I am trying to solve this problem but getting stuck in the general case. Here is the problem statement:
A bag contains a total of N balls. Out of which M balls are red, and N-M balls are blue. Here M < (N-M).
Now, consider sampling with replacement. In each sampling step, the probability of selecting a red ball is 0.9/M, while the probability of selecting a blue ball is 0.1/(N-M).
What is the probability that after M*k rounds of sampling, only (M-i) red balls are selected (Here 0 <= i <= M )?
If M = 1, this reduces to a simple instance, and the required probability is (1/N)^k, but not sure for M > 1.
Here is what I think the answer should be for M>1: $$ \sum_{i=0}^{M} \left( \frac{0.9}{M}\right)^{k(M-i)} \left( \frac{0.1}{N-M} \right)^{ik} $$
Is the above equation correct? Any suggestions?