July 7, 2011

A Passion for Tossing Dice

Ordinary dice—those sold in novelty stores and with many board games—have rounded edges and little hollows for each of the pips denoting the numbers from one to six.

But the hollows mean that such ordinary dice are somewhat biased. A little more material has been removed from sides with a larger number of pips, so a die with six hollows on one side and only one hollow on the opposite would have a slightly greater tendency to end up with the six side up.

Casino dice differ in crucial ways from everyday dice. Their edges are generally perfectly square and sharp. Moreover, the sides are flat, with no recesses.

Magician and gambling expert John Scarne once described the process of making casino dice in the following terms: Casino dice are often 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 filled in with paint of the same weight as the plastic that has been drilled out. The resulting transparent cubes are then buffed and polished.

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. You 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 a chance to test the model against the behavior of real dice tossed by a person. A man named Willard 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 manila 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. Altogether, Longcor had tested 219 dice of four different brands for a total of 4,380,000 throws.

"The only way to check the work was by checking the runs and then comparing the results with theory," Mosteller once explained. "It turned out [Longcor] was very accurate." Indeed, the results even highlighted some errors in the then-standard theory of the distribution of runs.

"The main formulas were correct, but the endpoints of the formulas were not quite right," Mosteller observed.

"We found some aberrant results that suggest that things a little unusual happen more often than the classical theory would suggest," he added. "Consequently maybe we should be a little more careful than we are when we interpret tests."

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.

Longcor's data were important enough that his name appears on the paper that Mosteller and his colleagues eventually published recounting these investigations: "Bias and runs in dice throwing and recording: A few million throws" by Gudmund R. IversenWillard H. LongcorFrederick MostellerJohn P. Gilbert, and Cleo Youtz, published in Psychometrika, Vol. 36, No., 1, pp. 1-19.


Albers, D.J., G.L. Alexanderson, and C. Reid, eds. 1990. More Mathematical People: Contemporary Conversations. Academic Press.

Scarne, J. 1986. Scarne's New Complete Guide to Gambling. Simon & Schuster.

Photos by I. Peterson

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