DE MOIVRE. 113 



258. De Moivre notices the advantage arising from employing 

 a single letter instead of two or three to denote the probaljility of 

 the happening of one event. Thus if x denote the probability of 

 the happening of an event, \ —x will denote the probability of its 

 failing. So also y and z may denote the probabilities of the hap- 

 pening of two other events respectively. Then, for example, 



x{\-y){\-z) 



will represent the probability of the first to the exclusion of the 

 other two. De Moivre says in conclusion, '^ and innumerable cases 

 of the same nature, belonging to any number of Events, may be 

 solved without any manner of trouble to the imagination, by the 

 mere force of a proper notation." 



259. In his third edition De Moivre draws attention to the 

 convenience of approximating to a fraction with a large numerator 

 and denominator by continued fractions, which he calls "the 

 Method proposed by Dr Wallis, Hiiygens, and others." He gives 

 the rule for the formation of the successive convergents which is 

 now to be found in elementary treatises on Algebra ; this rule he 

 ascribes to Cotes. 



2G0. The Doctrine of Clicuices contains 7-i problems exclusive 

 of those relating to life annuities ; in the first edition there were 

 53 problems. 



261. We have enunciated Problems I. and ii. in Art. 240. 

 Suppose p and q to represent the chances of A and ^ in a single 

 game. Problem I. means that it is an even chance that A "wall win 



1 1 



three o^ames before B wins one : thus p^ = cv- Hence x> = -^^^ , and 



7 = 1 — 7777 . Problem li. means that it is an even chance that A 



will win three games before B wins two. Thus p^ + Aip^q = ^ ; which 



must be solved by trial. 



These problems are simple examples of the general formula in 

 Art. 172. 



262. Problems ill, IV, and V. are included in the followin 





