Poisson Probability Calculator

The Poisson distribution models the number of events occurring in a fixed interval of time, space, area, or exposure when events happen independently at an average rate. It is often used for arrivals, defects, calls, incidents, mutations, and other count data. The single parameter lambda represents the expected number of events in the interval.

If a help desk receives an average of four tickets per hour, a Poisson model can estimate the probability of seeing zero, three, or ten tickets in a particular hour. The exact count varies, but the average rate anchors the distribution.

A defining property of the Poisson distribution is that its mean and variance are both lambda. That makes the model compact, but also easy to violate. Real count data often show overdispersion, where variance exceeds the mean because events cluster, rates change over time, or observations are not independent.

When overdispersion is present, a simple Poisson model can underestimate uncertainty. Alternatives such as negative binomial models or rate models with covariates may be more appropriate. The Poisson distribution is a starting point, not a guarantee.

Poisson reasoning depends on matching the rate to the exposure interval. An average of two events per minute is not the same as two events per hour. If the interval changes, lambda scales with it when the underlying rate is constant.

Exposure can be time, distance, population size, machine hours, web sessions, or manufactured units. Comparing counts without exposure can mislead. Ten defects in 1,000 units and ten defects in 1,000,000 units describe very different processes.

The Poisson distribution works best when events are individually uncommon, independent, and governed by a stable average rate over the interval. It is less suitable when one event makes another more likely, when capacity limits exist, or when the rate shifts during the observation window.

A good practical check is to compare observed count frequencies with what the model predicts. If zeros, extremes, or clusters appear far more often than expected, the process may need a richer model.