BEYOND INTUITION

Mathematician Snell on life's gambles.

In 1812 the great French scientist and probability pioneer Marquis de Laplace trenchantly observed, "The most important questions of life are, for the most part, really only problems of probability."

His words ring just as true for modern times. Today doctors prescribe medical treatment according to estimates of the probability of success. Insurers determine life insurance costs by estimating the probability that you will die at a particular age. Opinion polls work because we have learned how to change the uncertainty of chance into near certainty. And, despite Einstein's statement that "God does not roll dice," many modern theories of physics, most notably quantum theory, are probabilistic in nature.

Dartmouth has its own uses; for probability. For example, every year the College makes a huge chance experiment in accepting about 1,700 students, though it hopes to get a class of 1,050. If, on the average, 60 percent of the students accept, by the laws of probability there is only a two percent chance that more students will accept than the College can handle. Only once was former Admissions Director Eddie Chamberlain the victim of such bad luck, when, for the class of 1960, 60 more students than expected accepted. Thereafter Chamberlain played a less risky game: employing a larger waiting list.

When mathematician John Kemeny became president of Dartmouth, he applied probability to various problems, including the College's tenure process, which had been leading to too high a proportion of the faculty on tenure. Kemeny devised a probabilistic model to maximize a professor's chance of getting tenure while keeping the proportion of tenured faculty at a desired 60 percent.

Students at Dartmouth Medical School are taught how to analyze probability experiments by one of our former mathematics majors, Dr. Robert Beck '75. They learn, for example, how to estimate the probability that a person who tests positive for AIDS actually has the AIDS virus:

Assume that 15 people per thousand now have this virus. Assume further that if a person has the AIDS virus the test will be positive 99.1 percent of the time, and if he does not have the virus the test will be negative 99.5 percent of the time. (These are realistic percentages.) What is the probability that a person who tests positive for AIDS actually has the virus?

The answer is only about 75 percent, even though the test is accurate more than 99 percent of the time. This may seem puzzling to the doctor who has not studied probability, but what patient would not want to know that there is a 25 percent chance that a positive indication is wrong? Incidentally, the patient who tests negative can be assured that there is a 99.99 percent chance that the test is correct and he does not have AIDS.

The explanation for the surprising result for a positive test lies in the fact that the AIDS virus is still relatively rare. In a population of 100,000 who take the test, we expect about 1,500 to have the virus. We expect about 1,486 (99.1 percent) of these to test positive, and about 98,007 (99.5 percent of the 98,500 who don't have the virus) to test negative. Now since 493 of those who don't have the virus tested positive, there are in all 1,486 plus 493, or 1,979 who tested positive, of whom the 1,486 who actually have the virus constitute just 75 percent.

Wall Street offers a different example of the cost of misinterpreting the laws of probability. Any student of probability watching the Dow Jones average will notice the similarity of its graph to that of the fortune of a player of "heads or tails," in which a coin is tossed a series of times and on each toss the player wins $1 if heads turns up and loses $1 if tails turns up. A resulting graph has the pleasant property that if the player just waits long enough he can be sure to increase his fortune substantially. Unfortunately, he needs infinite resources to ride out possible large losses, and some gamblers (stock buyers) forget this.

The graph of a player's fortune in "heads or tails" is also the graph of the position of a person who walks on a long street and tosses a coin at each corner to decide whether to go forward or back one block. The interpretation of this process is called a "random walk" or a "drunkard's walk." The famous mathematician George Polya relates that when he was a young man he took a friend to meet a girl in a Zurich park. After he left them, he continued to walk randomly through the park, yet he kept meeting them. He was sufficiently concerned to go home and prove a remarkable fact: that two people walking randomly on an infinitely long street (a one-dimensional random walk) or through the streets of an infinite city (a two-dimensional random walk) would be certain to meet, but that this would not be the case of two random fliers in space (a three-dimensional random walk).

This is not just random trivia; Polya's findings have many practical applications, from electronics to aeronautics. Unfortunately, there is no intuitive way to understand Polya's startling conclusion.

Sometimes probability is our best—or only—bet.

Laurie Snell and needles.

Sitting in front of the color monitor of a Macintosh computer, Laurie Snell watches a program he designed for teaching probability. "I remember that as a ten-year-old boy at the 1935 Chicago World's Fair I was fascinated by a mechanical machine dropping needles on a ruled floor," he says. "Some of the needles fell across a line, and some fell between the lines. The machine was estimating the value for IT (the ratio of the circumference of a a circle to its diameter; IT = 3.1415 .. .) from the fraction of the needles that crossed a line. The French naturalist Buffon had invented this problem, but he dropped loaves of French bread onto a wide-board floor." And now here was Snell, using his Macintosh to drop 100 million'needles" in a fast, high-tech version of the same experiment. "There's no obvious reason why this should estimate the value of TT, but it does," he says with a hint of glee. "The explanation lies in probability." This is a man who loves probability. "My favorite probability paradox is the birthday problem: how many people do you need to make it a fair bet that two or more people have the same birthday? The answer, surprisingly, is 23. Try it sometime," he suggests, adding, "President Dickey tried it once at a meeting with Arthur Sulzburger, publisher of the New York Times, and won a $1 bet." Snell, who trained at the University of Illinois, came'to Dartmouth in 1954 on the advice of the same Princeton professor who had urged Dartmouth's President Dickey to hire John Kemeny. A rich collegiality between Snell and Kemeny resulted in their development of courses and texts on what they termed finite mathematics the various non-calculus branches of mathematics, including elementary logic, matrix algebra, and probability, that are especially useful to social' scientists and other non-mathematicians. For the last two years Snell has chaired the Master of Liberal Studies Program at the College. He has indulged his love of music by coteaching a MALS course on opera. But even his long involvement in music, including Dartmouth's Handel Society chorus, has a link to math: at the start of his teaching career he took up singing to improve his lecture voice. Karen Endicott