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



Similarly, Prob. that it does happen the nth. time 



1 



This is Boole's result. 2 



Let us now consider the general problem. 



I. If an event has occurred n — r times in n times, what is the 

 probability that it has occurred ; 1, 2, . . . r times of the remain- 

 ing times ? 



Prob. that it has happened all the r times 



Number of constitutions x n ~ r x r (i. e. x n ) 

 ~ No. of const. x n + No. of const. x n ~ r of- J x + . . . + No. of const. x n ' r 



1 



1.2 \r 



=Pr 



This denotes a very small chance, and one that decreases with 

 the increase both of n and r. 

 Similarly we have 



Prob. that it has happened r — 1 times and not happened once 



n 



1 + 7l + 1,2 + 



=i>r-i ; 



Prob. that it has happened r — 2 times and not happened twice 



n . (n — 1) 



7— Tx Pr-2) 



1 n.(n — 1) 

 1+^+ j. 2 + . . . 



&c. = &c; 

 Prob. that it has happened in none of the r times 

 n.(n — 1) . . . (?2— r + 1) 



Therefore 



n : 1 



\n . |» 



" |r /i — r r* — 1 \n — r+1 



1 1 



r |n— r * r— 1 n—r + 1 ' 



2 7* — 2 * 7i — 1 * ItT 



