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).
A long-windedAnother way of asking this question might be as follows... -
I'm a mushroom picker, and because the economy is terrible,Let's say that I have a small business on the side selling bundlesmatrix composed of mushrooms for making soup. Each bundle contains exactly$S$ blocks of $L$, randomly selected cells (pick whatever geometry you'd like), intact mushrooms. And for some reason it's extremely important to me that thiswhere $S > 10^3$ and $L$ is approximately $10^1$ ~ $10^2$ or so.
One day, I receive orders for $S$ bundles Some fixed number of mushrooms, all from separate customers. Pressed for time to collect the $S*L$ mushrooms I needcells, I collect $A$ of my mushrooms from the usual place in, store the forestvalue '0', but collectwhile the remaining $B$ from a nearby field where no grass can growcells store the value '1'. Once home To be clear, I randomly shuffle allthere are a total 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$A + B = S*L = N$ cells in the nearby fieldmatrix.
Oops. Turns out that there's industrial waste buried inWhat is the field leeching a toxin with an extremely sharp dose-response curve. If more thanprobability that all $S$ blocks contain at most $k$ ofcells with the storage value '1' $B$ mushrooms I harvested ends up in anyone's soup, that person will incur serious liver damage and possible die(i. However, if the number is less thane. at most $k$, 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$B$ cells)?