Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern

Brian Hayes in American Scientist:

ScreenHunter_02 May. 27 23.41 Timothy Gowers, a distinguished mathematician at the University of Cambridge, recently conducted an experiment in collaborative mathematics. He was puzzling over an interesting problem, and rather than go off to work on it in solitude, he posted a note on his blog, inviting others to join him in seeking a solution. There have been hundreds of responses, and the voluminous discussion has spread to other blogs as well as a wiki where participants coordinate efforts and summarize progress.

If Blaise Pascal had had a blog or a wiki, perhaps he would have tried the same strategy when he took up a mathematical challenge in 1654—a problem concerned with figuring the odds in a gambling game. Instead, Pascal wrote a letter to an older colleague, Pierre de Fermat, and the two of them batted the problem back and forth in a correspondence that went on for several weeks, with occasional input from a few others. Most of the letters were later published—after the deaths of both authors—and they became foundation documents in the theory of probability. Keith Devlin now gives us a helpful guidebook to this famous episode of epistolary mathematics.

Here is the essence of the wagering problem that caused all the fuss, as Devlin presents it in a slightly simplified form:

The players, Blaise and Pierre, place equal bets on who will win the best of five tosses of a fair coin. We’ll suppose that on each round, Blaise chooses heads, Pierre tails. Now suppose they have to abandon the game after three tosses, with Blaise ahead 2 to 1. How do they divide the pot?

Since Blaise is leading, it seems he deserves a larger share of the wager. But how much larger? Gamblers and scholars had taken up this question before, at least as far back as Luca Pacioli and Girolamo Cardano a century earlier, but they had failed to settle it. Pascal and Fermat not only got the answer; they also set forth with reasonable clarity how they derived it and why it’s right.

More here.