CONDORCET. 399 



p + q trials it will happen p times and fail q times. The required 

 probability is 



j^ + q ff'*'{i-^r''i^ 



'' 



as we have already remarked in Art. 704. 



Condorcet quotes this result ; he thinks however that better 

 formulae may be given, and he proposes two. But these seem 

 quite arbitrary, and we do not perceive any reason for preferrino- 

 them to the usual formula. We will indicate these formulae pro- 

 posed by Condorcet. 



I. Let t = 7n+ 71 + 2) + 2' ^^^ P^"^ 



3?, ~r ^o I "^Q ~r • • • I '^t 



U = -^ 2 3 . 



t 



then the proposed formula is 



\p + q j • ' ' ^"'"^ (^ ~ ^^y^' ^^1 ^^2 • • • ^^t 

 L^ Li jj[..,u'^ (1 - uy dx^ dx^ .,.dx 



The limits of each integration are to be and 1. 



II. Suppose an event to have happened n times in succession, 

 required the probability that it will hap|)en p times more in suc- 

 cession. 



_ X^ "p Xr, X^ -p X^^ -f- X^ 3/j ~t" X^ i~ • • • I Xf^ 



XjQXi U ^— X^ ;:: ;;; • t . • 



^23 n 



let V be an expression similar to u but extended to n +^:> factors ; 

 then Condorcet proposes for the required probability the formula 



I I I • • I C/ U/tX'j Cf/Xn • • • ^Xf^ip 



III ...u dx^ dx,^ . . . dx^ 



The limits of each integration are to be and 1. 



Condorcet proposes some other formulae for certain cases ; tliey 



