384^ CONDORCET. 



This important result had been given in effect by Laplace in 

 the memoir which we have cited in Art. 551 ; but in Laplace's me- 

 moir we must suppose the ^-? + </ events to be required to happen 



\P -^ ^ . 

 in an assiqned order, as the factor , — :' is omitted. 



We shall see hereafter in examining a memoir by Prevost 

 and Lhuilier that an equivalent result may also be obtained by an 

 elementary algebraical process. 



705. The remaining problems consist chiefly of deductions 

 from the seventh, the deductions being themselves similar to the 

 problems treated in Condorcet's first part. We will briefly illus- 

 trate this by one example. Suppose tliat A has occurred m times 

 and B has occurred n times ; required the probability that in the 

 next 2q + l trials there will be a majority in favour of A. Let 

 F{q) denote this probability ; then 



[ x''' (1 - xY cj, (q) dx 



ic'" (1 - xY dx 



^ 



where <^ (q) stands for 



x'^-"' + {2q + 1) x'' {l-x) + ^^^ + 1^^^^ x'^-' (1 - xy+ 



\2q+l 

 ~==r X'^' (1 - xy. 



...+ 



Hence if we use, as in Art. 663, a similar notation for the case 

 in which q is changed into q + 1, we have 



[ x'''{l-xy^(q + l)dx 



x'"" (1 - xy dx 



•^ 



Therefore, as in Art. 663, 



Cx'^l-xyU{q+l)~cl>{q)\dx 



F(q+1)-F(q)=i^ ^.r-^ ^ , 



x'''{i-xydx 



J n 



