Suppose there are n apples, where l of them are good and N-l are bad. From the n apples, I am sampling r apples (uniformally, without replacement). What's the probability that after the sampling, at least k out of the r apples are good?
So I got the following formula: $\sum_{i\geq k}\frac{\binom{r}{i}\binom{n-i}{l-i}}{\binom{n}{r}}$.
Explanation: The first step I took is deriving the probability for getting only i good apples out of r. There a $\binom{n}{r}$ different possibilities to sample of r apples. From that if there are i good apples, there are $\binom{r}{i}$ ways to select them and $\binom{n-i}{l-i}$ way to (not) select the rest of the (l-i) good apples. And so to get "at least k", I simply summed over all i greater than k.
Is my intuition correct?
This is the quick answer:
Let's set
$k \leq r \leq l \leq N$
The probability you are lookin for is the following:
$$\sum_{i=k}^r\frac{\binom{l}{i}\binom{N-l}{r-i}}{\binom{N}{r}}$$
for the future, please show you attempts to solve the problem.