Problem of points

The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.

The problem concerns a game of chance with two players who have equal chances of winning each round. The players contribute equally to a prize pot, and agree in advance that the first player to have won a certain number of rounds will collect the entire prize. Now suppose that the game is interrupted by external circumstances before either player has achieved victory. How does one then divide the pot fairly? It is tacitly understood that the division should depend somehow on the number of rounds won by each player, such that a player who is close to winning will get a larger part of the pot. But the problem is not merely one of calculation; it also involves deciding what a "fair" division actually is.

s, the game will surely have been won by someone after 2r+s−1 different possible outcomes. In some of these possible futures the game will actually have been decided in fewer than 2r+s−1 possible continuations and counting how many of them would lead to each player winning. Fermat now considered it obviously fair to divide the stakes in proportion to those odds.

Fermat's solution, certainly "correct" by today's standards, was improved by Pascal in two ways. First, Pascal produced a more elaborate argument why the resulting division should be considered fair. Second, he showed how to calculate the correct division more efficiently than Fermat's tabular method, which becomes completely impractical (without modern computers) if (r+s−1k) term represents the combination operator.

The problem of dividing the stakes became a major motivating example for Pascal in his 1665 Treatise on the arithmetic triangle.

Though Pascal's derivation of this result was independent of Fermat's tabular method, it is clear that it also describes exactly the counting of different outcomes of n−1 rounds, as readily expressed and calculated using Pascal's binomial form. If desired, terms pertaining to like outcomes can be summed to obtain Pascal's original result, taking account of distinct denominators p is unknown? Laplace solved this problem using a Bayesian approach, assuming a uniform prior distribution for "History of Probability | Research Starters | EBSCO Research". EBSCO. Retrieved 2025-09-02.

↑ (in French) with short English summary

↑ Laplace, Pierre Simon. 1774. “Memoire sur la probabilite des causes par les evenemens.” English translation by Stephen M. Stigler, appearing in Statistical Science, 1(3): 364-378.)