How this calculation works
λ is the expected count in the interval and k is a nonnegative integer. The tool calculates the exact-count probability and the probabilities of at most k and more than k. Recurrences handle cumulative tails without forming large factorials.
P(X = k) = e⁻λ λᵏ ÷ k!A worked example
For λ = 3, exactly 2 events has probability about 22.404181%.
Good to know
A mathematical probability model, not a fit test or an empirical guarantee. Inputs must match the model assumptions. Floating-point arithmetic is used; extremely small probabilities may underflow or round to zero.
Reference: NIST: Poisson distribution (opens in a new tab) ↗
The interval is part of the question
An average of three arrivals per hour is not the same input as three arrivals per minute. If the process has a constant average rate of three per hour and you are asking about two hours, the expected count for that interval is six. Convert the rate before entering λ.
An exact count and a cumulative count answer different questions. “Exactly two” excludes zero and one; “at most two” includes them. The more-than-k result is the complement of the cumulative result, so those two values should sum to 100% apart from display rounding.
A Poisson count has equal model mean and variance. If observed counts fluctuate much more than their average, that can be a reason to investigate clustering or varying rates rather than relying on this model automatically.
Check whether independent, constant-rate events are plausible
Customer arrivals during a quiet period may behave differently from arrivals during a scheduled event or a rush hour. Combining such periods into one average can hide a changing rate. A calculator can evaluate the supplied λ without establishing that the model is a good description.
The model has no fixed maximum count. Even if the average is small, a larger count can have positive probability. Conversely, an average of three does not require exactly three arrivals in any one interval.
At λ = 0, this tool uses the limiting zero-count model: zero events occurs with probability one. For positive λ, very small computed tails can round to zero. The numerical display should not be confused with proof that an event is impossible.
A few questions, answered
Does λ have to be an integer?
No. It is an expected count and may be fractional. The observed event count k must be a nonnegative whole number.
Can I use it for sampling without replacement?
A fixed finite-population sampling problem is usually better described by a hypergeometric model. The Poisson model uses independent event counts rather than depletion of a fixed pool.
How large are the supported inputs?
The expected count is limited to 1,000 and k to 5,000. The tool uses floating-point logarithms and recurrences, with explicit underflow caveats.