It is customary to date the birth of probability theory to 1654. That year, Blaise Pascal and Pierre de Fermat exchanged numerous letters about games of chance. Although it was published later, this was also when Blaise Pascal wrote his Traité du triangle arithmétique avec quelques autres petits traitez sur la mesme matière (see the article "Properties of the arithmetical triangle"), in which he presents solutions to the same questions. The best known of these questions, known as the "problem of points," was not entirely new. Several Italian scholars had already studied it, including Luca Bartolomeo Pacioli (c. 1447–1517), Girolamo Cardano (1501–1576) and Niccolò Fontana, known as Tartaglia (1499–1557). Pierre de Fermat (c. 1601–1665) had proposed a fairly laborious computational solution. But Blaise Pascal offered a new perspective, a "geometry of chance" whose clarity struck his contemporaries.
A thousand-pistole question -----------------------------
What, then, is this problem of points, put to the already famous Blaise Pascal, then aged 31, by Antoine Gombaud, Chevalier de Méré? Two players, A and B, are pitted against each other in a game of pure chance. Think, for example, of heads or tails, the card game War, the Game of the Goose, or even rock paper scissors. Each stakes the same sum of money. The winner is the first to win a given number of rounds—say, three. But how should the pot be divided if the two players have to leave before either has won the game? What if, for example, A leads by two rounds to one? We naturally feel that A should receive more than B, but exactly how much?
In the section of his treatise devoted to the "use of the arithmetical triangle to determine how the stakes should be divided between two players playing a series of rounds", Blaise Pascal sets out the context clearly. "The first thing to consider is that the money the players have staked no longer belongs to them; […] in return, they have acquired the right to await whatever chance may yield them under the terms they agreed at the outset. But since this is a voluntary rule, they may break it by mutual consent. Thus, whatever stage the game has reached, they may leave it and, undoing the terms on which they entered it, renounce what they expect from chance and each regain ownership of a share. In that event, what is to belong to each must be apportioned according to what each was entitled to expect from fortune, so that neither player has any preference between accepting what is assigned to him and continuing the game; this fair distribution is called the division."