146 DE MOIVRE. 



And if the number of Events contended for, as well as the number 



q be pretty large in respect to Unity; the number of Trials requisite for 



.... ^n—\ . , 



those Events to happen n times will be — ^ q^ or barely nq. 



De Moivre seems to have inferred the general result enun- 

 ciated in the last sentence, from observing the numerical values 

 obtained in the six cases which he had calculated, for he gives no 

 further investigation. 



265. In Art. 263 we have seen that De Moivre concludes 



that - always lies between 5 and 2"675. This may appear very 



probable, but it is certainly not demonstrated. It is quite con- 

 ceivable, in the absence of any demonstration to the contrary, that 



- should at first increase with q, and so be greater than 5, and 



then decrease and become less than 2 675, and then increase 

 again to its limit 2-675. The remark applies to the general pro- 

 position, whatever be the value of r, as well as to the particular 

 example in which r =- 3. 



It would not be very easy perhaps to shew from such an 

 equation as that in Art. 263, that x increases continually with q ; 

 and yet from the nature of the question we may conclude that 

 this must be the case. For if the chance of success in a single 

 trial is diminished, it appears obvious that the number of trials 

 must be increased, in order to secure an even chance for the event 

 to happen once at least. 



266. On pages 39 — 43 of the Doctrine of Chances, we have 

 the Lemma of which we have already given an account ; see 

 Art. 242. 



267. Problem VI. of the Doctrine of Chances is an example 

 of the Problem of Points with three players. De Moivre gives 

 the same kind of solution as Fermat : see Arts. 16 and 18. In 

 the third edition there is also a discussion of some simple cases 

 according to the method which Pascal used for two players ; see 

 Art. 12. De Moivre also gives here a good rule for solving the 

 problem for any number of players ; the rule is founded on 



