174 The Rules of Inference in Probability. [CHAP. vu. 



1080, 90 are short-sighted, and 990 are not. On the whole, 

 then, the 1200 are thus divided : black-haired short-sighted, 

 10 ; short-sighted without black hair, 90 ; black-haired men 

 who are not short-sighted, 110; men who are neither short 

 sighted nor have black hair, 990. 



This rule, expressed in its most general form, in the 

 language of Probability, would be as follows : If the chances 



of a thing being p and q are respectively and - , then the 



1 n 1 



chance of its being both p and q is , p and not q is , 



. m1 . (m 1) (n 1) 



&amp;lt;j and not p is - - , not p and not q is - , 



tnn mn 



where p and q are independent. The sum of these chances 

 is obviously unity; as it ought to be, since one or other of 

 the four alternatives must necessarily exist. 



6. I have purposely emphasized the distinction be 

 tween the inference in this case, and that in the two preced 

 ing, to an extent which to many readers may seem unwar 

 ranted. But it appears to me that where a science makes 

 use, as Probability does, of two such very distinct sources of 

 conviction as the necessary rules of arithmetic and the 

 merely more or less cogent ones of Induction, it is hardly 

 possible to lay too much stress upon the distinction. Few 

 will be prepared to deny that very arbitrary assumptions 

 have been made by many writers on the subject, and none 

 will deny that in the case of what are called inverse proba 

 bilities assumptions are sometimes made which are at least 

 decidedly open to question. The best course therefore is to 

 make a pause and stringent enquiry at the point at which the 

 possibility of such error and doubtfulness first exhibits itself. 

 These remarks apply to some of the best writers on the sub 

 ject; in the case of inferior writers, or those who appeal to 



