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Who's the Goat . . . Marilyn or the Mathematicians? . . . by Junpei Sekino, 1994.

On one Sunday of September 1990, the following question appeared in the Ask Marilyn column in Parade, a Sunday supplement of local newspapers.

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 has a goat. He then says to you, "Do you want to pick door No.2?" Is it to your advantage to switch your choice? -Craig F. Whitaker, Columbia, Md.

Well, let's think. If Monty Hall, the game show host of Let's Make a Deal (remember this game?), opened a door and asked you if you wanted to switch, you still don't know which door holds the car and which door holds the goat. Since there are only two doors left for consideration, you now have a fifty-fifty chance of guessing the correct door. After analyzing this far, my wife, Anya, who received a B in Technical Mathematics (not Math 130: Techniques of Mathematics) in St. Petersburg University (in Russia, not Florida), added: "Besides, you never know what motivates Monty to reveal a door in the actual game. I would definitely stick with my first choice."

The byline in Ask Marilyn notes that Marilyn vos Savant, the columnist, is listed in the 'Guinness Book of World Records Hall of Fame' for 'Highest IQ'. Her answer was different from Anya's, "Yes, you should switch." She explained that, obviously, there is a one-third chance that the original choice, door No.1, is the correct one and there must be a two-third chance that the car is behind door No.2 or No.3. Since door No.3 has now been eliminated as a possibility, the second door has a two-third chance. To emphasize her point, she wrote: "Here's a good way to visualize the problem. Suppose there are a million doors, and you pick door No.1. Then the host, who knows what's behind the doors and will always avoid the one with prize, opens them all except door No.777,777. You'd switch to that door pretty fast, wouldn't you? Since you choose first, it's unlikely that you picked the door that hides the car. The same logic applies whether there are three doors or a million doors."

This solution brought an outcry from the reading public, including many professional mathematicians! In a follow-up December column, Marilyn published some examples:

"I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error. . . ." –Robert Sachs, Ph.D., George Mason University.

"You blew it, and you blew it big!. . . You seem to have difficulty grasping the basic principle at work here. . . There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!" -Scott Smith, Ph.D., University of Florida.

"Your answer to the question is in error. But if it is any consolation, many of my colleagues have also been stumped by this problem." -Barry Pasternack, Ph.D., California Faculty Association.

So, who is right? In an effort to even more clearly illustrate the correctness of her original answer, Marilyn invoked the classic shell game, in which a pea is placed beneath one of three shells. The gambler is asked to place a finger on one shell, and the house then lifts away one of the others, leaving behind two shells, one of which covers the pea. As she explained, by removing one empty shell "we've learned nothing to allow us to revise the odds on the shell under your finger." She then presented a "probability grid" containing all possible permutations of the game, showing how "when you don't switch, you win one in three times and lose two in three. Try it yourself."

The shell game is, in fact, isomorphic to Marilyn's problem. There is a story about Charles E. Ford of St. Louis University who used the shell game to raise money for his church at a picnic. While Ford was not looking, the contestant was to slip a quarter under one of three inverted cups. Ford put his finger on one cup but did not pick it up; he told the contestant to pick up one of the other two which did not cover the quarter. Ford always switched his choice, and more often than not chose the cup with the quarter. Did Marilyn win? Not exactly, as the following appeared in her February 1991 column:

"I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake." -Kent Ford, Dickinson State University.

". . . Albert Einstein earned a dearer place in the hearts of the people after he admitted his errors." -Frank Rose, Ph.D., University of Michigan.

". . . Your answer is clearly at odds with the truth." -James Rauff, Ph.D., Millikin University.

"May I suggest that you obtain and refer to a standard textbook on probability . . . ." -Charles Reid, Ph.D., University of Florida.

". . . I am sure you will receive many letters from high school and college students. Perhaps you should keep a few addresses for help with future columns." -W. Robert Smith, Ph.D., Georgia State University.

