CONNECTED WITH GAMES OF CHANCE. 167 



series of bets be made plus or minus, denoting their success or 

 failure : 



-f- 100 

 + 100 

 + 90 

 — 80 

 -f- 80 

 -f 80 

 -f- 70 



The profit is 440. Here p = 6, q = l 9 k = 2, v = 10, 

 u = 100 ; and these values being substituted in (2) give 



W = 440. 



The next case I shall examine is one in which the amount 

 of each stake depends on the result of the three preceding 

 events. 



Let the law by which the stakes are regulated be such, that 

 when the three preceding have been won, a certain sum v is 

 taken from the last stake ; if they have been lost, the sum v is 

 added to the last ; but if only two of these three have been 

 gained, or only two lost, in the first case one-third of v is sub- 

 tracted, and in the second case is added to the last stake. 



The first stake being w, the three first bets will be repre- 

 sented by u ( — l) a , u ( — l) 6 , u ( — l) c . In order to fulfil the 

 conditions of the question, the fourth must be increased by 



the quantity — v + 3 ~ > an( l the profit re- 



sulting from it is represented by 



I ,_ p (-i)-+(-») t +(-»y j (_i)* 



that of the fifth is equally expressed by 



{ M _ M ( (-!)« +(-iy+(-i)° + (-p» + (-i). +( - 1)*) J ( _ 1)c 



The 



