It's Probably True
What are the chances of great minds thinking alike?
Jun 22, 2009, Vol. 14, No. 38 • By DAVID GUASPARI
The Unfinished Game
When Keith Devlin, a mathematician and skilled popularizer, was invited to write about a world-changing mathematical document, he chose a letter from Blaise Pascal to Pierre de Fermat. Written in 1654, it's part of a famous correspondence usually credited as the birth of the modern notion of probability. That, Devlin says, has "made the world modern" by enabling reasoned predictions about a future that cannot be known with certainty. Probability theory has a massive effect on everyday life--Devlin emphasizes the rational management of risk in, among other things, medicine, engineering, and finance--and requires a new way of thinking about the world.
As Ian Hacking stressed in his remarkable book The Emergence of Probability, great scientists of the era such as Galileo and Newton accepted an ancient philosophical distinction that sharply divided knowledge from opinion, as things differing not in degree but in kind. Knowledge concerned universal and necessary truths and their demonstratively certain consequences: Newton offered a proof that the planets must move in elliptical orbits. Opinions concerned things that could be discussed and debated but not demonstrated (or quantified).
So, Hacking argues, what we call probability was not an object of thought that existed in embryo to become fully developed when some social need or technological opportunity encouraged mathematicians to study it more carefully. Rather, it was something new, born into a world that seemed to have no place for it.
Pascal began the correspondence, it is said, to discuss problems posed to him by a nobleman fond of gambling. The letter that Devlin selected concerns an unsolved puzzle (already old in 1654) called "the problem of points" and seems, at first, a surprising choice since it does not offer Pascal's solution. Rather, it shows Pascal struggling to understand Fermat's and not quite succeeding--even though a modern reader, who has always lived in a mental world saturated with probabilistic thinking, may find Fermat's reasoning straightforward.
The letter provides an occasion to discuss Fermat's solution, illustrates how difficult is the birth of a new idea, and makes good on the book's promise to describe "how mathematics is really done."
The Unfinished Game begins with a leisurely explication of Pascal's letter, interpolating brief biographies of Pascal, Fermat, and other participants in probability's prehistory. Its second half sketches how mathematics that arose in the simple and artificial setting of gambling games--whose underlying mechanisms (coins, dice, cards) are easily understood--was later brought to bear on messy real-world situations in which the underlying mechanisms are quite unknown.
Here is a sample problem of points: Harry and Tom bet on tosses of a fair coin. Harry gets a point for each head, Tom for each tail, and the first to reach three wins the pot. If the game is discontinued partway through, how should the pot be split?
Sometimes the answer is obvious: an even split if they quit when the score is tied. But what if Harry leads two to one? The "obvious" case will be misleading if it fixes attention on the current score, but Pascal and Fermat saw past that to the key insight: What matters is not what has happened but what could happen from now on. They develop that insight in different ways, but with sound arguments.
Devlin describes Fermat's solution--superior, he says, because it is simpler and reaches the heart of the matter. Consider all the ways the game could continue. The two remaining tosses could play out in four different ways: H‑H, H‑T, T‑H, T‑T. Harry wins in the first three, and Tom in the last, so the pot should split 3‑to‑1 in favor of Harry.
Pascal struggles with the fact that, in practice, the game has just three possible futures (H-H, T‑T, T‑H) since Harry will win and the game will end if the fourth toss is heads. He tests his understanding of Fermat's analysis in terms of four possible futures by trying it on a more complex game in which Tom, Harry, and Dick toss a three‑sided coin and play a game to three.