332 PROBLEMS ON CAUSES. [CHAP. XX. 



above formula with the result proper to some known case. For 

 instance, if it were certain that the event A is always, and the 

 event B never, associated with the event E, then it is certain that 

 the events A and B are never conjoined. Hence if p = 1, g = 0, 

 we ought to have u = 0. Now the assumptions p- 1, q = 0, 



give 



h = a-l, h' = b-l, Z=0, m = a + b-2. 



Substituting in (10) we have 

 p rob _(-l)-l 



and this expression vanishes when the lower sign is taken. 

 Hence the final solution of the general problem will be expressed 

 in the form 



Frob.03/ lh' + h (h'm -l)- m ^{(l- M') 2 + 4hh'abpg} 

 "Prob. or = 2ahh' 



wherein k = ap + bq - 1, h' = a (1 - p) + b (1 - q) - 1, 



/ = ab (p + q - 1), m = a + b - 2. 



As the terms in the final logical solution affected by the co- 

 efficient - are the same as in the first problem of this chapter, 

 the conditions among the constants will be the same, viz., 

 ap 5 1 - b (1 - q), bq < 1 - a (1 - p). 



7. It is a confirmation of the correctness of the above solution 

 that the expression obtained is symmetrical with respect to the 

 two sets of quantities/?, q, and 1 -p, 1 - q, i. e. that on changing 

 p into 1 - p, and q into 1 - q, the expression is unaltered This 

 is apparent from the equation 



ftob.q, -06(12 + ^ P)0-g)| 



I fJL V 



employed in deducing the final result. Now if there exist pro- 

 babilities p, q of the event E, as consequent upon a knowledge 

 of the occurrences of A and B, there exist probabilities 1 -/?, 1 - q 

 of the contrary event, that is, of the non-occurrence of E under 

 the same circumstances. As then the data are unchanged in 



