CONNECTED WITH GAMES OF CHANCE. 175 



SO that the whole profit is 



+ P±n,n, + Pfn,n, + ..] (8) 



in the case of Wj = 1, n^ — — 1, n^ z= 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 w^, n^, n.,. . 

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

 and on the second drawing he receives Wj, ra„, ra,, . . times 

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

 on extracting any ball he receives n^, w^, w.,, . times the sum 

 of M 4- 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, q, r, . . what is the whole 

 profit ? 



Adopting the same notation as in the last problem, 

 Va = n^Sa + n^ Sa+1 + M3 Sa42 -f . . . wlll represent either 

 n^, n^, n^, . . and the first profit is mP„, the second is 

 (m + M P„) Pi ; that in the third is (m + m Pa + m Pj + w Pa Pj ) P<, , 

 and the sum of all tlie profits is 



mP. 



ttPi + mPo Pi 



M Pc + M Pa Pc + M Pa Pc + M Pa P* Po , 



&c. &c. &c. 



On 



