512 Mrs. S. Bryant on the Deduction of Inductive Principles 



is not last A 2 . X 1 + X 2 = l, and the X's, otherwise, are indeter- 

 minate. Hence if x n ~ l x occurs, 



(1) xx occurs n—2 times with prob. \ ly 



(2) „ n — 3 times with prob. \ 2 . 



Also if x n ~ 2 x 2 occurs, x x may come last in the series, in 

 which case x x occurs ^— 3 times ; or it may come any- 

 where else, or only one x come at the end, in both which 

 cases x x occurs w— 4 times ; or the two x's may occur sepa- 

 rately and neither at the end, in which case xx occurs n— 5 

 times. Let the probabilities of these three cases be fa, /a 2 , ^ 

 respectively, where fa + fa + Pz = 1. Then, if x n " 2 x 2 occurs, 



(1) xx occurs Ti — 3 times w r ith prob. fa, 



(2) „ n-4 „ „ ft* 



(3) „ w-5 „ „ fa. 



The other cases can similarly be analysed, but it is not 

 necessary for our purpose to go further. 



Now, by hypothesis, x n , x n " x x, &c. are all equally probable. 

 Let p = their probability. Then it follows at once from the 

 above that 



p= prob. that xx occurs n— 1 times, 



*!/>= 99 99 99 ^~ 2 99 



Ol + A*)P= 99 99 99 ^-3 99 



(y 1 + /i 2 )p= „ „ „ n-A „ 



&C. ss &C 



With the exception of the first, these are all indeterminate, 

 and they cannot be equal unless X 1 = /^ 1 =j/ 1 = . . , = 1, a sup- 

 position which is plainly inadmissible, as perfectly arbitrary. 

 The hypothesis is therefore, in the first place, inadequate, and, 

 in the second place, inconsistent as applied to all events indif- 

 ferently. It must therefore be rejected, 



II. 



Accepting, then, Boole's hypothesis, the question arises — 

 How does our mere knowledge of the past enable us to deter- 

 mine our expectancy of the future? and, further, what warrant 

 in general does the known occurrence of an event give us for 

 inferring its occurrence in cases unknown? From the hypo- 

 thesis of equally probable probabilities, it was, as is well 

 known, inferred that if an event has occurred m times in suc- 

 cession, the probability of its occurrence the (m + l)thtime is 



m + 1 

 m + 2 



