The Unfinished Game [Paperback]

Many textbooks on mathematical probability mention as a brief aside the correspondence between Pascal and Fermat on the subject of settling fairly a wager on an unfinished game. And many of the popular science style books on probability which have substantial historical components (amongst my favorites, Against the Gods: The Remarkable Story of Risk and Chances Are: Adventures in Probability) devote a few pages to this topic. The first half of Devlin's book, whose style positions it slightly more toward the "serious" end of the popular science spectrum, presents and discusses the correspondence, accompanied by background about the lives of the two principals and their contemporaries. Having a detailed yet easy to read account of this subject is a very welcome addition to the literature.

I'm less enthusiastic about the second half, consisting of briefer accounts of the contributions of people such as Graunt, the Bernoullis, Gauss, Bayes and fast forwarding to DNA testimony and Black-Scholes. Much of this material is similar in spirit to that in existing books (such as the two mentioned above) which paint a broader and richer historical picture. Moreover the implication that there's some kind of meaningful direct line from Pascal-Fermat to the present mathematical understanding of probability, risk etc seems to me just misleading. In core areas of mathematics (geometry, algebra, calculus ..) there was a continuous historical development, in that people consciously learned and built upon what was known before. In contrast, pre 20th century mathematical probability was more a disjointed collection of small topics initiated by different individuals with different motivations -- metaphorically, an archipelago not a continent.

Note: The listing as 208 pages may be misleading (the pages are smallish and the typeface large), though the price is still very reasonable.

Many years ago, I can remember that the weatherman on television would give us a forecast for the next day, and he'd make his blunt prediction that it was going to rain or shine. That was it; you had his prediction, and he turned out either to be wrong or right. A couple of decades ago, this changed, and the weatherman started giving us percentage chances of rain. If he says there is a ninety percent chance of rain, you make your decision accordingly about whether to take the umbrella, and if it doesn't rain, the weatherman wasn't wrong; it was just that other ten percent chance creeping through. We take predictions about the weather, and stocks, and countless other things for granted, but that we can predict the future and take such predictions seriously represents a philosophical shift based on pure and applied mathematics. Keith Devlin wants us non-mathematicians to understand how important this shift was, and how it got started from a letter from one mathematician to another written on 24 August 1654. In _The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern_ (Basic Books), Devlin has given a quick history of the beginning of probability theory, and more important, has shown how the mathematics was done and how it really did change everyone's outlook on the way the world works. Devlin, well known as "The Math Guy" on National Public Radio who tries to make complicated mathematical ideas understandable for the rest of us, does much the same thing here, making complicated and sometimes counterintuitive mathematical themes understandable, and even more important, relevant.

Before the letter, Devlin says, scholars and even leading mathematicians believed that any attempt to predict the likelihood of future events was futile; the future was known by God alone. Gamblers would particularly have liked to have predicted the future, and Pascal and Fermat were taking on a gambling problem: Two players are betting on a game in which they are going to toss a coin five times, and the one who calls the most tosses correctly wins a pot. What happens if the game gets interrupted before the fifth toss? How should they divide the pot? It is a matter of examining the possible outcomes, figuring the odds of each, calculating the chance each player would have had of winning if the game had continued, and dividing the pot according to their respective odds. The problem is not complicated (looking back on it!), but it required subtle reasoning. At no point did they attempt to solve the problem empirically, tossing coins for many simulated games to find out how often each outcome might happen; this was an effort of pure mathematics, applied to a real-world problem.

Neither Pascal nor Fermat could have known how real-world it was. For them it was a puzzle, a bit of mathematics inspired by gambling. What they were laying down, though, was the basics of risk management, and directly because of their correspondence, people started behaving in different ways. Within only a few decades, the solution to the unfinished game was being applied to life-expectancy calculations, and the business of selling life annuities began. Such calculations and such businesses are still going on, with insurance being sold on far more than just people's lives. Engineers can calculate risk of bridges or airplanes falling down. Quantifying risks means that investors can calculate expected gain, or that pharmaceutical companies can compare different drugs. And of course, back to gambling, casinos know just how much they can expect to make for every dollar wagered, and they can mathematically plan on that outcome. There is still randomness; no one knows exactly what tomorrow will bring. Devlin's clearly-written and entertaining book, however, shows that the intellectual effort of two mathematical giants enabled us to quantify what might happen, and to plan accordingly. The future would never be the same.

In this most engaging book, the author focuses on correspondence between Blaise Pascal and Pierre de Fermat - two great mathematicians of the seventeenth century. The subject of this correspondence deals with a particular problem in gambling. The resolution of this problem, as detailed in this exchange between these two geniuses, is viewed by mathematicians as the birth of probability theory as we know it today. As another reviewer as already pointed out, the author's analysis of this exchange, one letter in particular, occupies the better part of about half the book. The rest involves subsequent developments in this field due to other great luminaries in mathematics, as well as resulting applications in everyday life. Throughout the book, the author has included historical/biographical snippets which add an important human element to what could otherwise be viewed by some as a rather dry subject. I have read other books by this author, and I find this one to be clearly his best thus far. The writing style is clear, friendly, authoritative, quite engaging and accessible to a wide audience. Nevertheless, this book will likely be savored the most by math and science buffs.