68 INDUCTION. 



must be laid for an evaluation of the chances, by possessing 

 ourselves of the utmost attainable amount of positive know- 

 ledge. The knowledge required is that of the comparative 

 frequency with which the different events in fact occur. For 

 the purposes, therefore, of the present work, it is allowable to 

 suppose, that conclusions respecting the probability of a fact 

 of a particular kind, rest on our knowledge of the proportion 

 between the cases in which facts of that kind occur, and those 

 in which they do not occur : this knowledge being either de- 

 rived from specific experiment, or deduced from our knowledge 

 of the causes in operation which tend to produce, compared 

 with those which tend to prevent, the fact in question. 



Such calculation of chances is grounded on an induction ; 

 and to render the calculation legitimate, the induction must be 

 a valid one. It is not less an induction, though it does not 

 prove that the event occurs in all cases of a given description, 

 but only that out of a given number of such cases, it occurs 

 in about so many. The fraction which mathematicians use to 

 designate the probability of an event, is the ratio of these two 

 numbers ; the ascertained proportion between the number of 

 cases in which the event occurs, and the sum of all the cases, 

 those in which it occurs and in which it does not occur taken 

 together. In playing at cross and pile, the description of cases 

 concerned are throws, and the probability of cross is one-half, 

 because if we throw often enough, cross is thrown about once 

 in every two throws. In the cast of a die, the probability of 

 ace is one-sixth; not simply because there are six possible 

 throws, of which ace is one, and because we do not know any 

 reason why one should turn up rather than another ; though 

 I have admitted the validity of this ground in default of a 

 better ; but because we do actually know, either by reasoning 

 or by experience, that in a hundred, or a million of throws, 

 ace is thrown about one- sixth of that number, or once in six 

 times. 



4. I say, "either by reasoning or by experience;" 

 meaning specific experience. But in estimating probabilities, 

 it is not a matter of indifference from which of these two 



