THE INDUCTIVE OR INVERSE METHOD. 297 



opening for the doubt which Boole has cast upon it b . 

 But it may be replied, (i) that the supposition of an infinite 

 number of balls treated in the manner of Laplace is less 

 arbitrary and more comprehensive than any other that 

 could be suggested. (2) The result does not differ much 

 from that which would be obtained on the hypothesis of 

 any very ]arge finite number of balls. (3) The supposition 

 leads to a series of simple formulae which can be applied 

 with ease in many cases, and which bear all the appearance 

 of truth so far as it can be independently judged by a 

 sound and practiced understanding. 



Rules of the Inverse Method. 



By the solution of the problem, as described in the last 

 section, we obtain the following series of simple rules. 



1. To find the probability that an event which has not 

 hitherto ~been observed to fail will happen once more, 

 divide the number of times the event has been observed 

 increased by one, by the same number increased by two. 



If there have been m occasions on which a certain event 

 might have been observed to happen, and it has happened 

 on all those occasions, then the probability that it will 



happen on the next occasion of the same kind is ^il 



m+2' 



For instance, we may say that there are nine places in 

 the planetary system where planets might exist obeying 

 Bode's law of distance, and in every place there is a 

 planet obeying the law more or less exactly, although 

 no reason is known for the coincidence. Hence the pro- 

 bability that the next planet beyond Neptune will 

 conform to the law is ~. 



2. To find the probability that an event which has not 

 hitherto failed will not fail for a certain number of new 

 occasions, divide the number of times the event has hap- 



b 'Laws of Thought,' pp. 368-375. 



