CHAP. XX.] PROBLEMS ON CAUSES. 321 



chapter, and other forms of the general inquiry will also be 

 noticed. I would remark, that although these examples are de- 

 signed chiefly as illustrations of a method, no regard has been 

 paid to the question of ease or convenience in the application of 

 that method. On the contrary, they have been devised, with 

 whatever success, as types of the class of problems which might 

 be expected to arise from the study of the relation of cause and 

 effect in the more complex of its actual and visible manifestations. 



2. PROBLEM I. The probabilities of two causes A l and^4 2 

 are Ci and c 2 respectively. The probability that if the cause A l 

 present itself, an event E will accompany it (whether as a conse- 

 quence of the cause A l or not) is p l , and the probability that if 

 the cause A 2 present itself, that event E will accompany it, 

 whether as a consequence of it or not, is p^ . Moreover, the 

 event E cannot appear in the absence of both the causes AI and 

 At-* Required the probability of the event E. 



The solution of what this problem becomes in the case in 

 which the causes A 19 A z are mutually exclusive, is well known 

 to be 



Prob. E = ^ i + c z2 ; 



and it expresses a particular case of a fundamental and very im- 

 portant principle in the received theory of probabilities. Here 

 it is proposed to solve the problem free from the restriction above 

 stated. 



* The mode in which such data as the above might be furnished by expe- 

 rience is easily conceivable. Opposite the window of the room in which I write 

 is a field, liable to be overflowed from two causes, distinct, but capable of being 

 combined, viz., floods from the upper sources of the River Lee, and tides from 

 the ocean. Suppose that observations made on N separate occasions have 

 yielded the following results : On A occasions the river was swollen by freshets, 

 and on P of those occasions it was inundated, whether from this cause or not. 

 On B occasions the river was swollen by the tide, and on Q of those occasions it 

 was inundated, whether from this cause or not. Supposing, then, that the field 

 cannot be inundated in the absence of both the causes above mentioned, let it be 

 required to determine the total probability of its inundation. 



Here the elements a, b, p, q of the general problem represent the ratios 



A P B Q 



AT' A' N' B' 



or rather the values to which those ratios approach, as the value of A" is indefi- 

 nitely increased. 



Y 



