DE MOIVRE. 18.3 



Thus we have to sum the series 



a%c + ndr-'l {he - 1) + !?L^LdO ^-252 (^c - 2) + . . . , 



the series extending so long as the terms in brackets are positive. 

 "We have 



a%c - noT-'h = a''~'h (ac -n)=- aJ'-'b he ; 

 thus the first two terms amount to 



{n-l)d;'''hhc. 



f2, (fi 1^ 



Now combine this with ^- — -~ a''~%^2 ; we ^et 



1.2 > & 



(n - 1) a'^-'h' (ac - 7i), that is - (?i - 1) a'^-'h'hc ; 



thus the first tJu^ee terms amount to 



1.2 



This process may be carried on for any number of terms ; and 

 we shall thus obtain for the sum of he terms 



(n-l)(n-2) (n-ic + l) ^,.-^,,...j^_ 

 \oc — l 



This may be expressed thus 



[n 



n I he I ae 



a'^'h^'aehc, 



which is equivalent to De Moivre's result. The expectation of S 

 from B will be found to be the same as it is from A. 



82-i. When the chances of A and B for winning a single game 

 are in the proportion of a to & we know, from Bernoulli's theorem, 

 that there is a high probability that in a large number of trials the 

 number of games won by A and B respectively will be nearly in 

 the ratio of a to h. Accordingly De Moivre passes naturally from 

 his Problem Lxxiii. to investigations which in fact amount to what 

 we have called the inverse use of Bernoulli's theorem ; see 

 Art. 125. De Moivre says, 



