The Calculus Diaries - Jennifer Ouellette [30]
His methodology served him well as a gambler, but Cardano’s analysis was rather flawed. He assumed that all outcomes were equally likely; in fact, different outcomes have different probabilities. Galileo Galilei demonstrated this in the early seventeenth century in a short paper entitled “Thoughts About Dice Games.” Galileo wasn’t especially interested in probability theory, preferring to roll balls down inclined planes and time their rate of travel. But his patron, the Duke of Tuscany, was an inveterate gambler and thus keenly interested in the question of why—for games played with three dice—the number 10 seemed to occur a tad more frequently than the number 9. Galileo concluded (correctly) that this occurred because there were more combinations that yielded a 10 than yielded a 9. There are twenty-seven ways to roll a 10 with three dice, compared to twenty-five possible combinations for a 9. It’s now an established tenet of probability theory that the odds of a particular outcome are dependent on the number of ways in which it can occur.
Galileo took his analysis no further; his research interests lay elsewhere. Yet wealthy and titled patrons with gambling problems continued to push for advances in probability theory, most notably a social-climbing French essayist named Antoine Gombaud, who adopted the title chevalier after the character in his many dialogues who represented the author: Chevalier de Mere.
Gombaud was a man of letters who fancied himself an amateur mathematician, and in 1654 he found himself pondering what is known as the problem of points: How do you determine how the stakes in a game of chance should be divided if, for some reason, the players were interrupted and never finished their game? It was first proposed in 1494 by an Italian monk named Luca Pacioli in his treatise Summa de arithmetica, geometria, proportioni et proportionalita. (Yes, even monks fell victim to the lure of gambling. They didn’t have television in the Middle Ages.) So this question had been knocking around gambling circles for nearly two hundred years by the time Gombaud decided enough was enough—he wanted a solution to the conundrum.
Gombaud turned to a young mathematician named Blaise Pascal, who had taken up gambling when his doctors advised him to abandon mental exertions for the sake of his health. Pascal suffered from chronic stomach pain, nausea, migraines, and partial paralysis of the legs, among other ailments. Intrigued, Pascal quickly realized he would need to invent an entirely new method of analysis to solve the puzzle, because the solution would need to reflect each player’s chances of victory given the score at the time the game was interrupted. Thus began his legendary correspondence with fellow mathematician Pierre de Fermat, which over the course of several weeks, laid the foundation for modern probability theory. They quickly realized that in order to solve the problem it would be necessary to list all the possibilities and then determine the proportion of times that each player would win.
Caltech mathematician Leonard Mlodinow gives one of the clearest explanations of how to solve the problem of points in his book The Drunkard’s Walk, using the example of the 1996 World Series, in which the Atlanta Braves beat the New York Yankees. Atlanta won the first two games, but what were the odds of a Yankee comeback at that point? To get the answer, you would need to count every scenario in which the Yankees could have won and compare that to the number of scenarios in which they could have lost. By that reckoning—which assumes that the Yankees and the Braves had equal chances of winning each subsequent game—the chance of an overall Yankee victory would have been 6 in 32, or around 19 percent, compared to 26 in 32, or about 81 percent, for an Atlanta victory. “According to Pascal and Fermat, if the series had been abruptly terminated, that’s how they should have split the bonus pot, and those are the odds that should