598 LAPLACE. 



Let ^ be the observed number of baptisms of boys during a 

 certain number of years, q^ the observed number of baptisms of 

 girls, 2w the annual number of baptisms. Let x represent the 

 chance that an infant about to be born and baptised will be a 

 boy. 



Let (a? + 1 — xf" be expanded in a series 



x^^ + ^^nx'^-' (1 - ^) + ^"^ ^f ' ~ ^^ x''^'' {I -xy +...', 



then the sum of the first n terms of this series will represent the 

 probability that in a year the number of baptisms of boys will 

 predominate. 



Denote this sum by ^; then f^ will be the probability that 

 the superiority will be maintained during i years. 



Hence we put x^ {1 — xY for y and f ^ for z in the formula of 

 the preceding Article, and obtain 



P = 



I x"" [1 - x)" t,' dx 

 1x^(1- xy dx 



Laplace applies his method of approximation with great success 

 to evaluate the integrals. He uses the larger values of p and q^ 

 given in Art. 902 ; and he finds that P= '782 approximately. 



1032. Laplace's seventh Chapter is entitled De V influence des 

 ineg allies inconnues qui peuvent exister entre des chances que Von 

 suppose parfaitement egales : it occupies pages 402 — 407. 



The subject of this Chapter engaged the attention of Laplace 

 at an early period ; see Arts. 877, 881, 891. Suppose the chance 



of throwing a head with a coin is either — ^ — or — ^ — , but it is 



as likely to be one as the other. Then the chance of throwing 

 n heads in succession will be 



, . 1 r 72 (n — 1) 2 _ 71 (w — 1) (^ — 2) (7^ — 8) 4 

 that IS, ^ jl + ^^ ^ ^ a^-f— ^ ^ I ^ ^ ^ ^a*+... 



