from the Mathematical Theory of Probabilities. 513 



Logicians naturally have taken exception to this result; and 

 Boole has deduced from his alternative hypothesis the very- 

 different result that the probability is \, which, as the expres- 

 sion of mere indifferent ignorance, brings us back to our 

 original position. I have arrived at the same conclusion by 

 a process of reasoning which is longer, but by which I ob- 

 tain an answer to the more general question, i. e. if an event 

 has occurred n — r times out of n times, ivhat is the probability 

 (1) that it has occurred or will occur 0, 1, 2 ... r times of the 

 remaining r times, the order of these occasions relatively to the 

 n— -r times being quite indeterminate*} and (2) that it will occur 

 0, 1, 2 . . . r of the remaining r times, these being after the n— r 

 times ? It is the second question that has special interest for 

 the problem of inductive logic, as bearing directly on the 

 inquiry — How does our knowledge of the past enable us to 

 have a knowledge of the future ? But the first question is its 

 necessary complement in the inductive inquiry. 



Let 1 denote the Universe in Time ; 



x denote a time of an event, i. e. its occurrence ; 



x denote a time of its non-occurrence ; 



n denote total number of times in the Universe. 



Then the Universe may be considered as developed into a 

 series of x and x with n terms, as, for instance, 



x+x+x+x+x + . .. ; 



or, for brevity, we may prefer the form 



x x x xx ... $ 



it being possible that any place in the series may be occupied 

 either by x or x. Now the hypothesis is, that any series of x 

 and x is as probable as any other. All series in which x 

 occurs the same number of times are of the same type. 



To determine the number of possible constitutions, the number 

 of types, and the number of each type. 



(1) Every constituent may be an x; 



No. of ways = 1. 



(2) n—1 constituents may be x, and 1 constituent x; 



No. of ways = n. 



(3) ft— 2 const, may be x, and 2 constituents x; 



xt j? \ n n.(n— 1) 



No. of ways = — ^— = | 9 7 , 



n-2 |2 1.2 



