Poisson Distribution Probability Calculator

The Poisson distribution models how many times a rare, independent event happens in a fixed interval, given only the average rate λ (lambda) — customer arrivals per hour, typos per page, or server errors per day. The exact probability of observing precisely k occurrences is P(X = k) = λᵏ × e^(−λ) ÷ k!, evaluated here in log space (k×ln(λ) − λ − ln(k!)) so it stays numerically accurate even for larger λ or k rather than overflowing k! or underflowing λᵏ directly.

"At most k" sums that exact-probability formula across every outcome from 0 to k. Because the Poisson distribution has no upper bound on how many events could occur, "at least k" is computed instead as 1 minus the probability of fewer than k occurrences — a finite sum — rather than summing an infinite tail. The mean and variance of a Poisson distribution are both exactly λ, so the standard deviation reported here is simply √λ.