DE MOIVRE. 173 



are Units in-d+l; then take as many of tlie Terms next following 



as there are Units in ^ d, and prefix to them in an inverted order the 



Coefiicients of the preceding Terms, omitting the last of them; and 

 those Terms taken all together will compose the Numerator of a Frac- 

 tion expressing the Probability required, the Denominator of which 

 Fraction ought to be {a + S)"^**. 



Example I. 



Supposing the number of Stakes, which A is to win, to be Three, 

 and the given number of Games to be Ten; let a+h be raised to the 

 tenth power, viz. a'' + lOa'h + ioa'bh + UOa' b' + 210a'b' + 252a'b' 

 + 210a'' 6' + 120a' 6' + 45aa&' + lOab' + b'". Then, by reason that n^3, 



and w + 6^=10, it follows that d is =7, and — ^r— = 4. Wherefore let 



Z 



the Four first Terms of the said Power be taken, viz. a^° + lOa^b 

 + 4:5a^bb + 120a''b^, and let the four Terms next following be taken 

 likewise without regard to their Coefficients, then prefix to them in an 

 inverted order, the Coefficients of the preceding Terms : thus the four 

 Terms following with their new Coefficients will be 120a^b^ + 4:5a^b^ 

 + 10a^b^+la^b^. Then the Probability which ^ has of winning three 

 Stakes of £ in ten Games or sooner, will be expressed by the following 

 Fraction 



a'o ^ iQ^,9 J ^ 45^8^5 ^ UOa'b^ + UOa'b' + iSa'b' + lOa'b' + a'b' 



' {a + by ' 



which in the Case of an Equality of Skill between A and B will be 



A w 352 11 



reducea to r— --r- or ^r^ . 

 1024 32 



808. In De Moivre's solution there is no difficulty in seeing 

 the origin of his fii'st set of terms, but that of the second set of 

 terms is not so immediately obvious. We will take his example, 

 and account for the last four terms. 



The last term is a^b\ There is only one way in which ^'s 

 capital may be exhausted while A wins only three games ; namely, 

 A must win the first three games. 



The next term is 10a'b\ There are ten ways in which B's 

 capital may be exhausted while A wins only four games. For let 

 there be ten places ; put h in any one of the first three places, 



