THE INDUCTIVE OR INVERSE METHOD. 291 



separate probabilities contributed by each distinct hypo 

 thesis. 



To illustrate more precisely the method of solving the 

 problem, it is desirable to adopt some concrete mode of 

 representation, and the ballot-box, so often employed by 

 mathematicians, will best serve our purpose. Let the 

 happening of any event be represented by the drawing of 

 a white ball from a ballot-box, while the failure of an 

 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, I should 

 expect that after a certain number of trials the future 

 results would agree more or less closelv with the past 



o 



ones. 



The mathematical solution of the question consists in 

 nothing 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 im 

 probable ; a preponderance of white balls is a more pro 

 bable hypothesis, and as a deduction from this more 



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