CONNECTED WITH GAMES OF CHANCE. 16"9 



Hence 

 Q^3 (£^i)!ri(ii+l)_2(_ir(p_^)_(_iy(p_y) + 3- 



— iac + bd + cd+..)~2{bc + cd + ...). 

 The latter of these series is 



I (_ 1)0+' + (_ l)Hc + (_ ly+d +._!_(_ la+6 _ 



z= p+q~2k—l — (~.iy+\ 



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

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

 The value of the series 



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 I, then the series 

 in question is equal to p + q — '2l — 2: These substitutions 

 and the necessary reductions being made, we have 



+ 7 + 2^ + 4A: + 2(— P+* 

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



W = (;,-9){z.-|(3^-2(-l)«-(_l)') } 



a p = q 



^^ = -1 {7-9;'+2^ + 4^ + 2(-l)«+*} (5) 



VOL. IX. P. I. Y Jn 



