Let's say I have a bag with $A$ red balls, $B$ blue balls, and a total number of balls $N = A + B$. With uniform probability, and sampling without replacement, I fill an integer number of bins, $S$, with exactly $L$ balls each, where $N = S*L$. As such, the bag of all $N$ balls should be empty by the end of the procedure.

What is the probability of having at least $k$ blue balls in every one of the $S$ bins?