CHAP. XXI.] PROBABILITY OF JUDGMENTS. 379 



which must be substituted in the above formula. Of course a 

 admits only of such values as make i an integer. If n is even, 

 those values are 0, 2, 4, &c. ; if odd, 1, 3, 5, &c., as is otherwise 

 obvious. 



The probability of a condemnation by a majority of at least a 

 given number of voices m, will be found by adding together the 

 following several probabilities determined as above, viz. : 



1st. The probability of a condemnation by an exact ma- 

 jority m ; 



2nd. The probability of condemnation by the next greater 

 majority m + 2 ; 



and so on ; the last element of the series being the probability of 

 unanimous condemnation. Thus the probability of condemnation 

 by a majority of 4 at least out of 12 jurors, would be 



8 ~t~ -A-9 + -^M2> 



the values of the above terms being given by (3) after making 

 therein n = 12. 



4. When, instead of a jury, we are considering the case of a 

 simple deliberative assembly consisting of n persons, whose sepa- 

 rate probabilities of correct judgment are denoted by x 9 the above 

 formulae are replaced by others, made somewhat more simple by 

 the omission of the quantity k. 



The probability of unanimous decision is 



X = a? + (1 - x). 



The probability of an agreement of i voices out of the whole 

 number is 



Of this class of investigations it is unnecessary to give any 

 further account. They have been pursued to a considerable ex- 

 tent by Condorcet, Laplace, Poisson, and other writers, who 

 have investigated in particular the modes of calculation and re- 

 duction which are necessary to be employed when n and i are 

 large numbers. It is apparent that the whole inquiry is of a very 

 speculative character. The values of x and k cannot be deter- 



