ON THE RISKS OP LOSS OB GAIN. 109 



chance that in the long run B will rum A, and that A 

 will ruin B ? 



RULE 1. If a and b be equal, it is m to n that A 

 will mm B, and n to m that B will ruin A. 



RULE II. If a and b be unequal, let M represent 

 the difference of the rath powers of a and of 6, multiplied 

 by the nth power of a, and let N represent the difference 

 of the nth powers of a and of b, multiplied by the mth 

 power of b. Then it is M to N that A ruins B, or 

 N to M that B ruins A. 



N. B. If m and n be considerable, it is almost im- 

 practicable to apply the above rule without the aid of 

 logarithms. VThen b is many times a, take the nth 

 power of a for M, and the nth power of b for N. 



RULE III. When m and n are equal, it is as the 

 mth power of b to the twth power of a that B will 

 ruin A. 



RULE IV. If the means of both players be unlimited, 

 then it is certain the player who has odds in his favour 

 on a single game will, in time, gain any sum, however 

 great ; if the means of the stronger player be unlimited, 

 then it is certain he must at last ruin the weaker. 



RULE V. If the means of the weaker player be 

 unlimited, and those of the stronger limited, the chance 

 that the latter will in time win any sum, however great, 

 from the former, is as follows. Let B be the stronger 

 player (that is, b greater than a), and let him begin 

 with n times his stake, while A has unlimited means : 

 then it is the excess of the nth power of b over that of 

 a to the nth power of a that B gains any sum, in the 

 long run, from A, and vice versa for A ruining B. 



EXAMPLE. A has 51. and B 31., and they play at a 

 game for which it is three to two that B shall win any 

 one game. What are the chances for the ultimate 

 success of each player ? 



a = 2, 6 = 3, m = 5, n = 3 



M or (3 5 2 5 ) x 2 3 is (243 - 32) x 8 or 1688 

 N or (3 3 2 3 ) x 3 5 is (27 - 8) x 243 or 4617 : 

 It is therefore 4617 to 1688 that B shall ruin A. 



A gaming bank must be considered as a player of 



