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Wagering with Zeno

Brian Hayes

Zeno's Favorite Numbers

An alternative to tracing a few very long games is to gather statistics on the outcome of many shorter games. The illustration at right gives the observed frequencies of various outcomes for games of length one through six, based on samples of several thousand trials.

The statistics of the Zeno gameClick to Enlarge Image Games of length one (a single coin toss) can have only two possible outcomes, namely 1/4 and 3/4, and these events are equally likely. Two-round games must end with a value of 1/8, 3/8, 5/8 or 7/8, and again all four choices have the same probability.

Things get interesting with games of three or more rounds. After the third coin toss, the score of the gambler (or the position of the random walker) must be a fraction that, when expressed in lowest terms, has a denominator of 16. There are eight such fractions, but only six of them ever turn up as results of Zeno games; 5/16 and 11/16 are simply not observed. Among the six values that do occur, two of them (3/16 and 13/16) are twice as common as the others.

Going on to four-round games, the pattern gets more peculiar. In this case all game values must be fractions with a denominator of 32. Of the 16 possibilities, only 10 are actually observed, and a few of these are two or three times more frequent than others. The likeliest game outcomes are 3/32 and 9/32 (along with the symmetrically related values 29/32 and 23/32, which are equal to 1–3/32 and 1–9/32). The differences in frequency are much too large to be an effect of statistical noise.

As the number of wagering rounds increases further, the patterns become even more pronounced. Wide gaps in the frequency distribution turn the graph into a snaggle-tooth smile. And certain numbers are dramatically more popular than the rest. For games of length six, only 24 of 64 possible outcomes are observed, and much of the probability is concentrated in just three values (and their symmetrical counterparts). The three favored fractions are 9/128, 27/128 and 3/128. Why does the Zeno process favor these particular numbers? The powers of 2 in the denominator have already been explained, but why do the most common game outcomes all have powers of 3 in the numerator? It can't be an accident.

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