Bloom Filter 2D: Occupancy & False-Positive Statistics

This is a 2D companion to the 3D Bloom filter simulator, built around a genuinely different piece of the same theory: instead of hashing real words into one filter with a real hash function, it simulates the idealized "balls-into-bins" occupancy process the classic Bloom filter formula assumes, across an ensemble of 250 independent filters at once. Two analytic curves — the exact combinatorial false-positive probability and the common exponential approximation used almost everywhere in practice — are plotted against a live Monte Carlo measurement taken directly from the simulated ensemble, so you can see the small but real gap between the exact formula and its familiar shortcut, and watch a live histogram of bit-occupancy across the ensemble narrow as the array grows — the concentration-of-measure effect that makes the formula trustworthy for any one real filter.