108 ESSAY ON PROBABILITIES. 



whether A shall ruin B, or B shall ruin A ; but that 

 one or other will ultimately be ruined is certain. Sup- 

 pose each party to have a hundred guineas, the stake 

 being one guinea, and suppose two millions of games 

 are to be played. The most probable individual case is 

 that each shall win a million of games ; but if the 

 fluctuation amount to 100 in favour of either, the other 

 is ruined. Now, page 81, the probability that the 

 number of games won by A shall lie between a million 

 -f- 100 and a million 100 is ! 12, which is therefore the 

 chance that neither player shall be ruined. Consequently, 

 it is about nine to one that one player or other is ruined 

 or more than ruined in two millions of games. And 

 the chance is even greater than this : for the preceding 

 method of treating the problem supposes the players 

 not to balance their account till two million of games 

 have been actually played, so that one player or the 

 other may have been repeatedly playing on credit. The 

 same rule may be easily applied to any inequality of play, 

 the fortunes of the players being equal ; and the result 

 is, I. that ultimate ruin to one or other player is certain ; 

 2. that, if the stake be a sufficiently small fraction of 

 the player's income, the number of games which must 

 be played to render probable the ruin of either may be 

 made as large as we please. There are but two con- 

 ditions under which gambling can be prudently followed 

 as an amusement small stakes and equal play. In 

 games of pure chance it is possible to obtain the latter, 

 and almost impossible in games of mixed skill and 

 chance. Unfortunately, the stimulus of -gambling, a 

 combination of suspense and hope of large gain, cannot 

 be obtained upon any terms which prudence would 

 sanction. 



When two players, of unequal fortunes, play together 

 for the same stake, however equal the play may be, the 

 larger fortune has an unfair advantage. To estimate 

 the amount of the disadvantage, proceed as follows. 



PROBLEM. Two players, A and B, having funds of 

 m and n times their stake, play a game, at which it is a 

 to b that A wins, or b to a that B wins. What is the 



