DE MOIVRE. 145 



element of truth in the mistake made by M. de Mdre ; he assumed 

 that such a result should hold for all values of q : see Art. 14. 



263. As another example of the general enunciation of 

 Art. 262, let r = S. 



The chance that the event will happen at least 3 times in x 

 trials is equal to the first x — 2 terms of the expansion of 



a h 



X 



+ 



\a + b a + bj * 

 and this chance by hypothesis is - . Hence the last three terms 



of the expansion will also be equal to ^ , that is, 



W + xV-' a + ^^^ I'-'' «' = I (« + W' 



If ^ = 1 we find x = o. 



X 



If q be supposed indefinitely great, and we put - = z, we get 



where e is the base of the Napierian logarithms. 



By trial it is found that 2 =2675 nearly. Hence De Moivre 

 concludes that x always lies between oq and 2675(;^. 



264. De Moivre exhibits the following table of results ob- 

 tained in the manner shewn in the two preceding Ai'ticles. 



A Table of the Limits. 



The Value of x will always be 



For a single Event, between \q and O'GOSg'. 

 For a double Event, between 2)q and l-GTS^-. 

 For a triple Event, between 5q and 2 675^. 

 For a quadruple Event, between ^q aud 2>(dl2q. 

 For a quintuple Event, between 9^' and i-GTO^-. 

 For a sextuple Event, between \\q and b-ij^Sq. 

 ka. 



10 



