190 DE moivrp:. 



but it involves an interesting fact in approximation, and we will 

 therefore explain it. 



Suppose two players A and B of equal skill ; let A have an 

 infinite number of counters, and B have the number j^- Let 

 (j) [n,p) denote the chance that B will be ruined in n games. Then 

 the required ratio is 1 — (/) (ii, p) ; this follows from the first form 

 of solution of Problem LXV; see Art. 307. Again, suppose that 

 each of the players starts with p counters ; and let -v/r [n, p) then 

 denote the chance that B will be ruined in 7i games ; similarly if 

 each starts with Zp counters let -v/r {71, Sp) denote the chance that 

 B will be ruined in n games ; and so on. Then De Moivre says 

 that approximately 



^ (??, p) = 'f (n, p)+'^ (n, Sp), 



and still more approximately 



The closeness of the approximation will depend on n being 

 large, and p being only a moderate fraction of n. 



These results follow from the formulse given on pages 199 

 and 210 of the Doctrine of Chances... The second term of 

 T^r {n, p) is negative, and is numerically equal to the first term 

 of A|r {n, Sj)), and so is cancelled ; similarly the third term of 

 'ylr[n,p) is cancelled by the first of — 'sfr [n, 5p), and the fourth 

 term of ^fr (n, p) by the first of -v/r (/z, 7p). The terms which do 

 not mutually cancel, and which we therefore neglect, involve 

 fewer factors than that which we retain, and are thus com- 

 paratively small. 



333. We now proceed to notice the Supplement to the Mis- 

 cellanea Analytica. The investigations of problems in Chances 

 had led mathematicians to consider the approximate calculation 

 of the coefficients in the Binomial Theorem ; and as we shall now 

 see, the consequence was the discovery of one of the most striking 

 results in mathematics. The Supplement commences thus : 



Aliquot post diebus qiiam Liber qui inscribitur, Miscellanea Analy- 

 tica, in lucem prodiisset, Doctissimus Stlrlingius me liteiis admonuit 

 Tabulam ibi a me exhibitam de summis Logarithmorum, non satis aii- 

 toritatis habere ad ea firmanda quse in speculatione nitercntur, utpote 



