166 AN EXAMINATION OF SOME QUESTIONS 



Still, however, the knowledge of the order in which the 

 events take place is not absolutely required. It will be sufficient 

 to know how many changes from odd to even, and vice versa, 

 occur between the quantities a, b, c, c?, e, &c. j for if there is 

 no change in going from a to b, the first term will be -f- 1 ; but 

 if there is a change it will be — 1 ; and generally — 1 will oc- 

 cur as often as there is a change from odd to even, or from 

 even to odd in the series 



a, 6, c, d, e, &c. 



If, therefore, k denote the number of transitions, since 

 the number of terms is p-\- q — 1, the value of the series 

 will be 



p-\- q — 2k — 1; 



and the whole value of the part dependent on v will be 



- I { (P-1)' - (P+9) - iP+1) + 2 * + 1- (-1)* (p-q) + 1 } 

 and the profit on the whole number of bets is 



W = (p-q) U— I { {p-qf—2 (p+q) + 2k + 2 — 



(-i)° ip-q) } (2) 



if p — q> then 



W 1 = + v{2p — k—l) (3) 



k in the first of these expressions may vary from to p+q— 1, 

 and in the second from 1 to 2p — 1. 



In order to compare this with an example, let the following 



series 



