paul Erdös, one of the leading mathematicians of the twentieth century, said: "That's impossible." Then, when presented with a formal mathematical proof of the correct answer, he still didn't believe it and grew angry. Only after a colleague arranged for a computer simulation in which Erdös watched hundreds of trials came out [...] in favour of [...] did Erdös concede he was wrong."

(Some words elided in order not to give away the correct answer too easily.)

If you got the problem wrong, then do not worry; if Paul Erdös also got the problem wrong initially, then the world still has a chance.

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Leonard Mlodinow cites Bruce Schechter's book "My Brain Is Open: The Mathematical Journeys of Paul Erdös" as his source. I did not read that one, but I went directly to the original first-hand source of the story.

It was Andrew Vazsonyi who tried to convince Paul Erdös about the game show problem solution. His personal account of the events is available in short format in Zentralblatt and in longer format in a Decision Line paper and in an even longer format in Vazsonyi's book "Which Door has the Cadillac: Adventures of a Real-Life Mathematician". The book contains more side details and other stylistic changes, but the heart of the story and the quotes about Erdös's reactions are the same as in the above Decision Line paper.

Vazsonyi's account seems to suggest that Erdös was genuinely puzzled about the problem.

(P.S. You may also be interested in Andrew Vazsonyi's obituary for Paul Erdös containing more (unrelated) stories about their encounters. The stories are also present in the above-cited Vazsonyi's book "Which Door has the Cadillac: Adventures of a Real-Life Mathematician" which is an otherwise interesting reading in itself.)