OF THE CALCULATION OF CHANCES. 



355 



is not less an induction, though it 

 does not prove that the event occiirs 

 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 mathe- 

 maticians use to designate the pro- 

 bability 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 plajdng 

 at cross and pile, the description of 

 eases concerned are throws, and the 

 probability of cross is one-half, be- 

 cause 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 thei-e are six possible 

 throws, of which ace is one, and be- 

 cause 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 actu- 

 ally know, either by reasoning or by 

 experience, that in a hundred or a 

 million of throws, ace is thrown in 

 about one-sixth of that number, or 

 <mce in six times, 



§ 4. I say, "either by reasoning or 

 by experience ; " meaning specific ex- 

 perience. But in estimating pro- 

 babilities, it is not a matter of in- 

 difference from which of these two 

 sources we derive our assurance. The 

 probability of events as calculated 

 from their mere frequency in past 

 experience affords a less secure basis 

 for practical guidance than their pro- 

 bability as deduced from an equally 

 accurate knowledge of the frequency 

 of occurrence of their causes. 

 ' The generalisation that an event 

 occurs in ten out of every hundred 

 cases of a given description is as real 

 an induction as if the generalisation 

 were that it occurs in all cases. But 

 when we arrive at the conclusion by 

 loerelj' coupting instances in victual 



experience, and comparing the num- 

 ber of cases in which A has been 

 present with the number in which 

 it has been absent, the evidence is 

 only that of the Method of Agree- 

 ment, and the conclusion amounts 

 only to an empirical law. We can 

 make a step beyond this when we 

 can ascend to the causes on which the 

 occurrence of A or its non -occurrence 

 will depend, and form an estimate 

 of the comparative frequency of the 

 causes favourable and of those un- 

 favourable to the occurrence. These 

 are data of a higher order, by which 

 the empirical law derived from a 

 mere numerical comparison of affirm, 

 ative and negative instances will b(^ 

 either corrected or confirmed, and in 

 either case we shall obtain a more 

 correct measure of probability than 

 is given by that numerical compari- 

 son. It has been well remarked that 

 in the kind of examples by which 

 the doctrine of chances is usually 

 illustrated, that of balls in a box, 

 the estimate of probabilities is sup- 

 ported by reasons of causation stronger 

 than specific experience. "What is 

 the reason that in a box where there 

 are nine black balls and one white, 

 we expect to draw a black ball nine 

 times as much (in other words, nine 

 times as often, frequency being the 

 gauge of intensity in expectation) as 

 a white? Obviously because the local 

 conditions are nine times as favour- 

 able, because the hand may alight in 

 nine places and get a black ball, while 

 it can only alight in one place and 

 find a white ball ; just for the same 

 reason that we do not expect to suc- 

 ceed in finding a friend in a crowd, 

 the conditions in order that we and 

 he should come together being many 

 and difficult. This of course would 

 not hold to the same extent were the 

 white balls of smaller size than the 

 black, neither would the probability 

 remain the same : the larger ball 

 would be much more likely to meet 

 the hand." * 



It i», in fact, evident, that when 

 * Prosptctive Review for February 185Q, 



