148 DE MOIVRE. 



In the Miscellanea Analytica, page 204, the problem is again 

 said to have been proposed by Thomas Woodcock, sjyectatissimo 

 viro, but he is not mentioned in the second or third edition of 

 the Doctrine of Chances ; so that De Moivre's infinite respect for 

 him seems to have decayed and disappeared in a finite time. 



The solution of the problem is as follows : 



Let R and S respectively represent the Probabilities which A and B 

 have of winning all the Stakes of their Adversary ; which Probabilities 

 have been determined in the vii*^ Problem. Let us first suppose that 

 the Sums deposited by A and B are equal, viz. G, and G : now since A 

 is either to win ihe sum qG, or lose the sum pG, it is plain that the Gain 

 of A ought to be estimated by EqG — SpG; moreover since the Sums 

 deposited are G and G, and that the proportion of the Chances to win 



one Game is as a to h, it follows that the Gain of A for each individual 



aQ _ hQ 



Game is ^ — j and for the same reason the Gain of each individual 



a + 



aG — hL 

 Game would be j- , if the Suras deposited bv A and B were re- 



spectively L and G. Let us therefore now suppose that they are L 

 and Gj then in order to find the whole Gain of A in this second cir- 

 cumstance, we may consider that whether A and B lay down equal 

 Stakes or unequal Stakes, the Probabilities which either of them has 

 of winning all the Stakes of the other, sufier not thereby any alter- 

 ation, and that the Play will continue of the same length in both cir- 

 cumstances before it is determined in favour of either; wherefore the 

 Gain of each individual Game in the first case, is to the Gain of each 

 individual Game in the second, as the whole Gain of the first case, to 

 the whole Gain of the second; and consequently the whole Gain of the 



/-7 7 7- 



second case will be Rq -^px or restoring the values of H and >S', 



a — b ° ' 



qa^xa^-b^-pb^xa'^-b^ i.. -,. -, , aG-bL 

 ^^^,_f^^, multiplied by ^_^ . 



270. In the first edition of the Doctrine of Chances, 

 pages 136 — 142, De Moivre gave a very laborious solution of the 

 preceding Problem. To this was added a much shorter solution, 

 communicated by Nicolas Bernoulli from his uncle. This solution 

 was founded on an artifice which De Moivre had himself used in 



