MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers
Exercise 37: Fermat, Pascal, and Interrupted Games

In 1654, the French mathematicians Blaise Pascal (who should be familiar to you for other reasons) and Pierre de Fermat corresponded about the problem of how to divide prize money in an interrupted gambling game. The first case that they discussed was a game in which either player has an equal chance of winning any round, and the first player to win 3 rounds wins the game and all of the money at stake. Assume for the sake of concreteness that the winner of the game will win sixty-four dollars.

Everyone agrees that if the game is interrupted after each player has won 2 rounds, then the 64 dollars should be divided evenly, with each player getting 32 dollars. Similarly, if the game is interrupted after each player has won 1 round, then the money should be divided evenly.

1. Suppose that the game is interrupted at a time when one player has won 2 rounds, and the other player has won 1 round. How should the sixty-four dollars be divided?
2. Suppose that the game is interrupted at a time when one player has won 2 rounds, and the other player has not won any rounds. How should the sixty-four dollars be divided?
3. Suppose that the game is interrupted at a time when one player has won 1 round, and the other player has not won any rounds. How should the sixty-four dollars be divided?

Make sure that you explain the reasoning behind each of your three answers. Compare your three answers to make sure that they are consistent.