The problem of points
Two players of equal skill play a series of fair rounds, and the first to win 6 rounds takes the whole stake. The game is interrupted when one player leads 5 rounds to 3. How should the stake be divided fairly?
History and references
Solved in 1654 by Blaise Pascal and Pierre de Fermat.
Luca Pacioli discussed the problem in his 1494 Summa and proposed dividing the stake in proportion to the rounds already won, ignoring how many each player still needs. In the 16th century Niccolò Tartaglia pointed out that this rule gives absurd results, for example when only one round has been played. Around 1654 the Chevalier de Méré brought the problem to Blaise Pascal, and Pascal and Pierre de Fermat solved it in their correspondence by dividing the stake according to each player's chance of winning if play continued. Here the leader needs 1 more round and the other player needs 3, so the trailing player wins only by taking the next three rounds, which has probability 1/8, and the fair split is 7 to 1. Their exchange is often taken as the beginning of probability theory and of the idea of expected value.
References
Solutions 0
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