DE MOIVRE. 147 



Fermat's metliod, and is intended to lighten as mucli as possible 

 the labour which must be incurred in applying the method to 

 complex cases. The rule was first published in the Miscellanea 

 Analytica, in 1730; it is given in the second edition of the 

 Doctrine of Chances on pages 191, 192. 



2G8. Problem vii. is the fifth of those proposed by Huygens 

 for solution ; see Art. 35. We have already stated that De Moivre 

 generalises the problem in the same way as James Bernoulli, 

 and the result, with a demonstration, was first published in the 

 De Mensura Sortis ; see Arts. 107, 245. De Moivre's demon- 

 stration is very ingenious, but not quite complete. For he finds 

 the ratio of the chance that A will ruin B to the chance that 

 B will ruin A ; then he assumes in effect that in the lonof nm 

 one or other of the players must be ruined : thus he deduces 

 the absolute values of the two chances. 



See the first Appendix to Professor De Morgan's Essay on 

 Prohahilities in the Cabinet Cyclopcedia. 



We have spoken of Problem viii. in Art. 246. 



269. Problem ix. is as follows. 



Supposing A and B, whose proportion of skill is as a to 6, to play 

 together, till A either wins the number q of Stakes, or loses the number 

 ;; of them ; and that B sets at every Game the sum G to the sum L ; it 

 is required to find the Advantage or Disadvantage of ^. 



This was Problem XLIII. of the first edition of the Doctrine 

 of Chances, in the preface to which it is thus noticed : 



The 43d Problem having been proposed to me by Mr. Thomas Wood- 

 cock, a Gentleman whom I infinitely respect, I attempted its Solution 

 with a very great desire of obtaining it; and having had the good 

 Fortune to succeed in it, I returned him the Solution a few Days after 

 he was pleased to jn-opose it. This Problem is in my Opinion one of 

 the most curious that can be propos'd on this Subject ; its Solution 

 containing the Method of determining, not only that Advantage which 

 results from a Superiority of Chance, in a Play confined to a certain 

 number of Stakes to be won or lost by either Party, but also that which 

 may result from an unequality of Stakes ; and even compares those two 

 Advantages together, when the Odds of Chance being on one side, the 

 Odds of Money are on the other. 



10—2 



