from the Mathematical Theory of Probabilities. 517 



Popularly stated, the conclusion is that, so far as considera- 

 tions of mere probability go, an event being known to have 

 happened n—r times, it is more probable than not that it has 

 not happened in any of the remaining r times, and less pro- 

 bable that it has happened a greater number of times than a 

 less. Also, since the _p's decrease with the increase of r, it 

 follows, if n be taken as constant, that the more the event is 

 known to have happened the less likely is its occurrence in the 

 unknown cases. 



II. If an event has occurred n—r times in succession, what 

 u the probability that it will occur 0, 1, ... r times in the fol- 

 lowing r times. 



Prob. that it will happen in all the following r times 



No. of constitutions (oc x . . . ) w _ r (x x . . . ) r 

 No. of (x . . . ) n + No. of ( x . . . ) w _! x + No. of (x . . . ) n _ 2 (S . • . )s + 



. . . + No. Of (x . . . ) n -r(% . . . ) r 



1 + 



+ 



r- 1 .1 \r-2\2 

 1 



+ ... + 



i . , r.(r-l) 



l.r-1 



+ 1 



+ r + 1 



= Prob. that it will happen in none of the r times. 

 And Prob. that it will happen once and once only 

 = Prob. that it will not happen once 

 r 



~r 



Prob. that it will happen twice 



= Prob. that it will not happen twice 

 r . (r — 1) 



&c. = &c. 

 Prob. that it will happen e times 



= Prob. that it will not happen e times 

 ' r . (y— 1) . . . (r—e+ 1) 



\e .2 r ' 



