Balls into Bins

Throwing balls uniformly at random into m bins — watch the phase transition from collisions to coverage
Bins n = 0, m = 30
1 ball
2–3 balls
4+ balls
empty
Statistics over time phase: sparse
empty bins
bins with ≤1 ball
max load
theory (E[·])
birthday √(2m·ln2)
coupon m·ln m
30
40ms
3.0×
empty: 30 ≤1 ball: 30 max load: 0
Birthday-paradox line marks E[n] at which the first collision is expected (n ≈ √(2m·ln2)). Coupon-collector line marks E[n] at which all bins are expected to be non-empty (n ≈ m·ln m). Circles mode caps at m = 40 for clarity; bars mode supports up to 80.