DE MOIVRE, 175 



that IS a -^ — \ — ^^ a^ ; 



[x ' 



see Art.. 804. 



The chance of A for winning at or he/ore the (n -f 2^)*^' game 

 is therefore 



a- I l+nah + '-^^-^^^ a'lf -\- ... 



niii^-x + 1) (?i + a? + 2) ... {7i^2x-l) ,-,, 

 + — ^ ^^ ^ ^^ -' ah 



\x 



Laplace, T]ieorie...des Proh., page 235. 



310. De Moivre says with respect to his Problem LXV, 



In the first attempt that I had ever made towards solving tlie 

 general Problem of the Duration of Play, which was in the year 1708, 

 I began with the Solution of this lxv^^ Problem, well knowing that 

 it might be a Foundation for what I farther wanted, since which time, 

 by a due repetition of it, I solved the main Problem : but as I found 

 afterwards a nearer way to it, I barely published in my first Essay on 

 those mattei'S, what seemed to me most simple and elegant, still pre- 

 serving this Problem by me in order to be published when I should 

 think it proper. 



De Moivre goes on to speak of the investigations of Montmort 

 and Nicolas Bernoulli, in words which we have akeady quoted ; see 

 Art. 181. 



311. Dr L. Oettinger on pages 187, 188 of his work entitled 

 Die Wahrscheinlichkeits-Rechnung, Berlin, 1852, objects to some 

 of the results which are obtained by De Moivre and Laplace. 



Dr Oettinger seems to intimate that in the formula, which we 

 have given at the end of Art. 309, Laplace has omitted to lay 

 down the condition that A has an unlimited capital ; but Laplace 

 has distinctly introduced this condition on his page 234. 



Again, speaking of De Moivre's solution of his Problem LXiv. 

 Dr Oettinger says, Er erhiilt das namliche unhaltbare Resultat, 

 welches Laplace nach ihm aufstellte. 



But there is no foundation for this remark ; De Moivre and 



