Statistics
Poisson Probability Calculator
Use this Poisson calculator for count events over a fixed interval when the average rate is known.
P(X = k)
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- Mean
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- Variance
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Student quick launch
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Study path
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What the poisson probability calculator solves
Use this Poisson calculator for count events over a fixed interval when the average rate is known.
Formula
How to use it
- Enter the average event rate.
- Enter exact event count k.
- Read the exact probability.
How to read the answer
For a Poisson distribution, mean and variance both equal lambda.
Common mistakes and edge cases
- The rate must match the interval being counted.
- Events should be rare-ish and independent for the model to fit.
- Exact probability is not cumulative probability.
Worked examples
Exactly 2 events
P(X = k)
14.6525%
Zero events
P(X = k)
22.3130%
Frequently asked questions
What is lambda?+
Lambda is the average number of events in the interval you are studying.
Can k be zero?+
Yes. P(X=0) is the probability no events happen in the interval.
When is a Poisson model reasonable?+
It is most reasonable when events are counted over a fixed interval, happen independently, and have a roughly constant average rate.
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Last updated: May 15, 2026