The Poisson distribution answers a specific, very common question: given that something happens at a known average rate, how likely is it to happen a particular number of times in the next interval? It's the standard tool for modeling call volumes, website errors, customer arrivals, insurance claims, and any other count of rare, independent events — this calculator runs the exact formula so you get a precise probability instead of a gut-feel estimate.
How the Poisson formula works
The Poisson probability mass function, P(X=k) = lambda^k * e^-lambda / k!, has exactly one input: lambda, the average rate. Everything else — the mean, the variance, the shape of the curve — falls out of that single number. Small lambda values produce a distribution heavily skewed toward 0 and 1 events; as lambda grows past about 10, the distribution starts to look increasingly like a symmetric bell curve, approaching a normal distribution. The calculator evaluates this formula exactly using log-space arithmetic, so it stays accurate even for large k where a literal factorial would overflow.
Exactly, at most, and at least
Most real questions aren't really about one exact count. "What's the chance of at most 2 defects" sums every probability from 0 through 2; "what's the chance of at least 1 outage" is 1 minus the probability of exactly 0. The Cumulative tab lays out P(X=k), P(X<=k), and P(X>=k) side by side for every nearby k so you can read off whichever framing your decision actually depends on, without re-running the calculator for each threshold.
When Poisson applies — and when it doesn't
Poisson assumes events happen independently, at a constant average rate, and that two events can't occur at literally the same instant. It breaks down when events cluster — a single power outage that causes ten simultaneous alerts isn't ten independent events — or when the rate itself changes over the interval, like call volume spiking during a promotion. In those cases the raw count will look over-dispersed relative to what Poisson predicts, and a different model (like a negative binomial distribution) usually fits better.