CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 281 



Prob.(l-g)(l-y) _ 2-p-q-r 

 l-r 2(1 -r) 



for the value here sought. 



These results are mutually consistent. For since it is certain 

 that the joint statement of A and B must be either true or false, 

 the second members of (3) and (5) ought by addition to make 1. 

 Now we have identically, 



p + q - r 2 -p- a - r 

 - 1 * * = 1 



2(1 -r) H 2(1 -r) 



It is probable, from the simplicity of the results (5) and (6), 

 that they might easily be deduced by the application of known 

 principles ; but it is to be remarked that they do not fall directly 

 within the scope of known methods. The number of the data 

 exceeds that of the simple events which they involve. M. Cour- 

 not, in his very able wor^ " Exposition de la Theorie des 

 Chances," has proposed, in such cases as the above, to select 

 from the original premises different sets of data, each set equal in 

 number to the simple events which they involve, to assume that 

 those simple events are independent, determine separately from 

 the respective sets of the data their probabilities, and comparing 

 the different values thus found for the same elements, judge how 

 far the assumption of independence is justified. This method 

 can only approach to correctness when the said simple events 

 prove, according to the above criterion, to be nearly or quite in- 

 dependent ; and in the questions of testimony and of judgment, 

 in which such an hypothesis is adopted, it seems doubtful whether 

 it is justified by actual experience of the ways of men. 



5. Ex. 4. From observations made during a period of gene- 

 ral sickness, there was a probability p that any house taken at 

 random in a particular district was visited by fever, a probability 

 q that it was visited by cholera, and a probability r that it es- 

 caped both diseases, and was not in a defective sanitary condition 

 as regarded cleanliness and ventilation. What is the probability 

 that any house taken at random was in a defective sanitary 

 condition ? 



With reference to any house, let us appropriate the symbols 

 x 9 y, z 9 as follows, viz. : 



