390 PROBABILITY OF JUDGMENTS. [CHAP. XXI. 



These equations involve the direct solution of the problem 

 under consideration. In (16) we have the type of n equations 

 (formed by giving to i the values 1, 2, . . n successively), from 

 which the values of # 1? # 2J #n> will be found, and those values 

 substituted in (15) give the value of Prob. X as a function of 

 the constants i , c l , &c. 



One conclusion deserving of notice, which is deducible from 

 the above solution, is, that if the probabilities of the compound 

 events X i9 . . X m .i, are the same as they would be were the 

 events x l , . . x n entirely independent, and with given probabi- 

 lities Ci, . . c n , then the probability of the event X will be the 

 same as if calculated upon the same hypothesis of the absolute 

 independence of the events %i , . . x n . For upon the hypothesis 

 supposed, the assumption x l = c x , x n - c n , in the quantitative 

 system would give Xi = cti, X m = a TO , whence (15) and (16) 

 would give 



Prob. X = (XXJ + (XX 2 ) . . + (XX n ), (17) 



(an X,) + (an X t )..+ (an X m ) = Ci . (18) 



But since Xi+ X 9 . .+ X m = 1, it is evident that the second 

 member of (17) will be formed by taking all the constituents that 

 are contained in X, and giving them an algebraic significance. 

 And a similar remark applies to (18). Whence those equations 

 respectively give 



Prob. X (logical) = X (algebraic), 



on = Ci . 

 Wherefore, if X = (#,, o? 2 , - #), we have 



Prob. X = $ (c l9 c 2 , . . c n ), 

 which is the result in question. 



Hence too it would follow, that if the quantities c l9 . . c n 

 were indeterminate, and no hypothesis were made as to the 

 possession of a mean common value, the system (15) (16) would 

 be satisfied by giving to those quantities any such values, 

 Xi , # 2 > %n 9 as would satisfy the equations 



X l = ! . . X m .i = ,n-i > X = , 



supposing the value of the element a, like the values of ,,. . a m .i, 

 to be given by experience. 



