CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 269 



Here Cj. = probability that if the event d occur, the event w will 

 occur, and so on for the others. Convenience must decide which 

 form is to be preferred. 



16. The above is the complete theoretical solution of the 

 problem proposed. It may be added, that it is applicable equally 

 to the case in which any of the events mentioned in its original 

 statement are conditioned. Thus, if one of the data is the proba- 

 bility p, that if the event x occur the event y will occur ; the 

 probability of the occurrence of x not being given, we must as- 

 sume Prob. x = c (an arbitrary constant), then Prob. xy = cp, and 

 these two conditions must be introduced into the data, and em- 

 ployed according to the previous method. Again, if it is sought 

 to determine the probability that if an event x occur an event y 

 will occur, the solution will assume the form 



-r, , '* i , Prob. xy 

 Prob. sought = _ . ^ 



Prob. x 9 



the numerator and denominator of which must be separately de- 

 termined by the previous general method. 



17. We are enabled by the results of these investigations to 

 establish a general rule for the solution of questions in probabi- 

 lities. 



GENERAL RULE. 



CASE I. When all the events are unconditioned. 



Form the symbolical expressions of the events whose proba- 

 bilities are given or sought. 



Equate such of those expressions as relate to compound events 

 to a new series of symbols, s, t, &c., which symbols regard as re- 

 presenting the events, no longer as compound but simple, to 

 whose expressions they have been equated. 



Eliminate from the equations thus formed all the logical sym- 

 bols, except those which express events, s, t, &c., whose respective 

 probabilities p, q, &c. are given, or the event w whose probability 

 is sought, and determine w as a developed function of s, t, &c. 

 in the form 



