The Real Problem

Toward the end of the storied Twentieth Century—when math was hard, priorities perverted, and tempers quick to flare—the well-known and
-regarded Parade magazine columnist Marilyn vos Savant (in)famously caused a ruckus when she answered a reader’s question about the so-called “Monty Hall problem”—named for the well-known and -liked host of the popular television game show Let’s Make a Deal (which just celebrated its 50th anniversary in February 2013; Hall himself is 91!). The question was posed in Parade as follows:

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 No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which [reveals] a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

The answer that infuriated many—including many professional, accredited mathematicians—is that the contestant for certain should switch doors.

And this answer is indeed plainly erroneous. Why? Because the suggestion that a contestant should take the opportunity to choose Door No. 2 is based on the fact—a fact demonstrable by way of explanations, simulations, formal mathematical proofs, and computer simulations—that the contestant who switches has a 2/3 chance of winning the car, while a contestant who sticks with Door No. 1 has only a 1/3 chance of doing so.

But this assertion, and the 75 papers on the topic published in academic journals and the popular press, all ignore the other fact that a contestant should take the opportunity presented to choose Door No. 3—the door that has been opened already to reveal a goat behind it—in order to win that goat. Because it goes almost, but not quite, without saying that a goat is a better prize than a car. At any rate, one shouldn’t need a computer simulation to be convinced of this.

Nevertheless, an earlier formulation of the so-called Monty Hall problem might help to make plain the correct solution. In 1959, in the “Mathematical Games” column in Scientific American written by recreational mathematician Martin Gardner, the “Three Prisoners Problem” appeared, thus:

Three prisoners—A, B, and C—are in separate cells and sentenced to death. The governor selects one at random to be pardoned. The warden knows who is to be pardoned, but he is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others to be executed. “If B is to be pardoned, give me C’s name. If C is to be pardoned, give me B’s name. And if I’m to be pardoned, flip a coin to decide whether to give me the name of B or C.”

The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C. Prisoner A tells C the news, who is also pleased, because he reasons that A still has a chance of 1/3 to be the pardoned one, but his own chance has gone up to 2/3. Who is correct?

Well, this much is plain: A asked the warden for the wrong thing. Instead of asking for the name of either B or C, A should have asked the warden for a goat. If A had gotten a goat, he might have been able to escape from the prison, leaving the goat in his place to be executed (or possibly pardoned; who can say?). The ages-old practice of using goats to facilitate prison breaks is actually the origin of the term “(e)scapegoat.” For real.

Still not convinced? Consider this, then: The Doors were an American rock band formed in 1965 in Los Angeles. The group consisted of singer Jim Morrison, keyboardist Ray Manzarek, guitarist Robby Krieger, and drummer John Densmore. They was a controversial act, due in large part to Morrison’s “wild, poetic” lyrics and his “charismatic but unpredictable” stage persona. But they were popular. Morrison died in 1971, but the remaining members of the band continued performing as a trio for another couple of years. At this point, the relevance of this musical group to the discussion at hand should be obvious: With Morrison gone, there were three Doors. They tried to carry on without their front man, but ultimately they decided to disband in 1973. This was a mistake. Rather than call it quits, they should have gotten a goat to replace the Lizard King. And then they might have gone on to be even bigger than the Cars.

Matthew David Brozik wrote this and many other short humor pieces, which have been published in print and online by The New Yorker, Adult Swim, McSweeney’s Internet Tendency, Grin & Tonic, The Big Jewel, and no one.

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