CHAP. XXI.] PROBABILITY OF JUDGMENTS. 377 



6th. The data furnished by experience, such as the relative 

 numbers of cases in which unanimous decisions have been arrived 

 at, or particular majorities obtained ; the number of cases in 

 which decisions have been reversed by superior courts, &c. 



Again, the class of questions under consideration may be 

 regarded as either direct or inverse. The direct questions of pro- 

 bability are those in which the probability of correct decision 

 for each member of the tribunal, or of guilt for the accused 

 party, are supposed to be known a priori, and in which the proba- 

 bility of a decision of a particular kind, or with a definite majority, 

 is sought. Inverse problems are those in which, from the data fur- 

 nished by experience, it is required to determine some element 

 which, though it stand to those data in the relation of cause to 

 effect, cannot directly be made the subject of observation ; as 

 when from the records of the decisions of courts it is required to 

 determine the probability that a member of a court will judge 

 correctly. To this species of problems, the most difficult and 

 the most important of the whole series, attention will chiefly be 

 directed here. 



3. There is no difficulty in solving the direct problems re- 

 ferred to in the above enumeration. Suppose there is but one 

 juryman. Let k be the probability that the accused person is 

 guilty; x the probability that the juryman will form a correct 

 opinion ; X the probability that the accused person will be con- 

 demned : then 



kx = probability that the accused party is guilty, and that the 

 juryman judges him to be guilty. 



(1- A) (!-#) = probability that the accused person is inno- 

 cent, and that the juryman pronounces him guilty. 



Now these being the only cases in which a verdict of con- 

 demnation can be given, and being moreover mutually exclusive, 

 we have 



X-kx + (l-k)(l- x). (1) 



In like manner, if there be n jurymen whose separate proba- 

 bilities of correct judgment are x l9 x z . . x n , the probability of an 

 unanimous verdict of condemnation will be 



X = kx l x 2 ..x n + (l- k) (1 - *,) (1 -*,)..(!- x n ). 



