606 LAPLACE. 



Suppose that a large number of trials, s, is to be made, and 

 that at each trial one of two cases will happen ; suppose that in 

 one case a certain sum of money is to be received, and in the 

 other case a certain other sum : determine the expectation. 



Laplace applies an analysis of the same kind as in his fourth 

 Chapter ; we shall deduce the required result from the investiga- 

 tion in Art. 1002. We supposed in Art. 1002 that all values of 

 a certain variable z were possible, and that fi {£) denoted the 

 chance at the ^"^ trial that the value would lie between z and 

 z + hz. Suppose however that only two values are possible which 

 we may denote by £ and ^^ ; then we must suppose that f^ {z) 

 vanishes for all values of z except when z is very nearly equal 

 to fi or to ^i, and we may put 



ra 

 J b 



where ^^ stands for the part of the integral arising from values 

 of z nearly equal to ^^ and qi stands for the part of the integral 

 arising from values of z nearly equal to f ^ ; and thus 



Pi+qi = l. 



fa ^ 



Again, zf^ (z) dz will reduce to two terms arising from values 



of z nearly equal to fj and ^^ respectively, so that we shall have 



I ^^(2;) J^=£i?i + ft2'^• 



Similarly, 





Suppose now in Art. 1002 that 7i = 72 = ••• = 7s = 1 ; then 

 = X [iSlp, + ^^q^) (p, + q^) - (£p, + ^,qy] 



= ^ptqi (S - ft)'. 