"You are utterly incorrect. . . How many irate mathematicians are needed to get you to change your mind?" -E. Ray Bobo, Ph.D., Georgetown University.

". . . If all those Ph.D.s were wrong, the country would be in very serious trouble." -Everett Harman, Ph.D., U.S. Army Research Institute.

"When reality clashes so violently with intuition," Marilyn responded, "people are shaken." She decided this time to invoke a more universally accepted phenomenon. Suppose, she says, after choosing door No.1, and having the host step in and "give a clue" by opening one of the two remaining doors, "at that point . . . a UFO settles down onto the stage. A little green woman emerges, and the host asks her to point to one of the two unopened doors. The chances that she'll randomly choose the one with the prize are 1/2 (as opposed to 2/3 chances of the original contestant winning by switching). 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. 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.

The green woman may have done something good. Mathematicians at the Massachusetts Institute of Technology came to Marilyn's defense. "You are indeed correct," wrote Seth Kalson, Ph.D. "My colleagues at work had a ball with this problem, and I dare say that most of them - including me at first - thought you were wrong!" The same thing happened with mathematicians at the University of Oregon. After 92 percent of the letters Marilyn received expressed belief she was wrong, Frank Anderson, head of the University of Oregon's Mathematics Department, said, "Consensus is not the issue. She is 100 percent right."

This controversy generated numerous articles on the car and goats to appear in newspapers, magazines and professional journals including the New York Times and the American Mathematical Monthly. Most of the papers published in mathematical journals also consider various problems with subtle differences but all conclude that Marilyn is correct provided that the host always opens an unselected door concealing a goat and the contestant is offered a chance to switch. Their arguments are essentially similar to the following which involves conditional probability:

Let Ci denote the event that the car is at door i, and Hj the event that the host opens door j . Then

P(You win the car if you switch)

= P(H3 C2) + P(H2 C3) = P(C2)P(H3|C2) + P(C3)P(H2|C3) = (1/3)·1+ (1/3)·1 = 2/3

and in similar manner we find that
P(You win the car if you don't switch) = (1/3)·p + (1/3)·(1 - p) = 1/3 where p = P(H2|C1). I should mention that Sam C. Saunders of Washington State University had the above computation in his paper, which appeared in his university publication [5] in April of 1990, preceding the controversial Ask Marilyn column. Saunders visited Willamette two years ago to give a talk on "the Design of the Japanese Sword."

Lenna Mahoney, an atmospheric scientist in Richland, Washington, writes that she too at first thought Marilyn was wrong. "It wasn't till I started writing a computer program to simulate the set of choices that I realized I was wrong. . . ." She says not one of the several technical professionals, including Ph.D. scientists, of her acquaintance who saw the thought problem got the right answer by immediate intuition. And several came up with "very fanciful ways of 'proving' that the odds were even." One "refused to accept the results of my computer program," arguing that "in short, if the program didn't agree with him it was wrong. This circular rationale illustrates what I find fascinating about the problem. . . Not even experts have the correct intuition, the incorrect answer is infinitely rationalizable using technical-type rhetoric, and everyone thinks the solution is obvious. . . ."

Well, what did I think, first? I too thought that the solution was obvious, but I did not tell this to Anya. I pretended that I knew the correct answer from the beginning, of course.

References

1. Ed Barbeau, "Fallacies, Flaws, and Flimflam", the College Mathematical Journal, vol. 24, 1993.
2. Leonard Gillman, "The Car and the Goats", the American Mathematical Monthly, vol. 99, 1992.
3. Jack R. McCown and Michael A. Sequeira, "Patterns in Mathematics", PWS, 1994.
4. Gary P. Posner, "Nation's Mathematicians Guilty of 'Innumeracy' ", Skeptical Inquirer 15, 1991.
5. Sam C. Saunders, "The Shell Game, Prisoner Paradox and Other Paradoxes of Conditional Probability", Mathematical Notes, Washington State University, vol. 33(#2), April, 1990.