101 MONTMORT. 



(2) Next suppose that each player starts with three counters ; 

 ox 



then 1 — — is the chance that the match will be ended in 2ic + 1 



games at most. This result had in fact been given by Montmort in 

 his first edition, page 184. It may be deduced from Montmort's 

 general expression, and involves a property of the Binomial Coeffi- 

 cients which we will briefly indicate. 



Let w^, u^, u^, ... denote the successive term-; in the expansion 

 of (1 + iy'^\ Let S denote the sum of the following series 



Then shall 8=2'"'- 3^. 



If w^ denote the r**^ term in the expansion of (1 + 1)'^''"^ we can 

 shew that 



w^ + 2m^_i + 2w^_2 + u^_s 



+ S (2^,_i + 2w;^_2 + 2w;^_3 + 2(7^ J. 



By performing a similar transformation on every successive 

 four significant terms of the original series we transform it into 



2 (1 + 1)'''"^ + 3S, where 2 is a series like S with x changed into 



x-1. Thus 



8 = 2^^-2 + 32. 



Hence by induction we find that /S^= 2^"" - S''. 



184. Suppose the players of equal skill, and that each starts 



with the same odd number of counters, say m ; let /= '^'^^ , 



Then Montmort says, on his page 276, that we may wager with 

 adva.ntage that the match will be concluded in 3/' - 3/+ 1 trials. 

 Montmort does not shew how he arrived at this approximation. 



The expression may be put in the form \m'^\, De Moivre 



4 4 



spoke favourably of this approximation on page 148 of his first edi- 

 tion; he says, "Now Mr de Montmort having with great Sagacity 

 discovered that Analogy, in the case of an equal and Odd number 

 of Stakes, on supposition of an equality of Skill between the 



