MONTMORT. IQ: 



SO arranged that only a single h is to come among the four letters 

 a : we might have such an arrangement as aaahhhhhhhhhhha. We 

 shall return to this point in our account of De Moivi'e's in- 



vestigations. 



On his page 272 Montmort gives a rule deduced from his 

 formula ; he ought to state that the rule assumes that the players 

 are of equal skill : his rule also assumes that p — m is an even 

 number. 



183. On his pages 275, 276 Montmort gives without demon- 

 stration results for two special cases. 



(1) Suppose that there are two players of equal skill, and that 



each starts with two counters ; then 1 — ^- is the chance that the 



match will be ended in 2x games at most. The result may be de- 

 duced from Montmort's general expression. A property of the 

 Binomial Coefficients is involved which we may briefly indicate. 



Let Wj, u^, u^, ... denote the successive terms in the expansion 

 of (I + l)'"^. Let >S' denote the sum of the following series 



w. + ^ii.-i+ Ux-i+ + u,_,+ 2it,_,+ u,_,+ + w^_3+ ... 

 Then shall S=r'-'-2'-\ 



For let V, denote the r^^ term in the expansion of (1 + 1)"''"S and 

 lOy the ?'"' term in the expansion of (1 + 1)"''"^ Then 



t'y t/j. "t" I- r—Xf 



Employ the former transformation in the odd terms of our pro- 

 posed series, and the latter in the even terms ; thus we find that 

 the proposed series becomes 



'^x + ^.r-1 + ^'x-2 + ^.r-3 + ^x--4 + ' ' ' 

 + 2 [W^_^ -h 2W^_^ + IC,_^ + + 10, _, + ...}. 



The first of these two series is equal to ^ (1 + I)'''"' ; and the 



second is a series of the same kind as that which we wish to sum 

 with X chanced into x-1. Thus we can finish the demonstration 

 hy induction ; for obviously 



