MONTMORT. 105 



Gamesters..." In his second and third editions De Moivre with- 

 drew this commendation, and says respecting the rule " Which tho' 

 near the Truth in small numbers, yet is very defective in large 

 ones, for it may be proved that the number of Games found by his 

 Expression, far from being above what is requisite is really below 

 it." Doctrine of Chances, third edition, page 218. 



De Moivre takes for an example m = 45 ; and calculates by his 

 own mode of approximation that about 1531 games are requisite 

 in order that it may be an even chance that the match will be 

 concluded ; Montmort's rule would assign 1519 games. We should 

 differ here with De Moivre, and consider that the results are 

 rather remarkable for their near agreement than for their dis- 

 crepancy. 



The problem of the Duration of Play is fully discussed by 

 Laplace, Theorie...des Proh. pages 225 — 238. 



185. Montmort gives some numerical results for a simple 

 problem on his page 277. Suppose in the problem of Art. 107 that 

 the two players are of equal skill, each having originally n counters. 

 Proceeding as in that Article, we have 



Hence we find u^= Cx+ C^, where C and (7^ are arbitraiy con- 

 stants. To determine them we have 



^0=0, %„ = !; 



hence finally, w« = ^ • 



Montmort's example is for ?i = 6 ; he gave it in his first edition, 

 page 178. He did not however appear to have observed the gene- 

 ral law, at which John Bernoulli expressed his sm-prise ; see Mont- 

 mort's page 295. 



186. Montmort now proposes on pages 278 — 282 four pro- 

 blems for solution ; they were originally given at the end of the 

 first edition. 



The first problem is sur le Jeu dii Treize. It is not obvious 

 why this problem is repeated, for Montmort stated the results on 

 his pages 130 — 143, and demonstrations by Nicolas Bernoulli are 

 given on pages 301, 302. 



