ii APPENDIX THE FIRST. 



former in his Theorie, &c., the latter in a tract entitled, 

 Considerations sur la Theorie Mathematique du Jeu, 

 Lyons, 1802. have also solved the problem: both so- 

 lutions are of the highest order of difficulty, and cannot 

 be rendered elementary. If my memory be correct, I 

 have seen references to other solutions. 



The problem is as follows : Two players, A and B, 

 the first possessed of m times and the second of n times 

 his stake,, play at a game so constituted that it is a to b 

 that A shall win any one game ; required the proba- 

 bility which each has of ruining the other, if the game 

 be indefinitely continued. 



I shall first take the case where one of the players, A, 

 is possessed of unlimited means, that is, in which m is 

 infinite. Let B w> m represent the probability that 

 B having n counters, shall ruin A who has m counters. 

 Then, if m be infinite, B Wj oo will after the first game, 

 become either B w + i, oo, or B w _ lf ^ according as' 

 that game is B's or A's ; of which the chances are 



b and a 



a + b a+b 



Consequently, 



B n oo = 

 which gives 



which, when C' and C" are determined, will represent 

 the probability that B will never be ruined, but will 

 continually gain more and more from A. But the 



mental (in my mathematical treatise on Probabilities, in the Encyclopaedia 

 Metropolitana) in attributing to Laplace more than his due, having been 

 misled by the suppressions aforesaid, I feel bound to take this opportunity 

 of requesting any reader of that article to consider every thing there given 

 to Laplace as meaning simply that it is to be found in his work, in which, 

 as in the Mecanique Celeste, there is enough originating from himself to 

 make any reader wonder that one who could so well afford to state what 

 he had taken from others, should have set an example so dangerous to his 

 own claims. 



