Pascal, Fermat and Probability

Probability theory grew out of a gambling question known as "the problem of points": when a game is stopped before anyone has won, how should the stakes be divided? In 1654, Blaise Pascal and Pierre de Fermat answered it in an exchange of letters, and their method of counting future outcomes became the foundation of modern probability.

An old puzzle about unfinished games

The problem was an old one by 1654. Back in 1494, Luca Pacioli posed a version in his Summa de Arithmetica, Geometria, Proportioni et Proportionalita. His example involved a game of balla played to six goals. If play stopped with one side at five goals and the other at three, how should the pot be divided fairly? Italian mathematicians after Pacioli, including Gerolamo Cardano, attempted the question without settling it.

In 1654 the puzzle reached Paris. Antoine Gombaud, the Chevalier de Méré, a well-known gambler, put dice questions to Pascal, and Pascal wrote to Fermat about them. The introductory letter from Pascal to Fermat is no longer extant, but the surviving correspondence of that year is described as fundamental to the development of modern probability theory.

The fair split, worked out

The key insight was to stop looking at the score so far and look instead at what could still happen. A fair division, in this view, gives each player the share of possible future outcomes that would lead to victory.

Take a simple case. Two players have each staked 50 coins, so the pot is 100. The winner is the first to take three rounds, and play stops with player A on two wins and player B on one. A needs one more round; B needs two. At most two more rounds would settle the game, so Fermat's approach is to list all four ways they could fall: A-A, A-B, B-A and B-B. Player A wins the game in the first three, and B wins only in the last. A should therefore take 3/4 of the pot, or 75 coins, and B should take 25.

Pascal reached the same answer by reasoning one round at a time. If A wins the next round, A takes all 100. If B wins it, the players are level and should split the pot evenly, 50 each. Each case is equally likely, so A is entitled to half of 100 plus half of 50, which is 75. This way of weighting each outcome by its chance is the idea now called expected value. Applied to Pacioli's balla game, the same method gives a split of 7 to 1, since the trailing side could win only by taking three straight goals.

Huygens and the first printed treatise

The ideas moved from private letters into print through Christiaan Huygens. In 1657 he published De Ratiociniis in Ludo Aleae, in Latin, and it is described as the first printed treatise on probability. Propositions four to seven of the book deal with problems of points, meaning how to distribute the prize of a game that ends early. Huygens did not invent the subject, and Pascal and Fermat did not create it from nothing, but together they turned a gambler's complaint into a method others could learn and repeat.

That method is the lasting part of the story. A puzzle about splitting a pot became a general way of putting a value on uncertain outcomes by weighing each possibility by its likelihood. The same logic still sits at the heart of how insurers price risk and how the house edge of a casino game is calculated, more than 350 years after two mathematicians traded letters about an unfinished game.