Poisson Probability

The Poisson distribution describes rare events that happen independently at a steady average rate λ per interval: emails per hour, goals per match, mutations per genome. With λ = 2 calls an hour, the chance of exactly 3 calls is 2³e⁻²/3! = 8 × 0.13534/6 ≈ 0.180. It is the limit of the binomial when n grows large and p shrinks with np held at λ, which is why it works for "many chances, each tiny".

Ladislaus Bortkiewicz made it famous in 1898 by counting soldiers kicked to death by horses in fourteen Prussian cavalry corps over twenty years: 122 deaths in 280 corps-years, λ ≈ 0.61, and the observed tallies of 0, 1, 2, 3 and 4 deaths matched the formula almost exactly. R. D. Clarke did the same in 1946 for V-1 impacts across South London, showing the clusters Londoners saw as targeting were what randomness looks like. The trap is applying it to events that cluster or inhibit each other — the rate must be constant and the events independent. There is no closed form for λ or k given P, so this calculator solves only for the probability; note P(X = 0) = e^(−λ), the chance of a quiet interval.