72 JAMES BERNOULLI. 



log c + log {s - 1) /^ ^ s \ s__ 



log (r + 1) - log r V r + 1/ r + 1' 



and logc + log(r-l) A^ 



loor(s+ l)-log5 V 



(S + 1) - log 5 V 5 + 1/ 5+1 



James Bernoulli's demonstration of this result is long but 

 perfectly satisfactory ; it rests mainly on the fact that the terms 

 in the Binomial series increase continuously up to the greatest 

 term, and then decrease continuously. We shall see as we proceed 

 with the history of our subject that James Bernoulli's demonstra- 

 tion is now superseded by the use of Stirling's Theorem. 



124. Let us now take the application of the algebraical result 

 to the Theory of Probability. The greatest term of (r + 5)"', where 

 t=r-\-s is the term involving r"''^"'. Let r and s be proportional to 

 the probability of the happening and failing of an event in a single 

 trial. Then the sum of the 2?i + 1 terms of (r + s)"^ which have the 

 greatest term for their middle term corresponds to the probability 

 that in nt trials the number of times the event happens will lie 

 between n{r—l) and n (r+ 1), both inclusive ; so that the ratio 

 of the number of times the event happens to the whole number of 



7* + 1 T ~— 1. 



trials lies between and . Then, by taking for n the 



t f 



greater of the two expressions in the preceding article, we have 



the odds of c to 1, that the ratio of the number of times the event 



7* + 1 



happens to the whole number of trials lies between and 



r-1 



t ' 

 As an example James Bernoulli takes 



r = 30, 5=20, t=50. 



He finds for the odds to be 1000 to 1 that the ratio of the 

 number of times the event happens to the whole number of trials 



31 29 . . 



shall lie between —r and ~r, it will be sufficient to make 25550 



t)0 50 



trials ; for the odds to be 10000 to 1, it will be sufficient to make 



31258 trials ; for the odds to be 100000 to 1, it will be sufficient 



to make 36966 trials; and so on. 



