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 a ratio of blue balls over red balls, $r_{B/R}$, in all $S$ bins such that for some $B_1$ and $B_2$, $0 \leq B_1 \leq r_{B/R} \leq B_2 \leq 1$?