The Unfinished Game.

I’m still playing a bit of catchup on stuff I read during March (and just finished Joe Haldeman’s The Forever War over lunch today), but one title I definitely want to bring to everyone’s attention is the delightful, short book by mathematician (and NPR’s “Math Guy”) Keith Devlin called The Unfinished Game, which explains how one specific letter in the correspondence between Blaise Pascal and Pierre de Fermat opened the door to the world of probability and everything that this branch of mathematics makes possible.

The unfinished game of the book’s title was based on a common, popular controversy of the time surrounding games of chance, which were largely seen as incalculable – our modern, simple way of calculating odds of things like throws of the dice just did not exist at the time. Pascal and Fermat discussed the question of how to divide winnings in a game of two or more players where the players choose to abandon the game before any one player has won the requisite number of matches. (So, for example, they’re playing a best-of-five, but the players quit after three rounds, with one player having won two times and the other one.) The controversy in question will seem silly to any modern reader who’s taken even a few weeks of probability theory in high school math, but Devlin is deft enough to explain the problem in 1600s terms, so that the logical confusion of the era is clear on the page.

The confusion stemmed from the misunderstanding about the frequencies of subsequent events, given that the game would not always be played to its conclusion: You may say up front you’re going to play a best of seven, but you do not always need to play seven matches to determine a winner. If you quit after three games, in the situation I outlined above, it is possible that you would have needed just one more match to determine a winner, and it is possible that you would have needed two more matches. Pascal’s letter to Fermat proposed a method of determining how to split the winnings in such an unfinished game; the letter was the start of modern probability theory, and the problem is now known as the problem of points. (You can read the entire surviving correspondence on the University of York’s website; it also includes their conversations on prime numbers, including Fermat’s surprising error in claiming that all numbers of the form 2⁽²ⁿ⁾+1, which is only true for 0 ≤ n ≤ 4. Those five numbers are now called Fermat primes; Euler later showed Fermat’s hypothesis was wrong, and 2⁽²⁵⁾+1 = 4294967297, which is composite.)

Fermat realized you must count all of the potential solutions, even ones that would not occur because they involved playing the fifth game when it was made unnecessary by the first player winning the fourth match and taking the entire set, so to speak. (The problem they discussed was slightly more involved.) Pascal took Fermat’s tabular solution, a brute-force method of counting out all possible outcomes, and made it generalizable to all cases with a formula that works for any number of players and rounds. This also contributed to Pascal’s work on what we now call Pascal’s triangle, and created what statisticians and economists now refer to as “expectation value” – the amount of money you can expect to win on a specific bet given the odds and payout of each outcome.

Devlin goes about as far as you can when your subject is a single letter, with entertaining diversions into the lives of Pascal and Fermat (who corresponded yet never met) and tangents like Pascal’s wager. At heart, the 166-page book is about probability theory, and Devlin makes the subject accessible to any potential reader, even ones who haven’t gone beyond algebra in school. Given how much of our lives – things like insurance, financial markets, and sports betting, to say nothing of the probabilistic foundations of quantum theory – are possible because of probability theory, The Unfinished Game should probably be required reading for any high school student.

Next up: I just started Eimear McBride’s A Girl is a Half-Formed Thing, winner of the 2014 Baileys Women’s Prize for Fiction.