378 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



Whence, if the several probabilities # 1? x z . . x n are equal, and are 

 each represented by #, we have 



X= kx n + (I- k) (l-x)\ (2) 



The probability in the latter case, that the accused person is guilty, 

 will be 



kx n + (l-k) (\-x) n ' 



All these results assume, that the events whose probabilities 

 are denoted by k, x ly x z , &c., are independent, an assumption 

 which, however, so far as we are concerned, is involved in the 

 fact that those events are the only ones of which the probabilities 

 are given. 



The probability of condemnation by a given number of voices 

 may be found on the same principles. If a jury is composed of 

 three persons, whose several probabilities of correct decision are 

 x, x, x" 9 the probability X 2 that the accused person will be de- 

 clared guilty by two of them will be 



X z = k [xx 1 (1 - x") + xx' (1 - a/) + x'x" (1 - *)} 



which if x = x = x" reduces to 



3&c 3 (1 - x) + 3 (1 - k) x (1 - x)\ 



And by the same mode of reasoning, it will appear that if 

 X{ represent the probability that the accused person will be de- 

 clared guilty by i voices out of a jury consisting of n persons, 

 whose separate probabilities of correct judgment are equal, and 

 represented by x, then 





If the probability of condemnation by a determinate majority a 

 is required, we have simply 



i - a = n - , 

 whence 



n + a 



t = ' 



