Consider a family $F$ of subsets of a probability space $\Omega$. Assume that $F$ has bounded VC dimension and that the measure of each subset in $F$ is at least $\epsilon$.

Drawing $n$ iid points from $\Omega$, can we find a good lower bound on the probability that all subsets in $F$ contain at least a point?

I tried using the uniform CLT (VC inequality), but it seems suboptimal in the regime where the expected number of points in each subset is small.