170 AN EXAMINATION OF SOME QUESTIONS 



In the first of these expressions, k may vary from to 

 p-\- q — 1, and I from to p-jr q — 2 ; in the second k may 

 vary from 1 to 2p — 1, and / may vary from to 2p — 2. 



The last question which I shall examine is one in which 

 each stake depends on all those which precede it. 



A gamester stakes a certain sum u on an event whose 

 chance of happening is one-half He regulates each succeeding 

 stake in this manner : To the constant sum u he adds the 

 wth part of all his previous winnings ; or if he has lost by 

 the previous stakes, he subtracts the nth part of his loss, it 

 is proposed to find his profit * at the termination of p-j- q 

 bets. 



The first stake being u, the profit is u ( — 1)" ; the ?ith part 



of this added to u, or «+ - (— 1)" will constitute the second, 

 and the profit will be «<(— 1)'+ \ (—1)" (—1)', the nth 



part of these two, or I ((- 1)" + (— 1)') + l^ (— 1)" (— 1)' 



added to the quantity u, will be the third stake, and the profit 

 on the determination of the third event will be 



a (_ i)c + ;; ((_!)<. + (_i/) (-1)^ + I {-\r (-1)' (-1)^ ; 



the 



* The lano-uage of analysis is so much more general than that in which we 

 usually convey our thoughts, that it is almost impossible to make the latter keep 

 pace with the former. This is more particularly manifest when we are treating of 

 games of chance. The words profit, winning, gain, &c. must, if we wish to avoid 

 perpetual repetition, frequently be understood to comprehend their very oppo- 

 sites. 



