Poisson Probability Calculator
By Santaji GadepatilinUpdated October 27, 2026
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How to use this calculator
Enter the average rate λ for your interval and the number of events k. The probability of exactly k events and of none are calculated.
Formula
P(X = k) = λk e−λ ÷ k!
Worked example
A help desk averages 3 calls per hour. The chance of exactly 5 calls in an hour:
P = 3⁵ × e⁻³ ÷ 120 = 10.08%
The chance of a quiet hour with no calls is e⁻³ = 4.98%.
Distribution for λ = 3
| k | P(X = k) |
|---|---|
| 0 | 5.0% |
| 1 | 14.9% |
| 2 | 22.4% |
| 3 | 22.4% |
| 4 | 16.8% |
| 5 | 10.1% |
Frequently asked questions
When is the Poisson distribution used?
For counts of independent events happening at a constant average rate — calls per hour, defects per metre, arrivals per minute.
What is λ?
The expected (average) number of events in the interval.
How is it related to the binomial distribution?
It approximates the binomial when trials are many and the success probability is small, with λ = n × p.