** CHANCE**

**A Guide to Gambling, Love, the Stock Market, & Just About Everything Else**

By Amir D. Aczel

Thunder's Mouth. 161 pp. $21

"Chance" is a pretty little book, a hand-friendly 8 inches by 5 inches, with a red-and-white cover designed to look like the back of a playing card. The implication is that Aczel has written a how-to on gambling, though in fact games of chance are only part of what he addresses.

Still, the cover is appropriate for another reason: probability theory (the modern, scientific name for chance) had its genesis in a series of letters between the 17th-century mathematicians Blaise Pascal and Pierre de Fermat about an odds problem brought to Pascal by a gambler acquaintance.

The titled young man (he was a chevalier) wondered which of two dice games had the better odds: rolling a single die four times and winning if a one came up on any of the rolls, or rolling a pair of dice 24 times and winning if two ones came up simultaneously. He thought the probabilities of a win in either game ought to be equal. Pascal and Fermat disabused him (for how, see Aczel's explanation in the book), in the process laying the foundation for a theory that now underlies, among other things, quantum physics.

The author of several other books on science for the layman, Aczel encourages his readers to take a can-do attitude. Cheerfully, he confides, "I want to tell you a secret: measuring probabilities is as simple as *counting*." Well, sort of. "Chance" contains a number of equations. Happily, the math-impaired can skip these and still glean the general idea.

The general idea of probability theory is illustrated by the now infamous bell curve. Actually a value-neutral form of measurement, the curve demonstrates that if you gather random factors and then graph them, the resulting line will be near-flat on the left, rise gradually to the rounded peak of a hill, then sink at exactly the same gradient to near-flat again on the right, resulting in the shape of a bell. In a graph of the heights of American men, for instance, the central point would convey the largest category (say, men 5-feet-9), while the descending lines on the left and right would testify to the lesser number of men either smaller or taller, with the extremes growing as the line flattens out. The result is that we can know the odds of any American man being taller than 5-feet-9 or shorter than, say, 4-feet-10. An accumulation of random data always generates this curve of probability, as if conjuring it out of the air. No one knows why. It just happens.

At this point, we begin to approach the mystical element of mathematics. Any time you ask a "why" question about math, you can find yourself in unmapped territory.

Why, in simple arithmetic, do one and one always equal two? They just do. Where do we get the idea, removed from any objects, of "twoness"? We don't know. (Plato's theory of transcendent ideas is out of favor.) The individual flip of a coin is a memoryless event -- the second coin flipped doesn't remember whether the last coin came up heads or tails, and the third coin doesn't remember how the second flip turned out. You start fresh every time. And yet, as Aczel illustrates, after about the first 120 tosses, the results begin to come up 50-50 (though not all at once: for example, from roughly 250 to 550 tosses, the coins will mostly land heads-up, a lead that tails will catch up to later). Why does this happen? How does it happen? What's going on here anyway?

And what about beating the house? Aczel explains the Gambler's Ruin theorem: "When you play against a much wealthier adversary, like a casino, given enough gamblers, the gambler will lose *with probability **1* -- or absolute certainty." Another bit of useful information: The chance of winning at roulette is 47 percent rather than 50 (the odds are skewed by the zero and double-zero places on the wheel).

But there are some surprises too. Given a choice to wager it all on one throw or carefully parcel out your money in a series of cautious bets, you should take the big chance; making only one bet minimizes your encounter with the unfair odds that favor the house, but each additional wager takes you further into a disadvantageous position. Aczel's final advice on gambling is the same as common wisdom: Quit while you're ahead.

Some more tidbits: Yes, according to probability theory, a group of monkeys pounding on typewriters really will, given enough time, produce "Hamlet." Aczel gives you the odds. The trick is that in this case, "long enough" is sometime off beyond the death of the universe. And to maximize the probability of finding the perfect mate, "date about thirty-seven percent of the available candidates in your life, and then choose to stay with the next candidate who is better than all the previous ones." The next time you visit an Internet dating service, take your calculator.