xii.] THE INDUCTIVE OR INVERSE METHOD. 257 



much from that which would be obtained on the hypothesis 

 of any large finite number of balls. (3) The supposition 

 leads to a series of simple formulas 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, bi, 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 m , 



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 

 probability that the next planet beyond Neptune will 

 conform to the law is yj. 



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- 

 pened increased by one, by the same number increased by 

 ane and the number of times it is to happen. 



An event having happened m times without fail, the 



probability that it will happen n more times is *? * . 



Thus the probability that three new planets would obey 

 Bode's law is ^ ; but it must be allowed that this, as well 

 as the previous result, would be much weakened by the 

 fact that Neptune can barely be said to obey the law. 



3. An event having happened and failed a certain 

 number of times, to find the probability that it will happen 

 the next time, divide the number of times the event has 



