396 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



probabilities of those causes a l9 . . a m from experience, but form 

 the conditional probabilities of the event as dependent upon such 

 causes, 



W S, &c . (XVII. Prop, i.) 



JLi JL Z 



on the hypothesis of the independence of individual judgments, 

 and so deduce the equation (7). I conceive this, however, to be 

 a less rigorous, though possibly, in practice a more convenient 

 mode of procedure than that adopted in the general solution. 



12. It now only remains to assign the particular forms which 

 the algebraic functions Xt, (xXi), &c. in the above equations as- 

 sume when the logical function Xi represents that event which 

 consists in r members of the assembly voting one way, and n - r 

 members the other way. It is evident that in this case the alge- 

 braic function Xi expresses what the probability of the supposed 

 event would be were the events rc 1? # 2 , . . x n independent, and 

 their common probability measured by x. Hence we should 

 have, by Art. 3, 



Under the same circumstances (xXi) would represent the pro- 

 bability of the compound event, which consists in a particular 

 member of the assembly forming a correct judgment, conjointly 

 with the general state of voting recorded above. It would, 

 therefore, be the probability that a particular member votes cor- 

 rectly, while of the remaining n - 1 members, r - 1 vote cor- 

 rectly ; or that the same member votes correctly, while of the 

 remaining n - 1 members r vote incorrectly. Hence 



+ (fi-1) (n-2) ..(n-r) 





1 . 2 . . r - 1 1 . 2 . . r 



PROPOSITION IV. 



13. Given any system of probabilities drawn from recorded in- 

 stances of unanimity, or of assigned numerical majority in the de- 

 cisions of a criminal court of justice, required upon hypotheses 

 similar to those of the last proposition, the mean probability c of 



