Poisson Probability Calculator

About this calculator

The Poisson distribution gives the probability of a specific number of events k occurring in a fixed interval, given a known average rate λ: P(X=k) = (λᵏ·e^(−λ)) / k!. It applies to events that happen independently and at a constant average rate — this calculator also gives the cumulative probabilities P(X≤k) and P(X≥k) for finding "at most" or "at least" that many events.

Statisticians, actuaries, and operations researchers use the Poisson distribution to model call-center arrivals, website traffic spikes, defects per manufactured unit, radioactive decay counts, and rare-disease incidence rates. It's especially useful when events are individually rare but there are many opportunities for them to occur — the classic textbook example is the number of Prussian cavalry soldiers killed by horse kicks per year, one of the earliest documented applications of the distribution.