|Ivars Peterson's MathTrek|
April 28, 1997
As it evolved from one hour to the next, the view from my hotel-room window was continually spectacular. Thirteen floors below, in the hotel's casino, the sights and sounds were quite different: the glare of neon lights, the jangle of coins erupting from slot machines, the clink of chips at blackjack tables, and the mutter of avid gamblers testing their luck.
It was the middle of April, and I was at the Horizon Casino Resort in Stateline, Nevada, attending the 2nd Annual Recreational Mathematics Conference, sponsored by the California Mathematics Council, Community Colleges. About 100 college math instructors had gathered for a day or so of fun and games -- mathematics style.
The program had started off with an evening banquet and a talk by Rudy Rucker, a computer science professor at San Jose State University and author of several science-fiction novels. His topic was "how flies fly." He described some curves that he had discovered, which appear to capture the intricately bent and twisted, three-dimensional flight paths of flies.
Before the banquet, I had taken the opportunity open to conference participants to learn how to play blackjack and roulette (craps looked too intimidating), without having to risk losing any money. I enjoy games of chance, but I can never bring myself to throw money away on such pursuits. The games themselves are entertainment enough. And this was my first visit to a real casino.
It didn't take long for me to learn the different kinds of bets possible in roulette, and it was fun experimenting with various combinations. I actually did quite well for a while, though inevitably, I ended up losing the equivalent of $100.
I learned that certain bets are really never worthwhile, especially a five-number bet placed on 0, 00, 1, 2, and 3, where the payoff odds are 6 to 1 and the house edge is an unhealthy 7.9 percent. On most other bets, the house advantage is merely 5.3 percent.
The Saturday program included hour-long talks on topics ranging from Lewis Carroll's puzzles to mathematical models of bad luck and patterns in contra dancing. I'll say a little more about the mathematics of contra dancing at a later date.
My role as an invited speaker at the meeting was to keep everyone awake after the Saturday luncheon with a keynote presentation on how randomness arises in the gambling world, the amusement park, and elsewhere. I had a chance to give a preview of some of the topics in my upcoming book The Jungles of Randomness. It was also the first time I had spoken from a dinner theater stage -- one occupied at various times by such luminaries as Engelbert Humperdinck and Suzanne Somers and apparently featured in an obscure movie called Showgirls.
The casino setting gave me an excuse to consider some of the details of how both electronic and mechanical slot machines work, particularly how they create the illusion of chance. That meant talking about the sensitive dependence on initial conditions that characterizes chaos and on the vagaries of random-number generators. Dice also represent an intriguing case of the interplay between randomness and physical laws.
At the Horizon, I finally got a good look at real casino dice, which differ in crucial ways from those sold in novelty stores and with many board games. According to gambling expert John Scarne, casino dice are made by hand, each cube typically 0.75 inch wide and precisely sawed from a rod of cellulose or some other plastic. Pits are drilled about 0.017 inch deep into the faces of a cube, and the recesses are then filled in with paint of the same weight as the plastic that has been drilled out. The transparent dice are then buffed and polished. The edges are generally perfectly square and sharp. In contrast, store-bought dice have recessed spots and distinctly rounded edges.
How fair are casino dice? A cubic die produces six possible outcomes. It makes sense to use a mathematical model in which each face has an equal probability of showing up. One can then calculate other probabilities, including the number of times a certain number is likely to come up in a row.
Several decades ago, Harvard statistician Fred Mosteller had an opportunity to test the model against the behavior of real dice tossed by a real person. A man named William H. Longcor, who had an obsession with throwing dice, came to Mosteller with an offer to record the results of millions of tosses. Mosteller accepted the offer, and some time later, received a large crate of manilla envelopes. Each envelope contained the results of 20,000 tosses with a single die and a written summary showing how many runs of different kinds had occurred.
"The only way to check the work was by checking the runs and then comparing the results with theory," Mosteller says. "It turned out [Longcor] was very accurate." Indeed, the results even highlighted some errors in the then-standard theory of the distribution of runs.
Because the data had been collected using both casino dice from Las Vegas and ordinary, store-bought dice, it was possible to compare their performance not only with theory but also with each other and with a computer that simulated dice tossing.
As it turned out, the computer proved to have a flawed random-number generator, whereas the Las Vegas dice were very close to perfect in comparison with theory.
In throwing a die, one knows from experience (or theory) that each face has an equal chance of turning up in a long sequence of tosses. So, if one sees ten 6s in a row, it might be the legitimate though improbable result of a genuinely random process. However, it might also be advisable to check whether the die is fair and to find out something about the person who's doing the tossing. The context determines how one interprets the data.
From slot machines and dice to wandering molecules and dripping faucets, chance pervades everyday life. Understanding precisely what randomness means and recognizing it when one sees it, however, prove to be no laughing matter.
Copyright © 1997 by Ivars Peterson.
Orkin, Mike. 1991. Can You Win? The Real Odds for Casino Gambling, Sports Betting, and Lotteries. New York: W.H. Freeman.
Scarne, John. 1986. Scarne's New Complete Guide to Gambling. New York: Simon & Schuster.
Wykes, Alan. 1964. The Complete Illustrated Guide to Gambling. Garden City, N.Y.: Doubleday.
Information about the California Mathematics Council, Community Colleges, and its recreational math meetings can be obtained from Michael Eurgubian of Santa Rosa Junior College at email@example.com.
Additional information about Rudy Rucker's space curves and his latest novel, Freeware, is available at http://www.mathcs.sjsu.edu/faculty/rucker.
Comments are welcome. Please send messages to Ivars Peterson at. firstname.lastname@example.org