I am dealing with a problem that resembles the following modeling:
Given a bag of $n$ balls constitute of $b$ black balls and $r$ red balls, we are choosing $\alpha$ many balls uniformly without replacement. What is the probability of having at most $k$ many black balls?
P.S: I know that the problem is broken down into the following formula. What we are seeking for is a very simplified version of the following formula to decide on $\alpha$ given $b$, $r$, $n$, and a target $p$.
$p = Pr(num_{black} < k) = \frac{\sum_{i=0}^{k-1} \binom{b}{i} \times \binom{r}{\alpha-i}}{\binom{n}{\alpha}}$