CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 265 



In the process of elimination we may, if needful, avail our- 

 selves of the simplifications of Props, i. and n. Chap. ix. 



13. Should the data, beside informing us of the probabilities 

 of events, further assign among them any explicit connexion, such 

 connexion must be logically expressed, and the equation or equa- 

 tions thus formed be introduced into the general system. 



PROPOSITION IV. 



14. Given the probabilities of any system of events ; to deter- 

 mine by a general method the consequent or derived probability of 

 any other event. 



As in the last Proposition, let S, T 7 , &c., be the events whose 

 probabilities are given, TFthe event whose probability is sought, 

 these being known functions of #, y, z, &c. Let us represent the 

 data as follows : 



Probability of S = p ; 



Probability of T= ? ; 



and so on, p, q, &c., being known numerical values. If then 

 we represent the compound event S by s, T by t, and W by w, 

 we find by the last proposition, 



w = A + QB + - C + - D\ C2^ 



ft 



^4, JB, C 9 and D being functions of s, t, &c. Moreover the data 

 (1) are transformed into 



Prob. s = p, Prob. t = q, &c. (3) 



Now the equation (2) is resolvable into the system 

 w = A + qC 1 



D-O < 4 > 



x/ u, j 



q being an indefinite class symbol (VI. 12). But since by the 

 properties of constituents (V. Prop, in.), we have 



A + B+ C+D= 1, 



the second equation of the above system may be expressed in the 



form 



A + B+ C= 1. 



