CONNECTED WITH GAMES OF CHANCE. 163 



points out another limit, beyond which if the number of fa- 

 vourable events increase, he will lose money. 



In the problem just examined, each stake depends on that 

 which immediately precedes it. Others may be proposed, in 

 which it is made to depend on the result of two or more of 

 the events which immediately precede, such is the following 

 one : 



A gamester begins by staking a given sum m, and regulates 

 his succeeding bets in this manner ; if one of the two immedi- 

 ately antecedent to that which he is at any time making, have 

 been gained, and the other lost, the new stake is the same as 

 that last made ; but if both those bets have been gained, he di- 

 minishes his stake by the sum v ; and if both have been lost, 

 he increases it by the same sum v. The first stake being u, 

 the first gain may be represented by u ( — 1)"; and as there 

 has only been one bet yet made, the second cannot be affected 

 by the law prescribed ; it will therefore be the same as the first, 

 and the profit arising from it will be u ( — 1)*. The third stake 

 depending on the two former, we must add to the quantity u 

 some function of a and b multiplied by v, which shall va- 

 nish if a and b are one odd and the other even, and which 

 shall become — 1 when they are both even, and + 1 when 



they are both odd* Such is the function — ^~^^°^^~^^ ; 



the profit from the third stake may therefore be represented 



by 



X 2 and 



