252 THE PRINCIPLES OF SCIENCE. [CHAP. 



event is represented by the drawing of a black ball. Now, 

 in the inductive problem we are supposed to be ignorant 

 of the contents of the ballot-box, and are required to 

 ground all our inferences on our experience of those con- 

 tents as shown in successive drawings. Rude common 

 sense would guide us nearly to a true conclusion. Thus, 

 if we had drawn twenty balls one after another, replacing 

 the ball after each drawing, and the ball had in each case 

 proved to be white, we should believe that there was a 

 considerable preponderance of white balls in the urn, and 

 a probability in favour of drawing a white ball on the next 

 occasion. Though we had drawn white balls for 

 thousands of times without fail, it would still be possible 

 that some black balls lurked in the urn and would at last 

 appear, so that our inferences could never be certain. On 

 the other hand, if black balls came at intervals, we should 

 expect that after a certain number of trials the black balls 

 would appear again from time to time with somewhat the 

 same frequency. 



The mathematical solution of the question consists in 

 little more than a close analysis of the mode in which our 

 common sense proceeds. If twenty white balls have been 

 drawn and no black ball, my common sense tells me that 

 any hypothesis which makes the black balls in the urn 

 considerable compared with the white ones is improbable ; 

 a preponderance of white balls is a more probable- hypo- 

 thesis, and as a deduction from this more probable hypo- 

 thesis, I expect a recurrence of white -balls. The mathe- 

 matician merely reduces this process of thought to exact 

 numbers. Taking, for instance, the hypothesis that there 

 are 99 white and one black ball in the urn, he can calcu- 

 late the probability that 20 white balls would be drawn 

 in succession in those circumstances; he thus forms a 

 definite estimate of the probability of this hypothesis, and 

 knowing at the same time the probability of a white ball 

 reappearing if such be the contents of the urn, he com- 

 bines these probabilities, and obtains an exact estimate 

 that a white ball will recur in consequence of this hypo- 

 thesis. But as this hypothesis is only one out of many 

 possible ones, since the ratio of white and black balls may 

 be 98 to 2, or 97 to 3, or 96 to 4, and so on, he has to 

 repeat the estimate for every such possible hypothesis. 



