392 PROBABILITY OF JUDGMENTS. [CHAP. XXI 



determine ,_! as a function of c l . . c n , a l . . a m _ 2s the numerical 

 data supposed to be furnished by experience. We now proceed 

 to their application. 



PROPOSITION III. 



10. Given any system of probabilities drawn from recorded 

 instances of unanimity ', or of assigned numerical majority in the 

 decisions of a deliberative assembly ; required^ upon a certain deter- 

 minate hypothesis, the mean probability of correct judgment for a 

 member of the assembly. 



In what way the probabilities of unanimous decision and of 

 specific numerical majorities may be determined from experience, 

 has been intimated in a former part of this chapter. Adopting 

 the notation of Prop. i. we shall represent the events whose pro- 

 babilities are given by the functions X 19 X 2 , . . X m _ l . It has 

 appeared from the very nature of the case that these events are 

 mutually exclusive, and that the functions by which they are re- 

 presented are symmetrical with reference to the symbols a? 19 x z > # 

 Those symbols we continue to use in the same sense as in Prop, i., 

 viz., by Xi we understand that event which consists in the for- 

 mation of a correct opinion by the i th member of the assembly. 



Now the immediate data of experience are 



Prob. Xi=a lt Prob. X 2 = 2 , . . Prob. A r m _ 3 = a ffl _ 2 , (1) 

 Prob. *.,_, = a*.,. (2) 



Xi . . Xm.i being functions of the logical symbols x l9 . . x n to the 

 probabilities of the events denoted by which, we shall assign the 

 indeterminate value c. Thus we shall have 



Prob. a?! = Prob. x z . . = Prob. x n = c. (3) 



Now it has been seen, Prop, i., that the immediate data (1) 

 (2), unassisted by any hypothesis, merely conduct us to a re- 

 statement of the problem. On the other hand, it is manifest that 

 if, adopting the methods of Laplace and Poisson, we employ the 

 system (3) alone as the data for the application of the method of 

 this work, finally comparing the results obtained with the expe- 

 rimental system (1) (2), we are relying wholly upon a doubtful 

 hypothesis, the independence of individual judgments. But 



