266 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



If we represent the function A + B + C by F, the system (4) 



becomes 



w = A + qCj (5) 



F- 1. (6) 



Let us for a moment consider this result. Since F is the sum 

 of a series of constituents of s, t, &c., it represents the compound 

 event in which the simple events involved are those denoted by 

 s 9 t, &G. Hence (6) shows that the events denoted by s, t, &c., 

 and whose probabilities are /?, q, &c., have such probabilities not 

 as independent events, but as events subject to a certain condition 

 F. Equation (5) expresses w as a similarly conditioned combi- 

 nation of the same events. 



Now by Principle vi. the mode in which this knowledge of the 

 connexion of events has been obtained does not influence the mode 

 in which, when obtained, it is to be employed. We must reason 

 upon it as if experience had presented to us the events s, t, &c., 

 as simple events, free to enter into every combination, but pos- 

 sessing, when actually subject to the condition F", the probabili- 

 ties p, q, &c., respectively. 



Let then //, q', . . , be the corresponding probabilities of such 

 events, when the restriction F is removed. Then by Prop. u. 

 of the present chapter, these quantities will be determined by the 

 system of equations, 



& c, (T) 



) 



and by Prop. i. the probability of the event w under the same 

 condition F will be 



wherein V 8 denotes the sum of those constituents in F of which s 

 is a factor, and [ FJ what that sum becomes when s, , . . , are 

 changed into p', </, . . , respectively. The constant c represents 

 the probability of the indefinite event q\ it is, therefore, arbitrary, 

 and admits of any value from to 1 . 



Now it will be observed, that the values of/?', q' 9 &c., are de- 

 termined from (7) only in order that they may be substituted in 

 (8), so as to render Prob. w a function of known quantities, p, q, 



