334 PROBLEMS ON CAUSES. [CHAP. XX. 



nomena, conditions which, it is felt, do not affect their mutual 

 independence. I apprehend therefore that the solution indicates, 

 that when a particular condition has prevailed through the whole 

 of our recorded experience, it assumes the above character with 

 reference to the class of phenomena over which that experience 

 has extended. 



8. PROBLEM IV. To illustrate in some degree the above 

 observations, let there be given, in addition to the data of the 

 last problem, the absolute probability of the event E, the com- 

 pleted system of data being 



Prob. x = a, Prob. y = >, Prob. z = c 9 

 Prob. xz = ap, Prob.yz = bq, 



and let it be required to find Prob. xy. 



Assuming, as before, xz = s, yz = t 9 xy = w, the final logical 

 equation is 



w = xystz + xysTz + (xystz + xy~tz + xyzsJ + xyzsJ 



xyz'sT). 



+ terms whose coefficients are -. (1) 



The algebraic system having been formed, the subsequent elimi- 

 nations may be simplified by the transformations adopted in the 

 previous problem. The final result is 



(2) 



The conditions among the constants are 



c > ap, c>bq 9 c < 1 - a (1 - p), c < 1 - b (1 - q). 

 Now if p = 1, q = 1, we find 



T> i. 



Prob. xy = , 



c not admitting of any value less than a or b. It follows hence 

 that if the event E is known to be an occasional one, its inva- 

 riable attendance on the events x and y increases the probability 

 of their conjunction in the inverse ratio of its own frequency. 



