DE MOIVRE. 1G7 



This problem was not in the first edition. The gain of A 



. 1 



IS 75. 



301. Problem LVir, which was not in the first edition, is as 

 follows : 



A and B playing together deposit £s apiece ; A lias 2 Chances to 

 win s, and B 1 Chance to win 5, whereupon A tells B that he will 

 play with him. upon an equality of Chance, if he B will set him 2s to I5, 

 to which B assents : to find whether A has any advantage or disad- 

 vantage by that Bargain. 



In the first case ^'s expectation is - s, and in the second, 



o 



it is ^ 5 ; so that he gains ^ s by the bargain. 



802. We now arrive at one of the most important parts of 

 De Moivre's work, namely, that which relates to the Duration of 

 Play ; we will first give a full account of what is contained in the 

 third edition of the Doctrine of Chances, and afterwards state how 

 much of this was added to the investigations originally published 

 in the De Mensura Sortis. 



De Moivre himself regarded his labours on this subject with 

 the satisfaction which they justly merited ; he says in his 

 Preface, 



When I first began to attempt the general Solution of the Problem 

 concerning the Duration of Play, there was nothing extant that could 

 give me any light into that Subject; for altho' Mr de Monmort^ in the 

 first Edition of his Book, gives the Solution of this Problem, as limited 

 to three Stakes to be won or lost, and farther limited by the Suppo- 

 sition of an Equality of Skill between the Adventurers; yet he having 

 given no Demonstration of his Solution, and the Demonstration when 

 discovered being of very little use towards obtaining the general Solu- 

 tion of the Problem, I was forced to try what my own Enquiry would 

 lead me to, which having been attended with Success, the result of 

 what I found was afterwards published in my Specimen before men- 

 tioned. 



The Specimen is the Essay De Mensura Sortis. 



