138 BE MOIVRE. 



that A can give B one game out of three : see Problems I. and ii. 

 of the Doctrine of Chances. 



24^1. The fifth problem is to find how many trials must be 

 made to have an even chance that an event shall happen once at 

 least. Montmort had already solved the problem ; see Art. 170. 



De Moivre adds a useful approximate formula which is now one 

 of the permanent results in the subject; we shall recur to it in 

 noticing Problem III. of the Doctrine of Chances, where it is repro- 

 duced. 



242. De Moivre then gives a Lemma : To find how many 

 Chances there are upon any number of Dice, each of them of the 

 same number of Faces, to throw any given number of points ; see 

 Doctrine of Chances, page 89. We have already given the history 

 of this Lemma in Art. 149. 



243. The sixth problem is to find how many trials must be 

 made to have an even chance that an event shall happen twice at 

 least. The seventh problem is to find how many trials must be 

 made to have an even chance that an event shall happen three 

 times at least, or four times at least, and so on. See Problems III. 

 and IV. of the Doctrine of Chances. 



244. The eighth problem is an example of the Problem of 

 Points with three players ; it is Problem VI. of the Doctrine of 

 Chances. 



245. The ninth problem is the fifth of those proposed for 

 solution by Huygens, which Montmort had enunciated wrongly in 

 his first edition ; see Art. 199. Here we have the first publication 

 of the general formula for the chance which each of two players 

 has of ruining the other in an unlimited number of games ; see 

 Art. 107. The problem is Problem vil. of the Doctrine of 

 Chances. 



246. The tenth problem is Problem viii. of the Doctrine of 

 Chances, where it is thus enunciated : 



Two Gamesters ^ and -5 lay by 24 Counters, and play with three 

 Dice, on this condition ; that if 1 1 Points come np, A shall take one 



