CONNECTED WITH GAMES OF CHANCE. 175 



so that the whole profit is 

 W-u{pn +qn +r« 5 +...) + v {-^-V + ^^V" + 

 + P±n i n i + P{-n 1 n 3 +..} (8) 



in the case of n x — 1, n 2 = — 1, w 3 := 0. This formula re- 

 duces itself to (1). 



Supposing the urn in the last question filled with the same 

 balls, and a person drawing out one receives n lJ rc 2 , » r> . . 

 times the sum w, according to the number of the ball drawn ; 

 and on the second drawing he receives n x , n e , n s , . . times 

 the sum of u + the profit by the last drawing : and generally 

 on extracting any ball he receives n x , n q , n 3 , . . times the sum 

 of u -j- the amount of the profit on all the preceding events, 

 if the number of times each of the balls marked 1, 2, 3, . . are 

 drawn, be respectively denoted by p, #, r, . . what is the whole 

 profit ? 



Adopting the same notation as in the last problem, 

 P a = n 1 S a + n 2 S a+ i + n 5 S a42 + . . . will represent either 

 n,, n 2 , n 3 , . . and the first profit is mP„, the second is 

 (u + u P a ) Pi ; that in the third is (u + u P a + u V b + u P a P* ) P c , 

 and the sum of all the profits is 



MPj+ uf a Yb 



U Pc + U P a Pc + U ?b Pc + U P a P* Pc , 

 &C. &C. &C. 



On 



