CHAP. XX.] PROBLEMS ON CAUSES. 357 



vious problems, let it be required to determine the probability 

 that if the event E present itself, it will be associated with the 

 particular cause A r ; in other words, to determine the a posteriori 

 probability of the cause A r when the event E has been observed 

 to occur. 



In this case we must seek the value of the fraction 



Prob. x r z c r p r 



~> , - , or ^^ , by the data. (1) 



Prob.z Prob. z 



As in the previous problems, the value of Prob. z has been as- 

 signed upon different hypotheses relative to the connexion or 

 want of connexion of the causes, it is evident that in all those 

 cases the present problem is susceptible of a determinate solution 

 by simply substituting in (1) the value of that element thus de- 

 termined. 



If the a priori probabilities of the causes are equal, we have 

 d = c 2 . . = c r . Hence for the different causes the value (1) will 

 vary directly as the quantity p r . Wherefore whatever the nature 

 of the connexion among the causes, the d posteriori probability of 

 each cause will be proportional to the probability of the observed 

 event E when that cause is known to exist. The particular case 

 of this theorem, which presents itself when the causes are mu- 

 tually exclusive, is well known. We have then 



vz c r p r p r 



Prob. z ^c r p r pi+ p 2 . . + pn 



the values of c x , . . c n being equal. 



Although, for the demonstration of these and similar theo- 

 rems in the particular case in which the causes are mutually ex- 

 clusive, it is not necessary to introduce the functional symbol 0, 

 which is, indeed, to claim for ourselves the choice of all possible 

 and conceivable hypotheses of the connexion of the causes, yet, 

 under every form, the solution by the method of this work of 

 problems, in which the number of the data is indefinitely great, 

 must always partake of a somewhat complex character. Whe- 

 ther the systematic evolution which it presents, first, of the logi- 

 cal, secondly, of the numerical relations of a problem, furnishes 

 any compensation for the length and occasional tediousness of its 



