144 DE MOIVRE. 



general enunciation. Suppose a the number of chances for the 

 happening of an event in a single trial, and h the number of 

 chances for its failing : find how many trials must be made to have 

 an even chance that the event will happen r times at least. . 

 For example, let r = 1. 



Suppose X the number of trials. Then the chance that 



If 

 the event fails x times in succession is -. tt^ . And by suppo- 

 sition this is equal to the chance of its happening once at least 

 in X trials. Therefore each of these chances must be equal 



to -X . Thus 



2 



}f 1 



{a + hy 2 ' 

 from this equation x may be found by logarithms. 



De Moivre proceeds to an approximation. Put - = q. Thus 



X log [ 1 4- - ) = log 2. 



If ^ = 1, we have x=l. If 5' be gi'eater than 1, we have by 



expanding log ( 1 + - J , 



where log 2 will mean the logarithm to the Napierian base. Then 

 if q be large we have approximately 



7 



x= q log 2 = zTTzq nearly. 



De Moivre says, page 87, 



Thus we have assigned the very narrow limits within which the ratio 

 of £c to q is comprehended ; for it begins with unity, and terminates at 

 last in the ratio of 7 to 10 very near. 



But X soon converges to the limit 0.7^', so that this value of x may 

 be assumed in all cases, let the value of q be what it will. 



The fact that this result is true when q is moderately large is the 



