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 from the $N$ balls, 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 most $k \leq B$ blue balls in every one of the $S$ bins?

Edit - I would be very much interested in approximate solutions! This problem should essentially come down to something akin to making sure a fixed number of points randomly placed on a matrix are at least some fixed distance apart (causing them to end up in different bins), so surely there have been problems like this tackled in the literature?

Edit 2 - As per Peter Shor's recommendation, I'm interested in examples where $S$ and $L$ are of fixed size, but where $S$ is large, at least $10^3$ to $10^4$ or so, and $L$ is typically significantly smaller than $S$ (around $10$ to $10^2$ or so).

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A long-winded way of asking this question might be as follows...

I'm a mushroom picker, and because the economy is terrible, I have a small business on the side selling bundles of mushrooms for making soup.  Each bundle contains exactly $L$, randomly selected, intact mushrooms.  And for some reason it's extremely important to me that this is so.

One day, I receive orders for $S$ bundles of mushrooms, all from separate customers.  Pressed for time to collect the $S*L$ mushrooms I need, I collect $A$ of my mushrooms from the usual place in the forest, but collect the remaining $B$ from a nearby field where no grass can grow.  Once home, I randomly shuffle all of the mushrooms while washing them, and randomly partition them into the $S$ bundles.  Only after packaging and mailing out the orders does it occur to me to check into the lack foliage in the nearby field.   

Oops. Turns out that there's industrial waste buried in the field leeching a toxin with an extremely sharp dose-response curve.  If more than $k$ of the $B$ mushrooms I harvested ends up in anyone's soup, that person will incur serious liver damage and possible die.  However, if the number is less than $k$, the person will only feel mildly ill.

As I'm from a village where you can be sued and face serious criminal charges for this sort of thing, I'd like to know the probability that none of the $S$ bundles contains $k \leq B$ of the poison mushrooms?