CONNECTED WITH GAMES OF CHANCE. 169 



Hence 



— (ac + bd-\-cd-\-..) — 2 (6 c -f- c d -}- . . .) . 

 The latter of these series is 



| (_ 1)«+* + (_ l) 6 +c _|_ (__ ly+d + . . . | _ (_ la+6 -- 



— p-j-2_2& — 1 — (~ i)«+* 9 



as we found in the last question, k being the number of 

 changes from even to odd, or vice versa, in the series a, 6, c, . . 

 The value of the series 



(_ l)«+c + (_ \)Hd + (_ 1)c+ e _|_ . # 



is not determined without some other data. If, however, we 

 are acquainted with the number of alterations from odd to 

 even, and the contrary, in the series 



a c b d c e d f . 

 we can assign its value ; let the number be /, then the series 

 in question is equal to p + q — % I — 2 : These substitutions 

 and the necessary reductions being made, we have 



Q=(i»-*){s^-2(-i)--(-i)*}--(p + 0)g 



+ 7 + 2/ + 4flr + 2(— -1 B +* 

 and W the profit on p -f- q events is 



W = (p-q){u-l(s i^ _ 2 (-1)« -(_!)*) } 



— I {7 — (p + ?)f +2l + 4* + 2(— 1)'+*} (4) 



if p = q 



W 1 = — | {7— 9j> + 2/ + 4* + 2(— 1)-+*} ( 5 ) 



VOL. IX. P. I. y J n 



