Say, a bag has 10 balls, in which 9 are red, 1 is black.
Each red ball is worth 1 point, each black is worth 4 points.
I have 8 picks from the bag to start with (the bag refills itself after each pick: returns to 9 red 1 black), once I get a black ball, I will get 8 additional picks, and go on.
Ultimately I would like to calculate the expected total points.
So, for example
1st pick red 7 picks remaining 2nd pick black 6 + 8 = 14 picks remaining ...
One way to solve the problem:
Assume the expected number of red balls is A, black balls is B. $A/B=9$ and $8+8*B=A+B$ So $A = 36,B = 4$
But I am looking for a more formal/generalized way (maybe markov/matrix or wald equation or stopping time) to this kind of problem.