CONNECTED WITH GAMES OF CHANCE. 159 



sion in which (—1) has only one index, it appears to be 

 u { l(-l) 6 +2(-l) c +3(-l) d +&c.} 5 



and the knowledge that p of the letters a, 6, c, &c. represent 

 even, and q of them odd numbers, will not assist us in deter- 

 mining the sum of this series ; we must also be acquainted 

 with the order of succession of the even and odd numbers. 

 Another reason will afterwards be assigned, why it is not suffi- 

 cient merely to be acquainted with the number of favourable 

 and of unfavourable events, which may with more propriety 

 be stated, when some other questions of a similar nature have 

 been examined. 



Let us now consider another case, in which the amount of 

 the sum staked on each event follows a different law, we will 

 suppose the following problem : 



A gamester begins a series of bets on an event whose chance 

 of occurring is one-half, by staking the sum u. Whenever he 

 wins he makes the next succeeding stake less than his last by 

 the quantity v ; but if he lose, he then increases his stake by 

 the same quantity. Supposing he should win p times, and 

 lose q times, what will he have gained or lost on the whole 

 number p -f q. 



His first stake being w, his first profit may be expressed by 



u ( 1)°, which will be won or lost according as u is even or 



odd ; in the first case his next stake would be u — v, and in the 

 second it would be u + v. « These two cases may be combined 

 into one, and thus expressed u — v( — l) a , and his second 



profit will be •! u — v( — 1)° [ ( — I) 6 -.; his next stake will 



be w— v{— l) a —v if b is even j but it will be u — v{ — l) a +v 



if 



