JAMES BERNOULLI. 61 



and c = 31 ; let e denote the number of ways in which seven can 

 be thrown, and /the number of ways in which seven can fail, so 

 that e = 6, and /= 30 ; and let a = 6 4- c = e +/ 



Now consider the expectations of the different players ; they 

 are as follows: 



For it is obvious that - expresses the expectation of the first 



player. In order that the second player may win, the first throw 



must fail and the second throw must succeed ; that is there are ce 



ce 

 favourable cases out of o^ cases, so the expectation is -2 . In 



order that the third player may win, the first throw must fail, 



the second throw must fail, and the third throw must succeed; 



that is there are cfh favourable cases out of a^ cases, so the ex- 



Icf 

 pectation is — . And so on for the other players. Now let a 

 a 



single player. A, be substituted in our mind in the place of the 



first, third, fifth,...; and a single player, B, in the place of the 



second, fourth, sixth.... We thus arrive at the problem proposed 



by Huygens, and the expectations of A and B are given by two 



infinite geometrical progressions. By summing these progressions 



we find that ^'s expectation is -3 — -, and 5's expectation is 



CB 



; the proportion is that of 30 to 81, which agrees with 



the result in Art. 31. 



107. The last of the five problems which Huygens left to be 

 solved is the most remarkable of all ; see Art. 35. It is the first 

 example on the Duration of Play, a subject which afterwards 

 exercised the highest powers of De Moi\Te, Lagrange, and Laplace. 

 James Bernoulli solved the problem, and added, without a demon- 

 stration, the result for a more general problem of which that of 

 Huygens was a particular case; see Ars Conjectandi page 71. 



