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This year marks the 350th anniversary of the correspondence between Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) that led to the formal development of the probability calculus. The letters were exchanged ...
This year marks the 350th anniversary of the correspondence between Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1665) that led to the formal development of the probability calculus. The letters were exchanged between July and October of 1654. The subject of the letters were two gambling problems that Pascal called the problem of points for dice and the problem of points for sets of games. At least one of the problems, the problems of points for dice, had been suggested to Pascal by Antoine Gombaud (the chevalier de Méré) and purportedly solved by him.
The correspondence, relating initially to the problem of points for dice, actually began in error on both sides. The initial question in an undated letter concerned a bet on seeing a six at least once in eight throws. How should the stakes be divided if the game is terminated after three throws without the six showing? Pascal thought that the fraction 125 / 1296 = ( 5 / 6 ) 3 ( 1 / 6 ) should go to the player making this bet, while Fermat thought that the fraction should be 1 / 6. With five tosses remaining, the correct probability of seeing a six at least once in five throws is 4651 / 7776 = 1 ( 5 / 6 ) 3. Soon after this initial correspondence, both Pascal and Fermat came to the correct solution for the dicing problem as well as the problem of points for sets of games, or what is now simply called the problem of points. In the latter problem, there are two players. They play some games until one of them has won a fixed number n of games. If play is terminated prematurely with one having won a games and the other b, how should the stakes be divided? The important concept was the expected winnings for each player, or dividing the stakes according to the probability of future outcomes. The problem of points remains a tricky combinatorial exercise for undergraduate students. Pascal found a short cut to the solution, commenting to Fermat that:
None of the correspondence or the results of Pascal and Fermat’s research were published in their lifetimes. Pascal did write a treatise on the arithmetic triangle, which included the problem of points as an application; it was published posthumously in 1665. The person responsible for making these new results in probability known was Christiaan Huygens (1629-1695), for whom my committee is named. He heard about the correspondence in 1655 during a visit to Paris. When he returned home to the Netherlands, Huygens worked out his own solutions to these problems. He published his book on probability De ratiociniis in ludo aleae in 1657. It remained a highly influential work well into the eighteenth century.
Ce théorème a été énoncé par Fermat, démontré dans le cas des nombres carrés par Jacobi et, indépendamment par Joseph-Louis Lagrange au XVIIIe siècle (Ce dernier se servant de résultats partiels obtenus par Euler). Gauss résolut le cas des nombres triangulaires en 1796. Une preuve complète a été proposée par Cauchy en 1813.