MOXTMORT. 101 



179. Montmort now passes to a problem of a more important 

 character which occupies his pages 268 — 277, and which is also 

 new in the second edition; it relates to the Duration of Play; 

 see Art. 107. 



Suppose A. to have m counters and i? to have n counters ; let 

 their chances of winning a single game be as a to ^ ; the loser in 

 each game is to give a counter to his adversary : required the chance 

 that A will have won all 5's counters on or before the x^^ game. 



This is the most difficult problem which had as yet been solved 

 in the sulyect. Montmort's formula is given on his pages 268, 269. 



180. The history of this problem up to the current date will 

 be found by comparing the following pages of Montmort's book, 

 275, 309, 315, 324, 344, 368, 375, 380. 



It appears that Montmort worked at the problem and also 

 asked Nicolas Bernoulli to try it. Nicolas Bernoulli sent a 

 solution to Montmort, which Montmort said he admired but 

 could not understand, and he thought his o^^TL method of investi- 

 gation and that of Nicolas Bernoulli must be very different : but 

 after explanations received from Nicolas Bernoulli, Montmort 

 came to the conclusion that the methods were the same. Before 

 however the publication of Montmort's second edition, De Moi\Te 

 had solved the problem in a different manner in the De Mensura 

 Sortis. 



181. The general problem of the Duration of Play was studied 

 by De Moivre with great acuteness and success ; indeed his inves- 

 tigation forms one of his chief contributions to the subject. 



He refers in the following words to Nicolas Bernoulli and 

 Montmort : 



Monsieur de Monniort^ in the Second Edition of his Book of Chances, 

 having given a very handsom Solution of the Problem relating to the 

 duration of Play, (which Solution is coincident with that of Monsieur 

 Nicolas Bemoully, to be seen in that Book) and the demonstration of it 

 being very naturally deduced from our first Solution of the foregoing 

 Problem, I thought the Reader would be well pleased to see it trans- 

 ferred to this place. 



Doctrine of Chances; first edition, page 122. 



