Poisson Distribution Explorer

The Poisson distribution models the probability of a given number of independent events occurring in a fixed interval of time or space: P(X=k) = (λ^k * e^{-λ}) / k!, where λ is the average event rate. It assumes that events occur independently and at a constant average rate, serving as the standard model for radioactive decay, call arrivals, and queueing theory.

Picture a huge number of independent opportunities, each one very unlikely on its own - typos on a page, lightning strikes in a county, flaws on a production line. Flip a coin n times with success chance p; let n grow large while p shrinks so that n × p = λ stays fixed, and the Binomial distribution smoothly turns into Poisson. Toggle the overlay here, raise n, and watch the jagged Binomial bars settle onto the Poisson curve. The lesson: whenever you have many tries, tiny odds each, Poisson is the shortcut.

In 1898 the statistician Ladislaus Bortkiewicz studied deaths from horse kicks across 10 Prussian army corps over 20 years. The yearly counts - mostly 0, occasionally 1 or 2, rarely more - matched a Poisson distribution with λ ≈ 0.61 almost perfectly. He named this pattern the law of small numbers. It was a striking demonstration that a distribution built for physics and telephone traffic also governs freak accidents, and it remains a textbook proof that Poisson describes the real world.