The Unfinished Game

Hey Everybody,

I spent two weeks at the hilly, and burning hot, Cornell campus this summer.

My high school, Hotchkiss, offers this scholar program where 12 rising seniors travel to Ithaca for two weeks to do intensive research on a mathematician of each of our choosing. Which is then followed by a normal spring semester course on the history of math.

In the summer part of the class, we spent a ton of time in the Cornell Math Library researching each of our mathematicians. I chose the infamous Pierre de Fermat, who was actually a lawyer by profession and studied math for fun, hence the name, "The Great Amateur."

One of Fermat's greatest findings was when he teamed up with Blaise Pascal to revolutionize probability theory as we know it today. Fermat solved the problem of the Unfinished Game, where two players betting on coin flips have to stop the game before it finishes and must split the money based on who's most likely to win.

For example, Pierre bets on heads and Blaise bets on tails. Whichever side is flipped 3 times first get the money. Heads has been flipped twice and tails has been flipped once before they have to stop the game. Since there can only be a maximum of 2 rounds played to "finish" the game, we can calculate who's most likely to win based on the four possibilities of 2 rounds of coin flips: HH, HT, TH, TT (H = heads, T = tails). In 3 of the 4 outcomes, heads is flipped 3 times before tails is reaches 3, and thus, Pierre with 3/4 of the pot while Blaise is left with 1/4 of the money.

Fermat predicts the probability of each player winning through direct enumeration, making the Unfinished Game the foundation of all probability theory and how prediction models work.

Probably your friend,

Noah