It’s a safe bet that the average shooter at a Mohegan Sun craps table is not thinking about the mathematical probability of rolling a seven and seeing his wager swept away, any more than the average blackjack player knows the odds of improving a weak hand with the next card.
Alan Stein may not know either, but he does know how to express precisely these kinds of dilemmas in formulas that are readily understandable to other students of mathematics.
During a lunch-time talk on Wednesday, Stein, an associate professor of mathematics, spent 40 minutes introducing students and faculty in Waterbury to “the wonderful world of binomial coefficients and probability.” To illustrate how binomials – the sum of two X and Y terms – work, he used a construct of a 17th-century French mathematician, Blaise Pascal.
“Pascal’s Triangle” consists of “binomial coefficients.” Each entry in the interior of the triangle is the sum of the two numbers above it.
“The entries in Pascal’s Triangle have tons of interesting properties and also have significant connections to the science of counting and to probability,” Stein said.
“If we add up the terms in each row, the successive sums are 1, 2, 4, 8, 16. Each is exactly double the one before. They are all powers
of 2,” he noted.
Stein engaged his audience with real-life conundrums that lend themselves to mathematical formulae. He explained that a deck of 52 cards, for example, equals the number of ways a card can be chosen: by face value (13 ways) multiplied by the number of suits (four), equaling 52.
In another example, he demonstrated that a light bulb with an “average life” of 1,000 hours has “zero probability” of lasting exactly that long. Then he created equations to express the likelihood that the bulb would last exactly, say, 1,000 hours and one minute.
“The probability that it lasts exactly 1,000 hours is 0, and a similar argument would clearly hold for every other possible time,” Stein said.
“But the light bulb must burn out at some point, so an event with probability zero will inevitably occur.”
Stein, whose sense of humor is evident on his website, http:// www.math.uconn.edu/~stein, offered any of his students in the audience an opportunity to curry favor by solving the following problem:
“Suppose you are standing in front of three chests which look exactly alike, each of the chests containing two drawers. We will call the chests A, B, and C. Each drawer of Chest A contains a gold piece. In Chest B, one drawer has a gold piece, one a silver piece.
In Chest C each drawer contains a silver piece. You choose a chest at random, choose one of its drawers at random, open the drawer, and find it contains a gold piece. What is the probability that, if you open the other drawer in the same chest, it will also contain a gold piece?”
Stein’s current course list includes calculus, differential equations, and probability theory.