608 LAPLACE. 



that already given. Denote these values by fj, ft, %» ... ; we shall 

 have 



-where pi + qi + Wi+ ... = !'; 



Laplace himself takes the particular case in which the function 

 fi(z) is supposed the same at every trial ; see his pages 423 — 425. 



1040. Laplace proceeds to a modification of the problem just 

 considered, which may be of more practical importance. Nothing 

 is supposed known a priori respecting the chances, but data are 

 taken from observations. Suppose we have observed that in /x^ 

 trials a certain result has been obtained v^^ times : if /jl more trials 

 are made determine the expectation of a person who is to receive f 

 each time the result is obtained, and to forfeit, f each time the 

 result fails. 



The analysis now is like that which we have given at the end of 



2 f'^ 

 Art. 1036. There is the probability -j- e'^"^ dt that the number 



of times the result is obtained will lie between 



But if the result is obtained a times in fx trials the advan- 

 tage is 



o-f- {fi - a) f, that is, c (f + f) - yt^f. 



Hence there is the probability above assigned that the advan- 

 tage will lie between 



This will be found to agree substantially with Laplace's 

 page 425. 



1041. Laplace passes on to questions connected with life in- 

 surances : he shews that the stability of insurance companies 

 depends on their obtaining a very large amount of business. It 

 has been pointed out by Bienaym^, that if the consideration of 



