LAPLACE. 535 



is connected with the game which is called Treize or Rencontre ; 

 see Arts. 162, 280, 286, 430, Q±Q. 



Laplace devotes his pages 217 — 225 to this problem ; he gives 

 the solution, and then applies his method of approximation in 

 order to obtain numerical results when very high numbers are 

 involved. 



980. Laplace takes next on his pages 225 — 238 the problem 

 of the Duration of Play. The results were enunciated by De 

 Moivre and demonstrated by Lagrange ; Laplace has made great 

 use of Lagrange's memoir on the subject ; see Arts. 311, 583, 

 588, 863, 885, 921. We may observe that before Laplace gives 

 his analytical solution he says, Ce probleme pent etre resolu 

 avec facilite par le precede suivant qui est en quelque sorte, 

 mecanique ; the process which he gives is due to De Moivre ; 

 it occurs on page 203 of the Doctrine of Chances. See also 

 Art. 303. In the course of the investigation, Laplace gives a 

 process of the kind we have already noticed, which is criticised in 

 the fourth Supplement ; see Art. 978. 



981. Laplace takes next on his pages 238 — 2-17 the problem 

 which we have called Waldegrave's problem ; see Arts. 210, 249, 

 295, 348. 



There are n-\-l players C^, G^, ... (7„^j. First C^ and Cg play 

 together ; the loser deposits a shilling in a common stock, and the 

 winner plays with C^ ; the loser again deposits a shilling, and the 

 winner plays with C^\ the process is continued until some one 

 player has beaten in succession all the rest, the turn of C^ coming 

 on again after that of (7„^j. The winner is to take all the money 

 in the common stock. 



Laplace determines the probability that the play will terminate 

 precisely at the x^^'^ game, and also the probability that it will 

 terminate at or before the ic^'* game. He also determines the 

 probability that the r^^ player will win the money j)recisely at the 

 x^^ game ; that is to say, he exhibits a complex algebraical func- 

 tion of a variable t which must be expanded in powers of x 

 and the coefficient of t'' taken. He then deduces a general ex- 

 pression for the advantage of the r*^ player. 



The part of the solution which is new in Laplace's discussion 



