284 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



indicated a defective sanitary condition, the probability that any 

 house would be in a defective sanitary state would be simply 

 equal to the probability that it was not found in that category 

 denoted by 2, the probability of which would, by the data, be 1 - r. 

 Perhaps the general verification of the above solution would be 

 difficult. 



The constants p, q, and r in the above solution are subject to 



the conditions 



/? + /<!, + r<l. 



7. Ex. 5. Given the probabilities of the premises of a hypo- 

 thetical syllogism to find the probability of the conclusion. 

 Let the syllogism in its naked form be as follows : 



Major premiss : If the proposition Yis true X is true. 

 Minor premiss : If the proposition Z is true Y is true. 

 Conclusion : If the proposition Z is true X is true. 



Suppose the probability of the major premiss to bep, that of the 

 minor premiss q. 



The data then are as follows, representing the proposition X 

 by x, &c., and assuming c and c as arbitrary constants : 



Prob. y = c, Prob. xy = cp ; 

 Prob. z = c', Prob. yz = c'q ; 

 from which we are to determine, 



Prob. xz Prob. xz 



__ r\\* _ . 



Prob.z c 



Let us assume, 



xy = u, yz = V) xz - w ; 



then, proceeding according to the usual method to determine w 

 as a developed function of y, z 9 u 9 and v, the symbols corres- 

 ponding to propositions whose probabilities are given, we find 



w = uzvy + OM (1 - z*) (1 - v) y + (1 - u) zvy 



+ (1 - u) (1 - z) (1 - v) (1 - y) + terms whose coeffi- 

 cients are - ; 



