By Dauxois J.-Y., Druihlet P., Pommeret D.

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**Sample text**

1). In 1655, a few months after Pascal and Fermat’s correspondence, Huygens visited France. There he became aware of the Problem of Points, although he did not meet any of the two Frenchmen. However, Huygens met the mathematicians Claude Mylon (1618–1660), who was a friend of Carcavi, and Gilles de Roberval, with whom Pascal had previously discussed de Mere’s questions. Back in the Netherlands, Huygens wrote a small manuscript on probability, which he sent to his mentor, the Dutch mathematician Franciscus van Schooten (1615–1660).

The latter would have been very tedious in those days even if the number of remaining rounds was small ($10). Pascal’s method is recursive in nature and it uses the concept of expectation. Although the latter concept is usually attributed to Huygens, Pascal was actually the first to use it. Hacking correctly points out (2006) Not until the correspondence between Fermat and Pascal do we find expectation well understood. Pascal realized that his recursive method would quickly become unwieldy for large a and b.

Suppose the total prize is 1 dollar and the person who wins a total of R fair rounds first collects the prize. 1 Discussion 27 game stop suddenly when A is short to win by a rounds and B is short to win by b rounds, and let A’s expected win be ea,b. Then, A can either win the next round with probability 1/2 (in which case his expected win would be eaÀ1,b) or lose the next round with probability 1/2 (in which case his expected win would be ea,bÀ1) 9 1 > > ea;b ¼ ðeaÀ1;b þ ea;bÀ1 Þ; a; b ¼ 1; 2; .