182 DE MOIVKE. 



The particular case of Problem LXXI, in which a = h, and 

 c = d, was given in the first edition of the Doctrine of Chances, 

 13age 152. 



823. Problems LXXII. and LXXIII. are important ; it will be 

 sufficient to enunciate the latter. 



A and B playing together, and having a different number of Chances 

 to win one Game, which number of Chances I suppose to be respectively 

 as a to h, engage themselves to a Sj)ectator S, that after a certain, number 

 of Games is over, A shall give him as many Pieces as he wins Games, 



over and above ^ n, and B as many as he wins Games, over and above 



the number n ; to find the Expectation of S. 



Problem LXXII. is a particular case of Problem LXXIII. obtained 

 by supposing a and h to be equal 



These two problems first appeared in the Miscellanea Ana- 

 lytica, pages 99 — 101. We there find the following notice respect- 

 ing Problem LXXII : 



Cum aliquando labente Anno 1721, Yir Clarissimus Alex. Cuming 

 Eq. Au. Pegi?e Societatis Socius, quaestionem infra subjectam mihi 

 proposuisset, solutionem problematis ei postero die tradideram. 



After giving the solution De Moivre proceeds to Problem LXXIII. 

 which he thus introduces : 



Eodem procedendi modo, solutum fuerat Problema sequens ab eodem 

 CI. viro etiam propositum, ejusdem generis ac superius sed multo latins 

 patens. 



We will give a solution of Problem LXXIII ; De Moivre in the 

 Doctrine of Chances merely states the result. 



Let n = c {a-\-h) ', consider the expectation of 8 so far as it 



depends on A. The chance that A will win all the games is 



a" 



ia + hy 



- , and in this case he gives ch to S. The chance that A will 



.n-lj 



win n—1 games is 7 r-r , and in this case he gives cl — l to S. 



And so on. 



