Poisson Probability Calculator
Compute Poisson probabilities for the number of events in a fixed interval given an average rate λ — exactly k, at most k, and at least k.
How it works
P(X = k) = (λᵏ · e⁻λ) ÷ k! · where mean = variance = λ
The Poisson distribution models how many independent events happen in a fixed span of time or space when they occur at a constant average rate λ. It is the limit of the binomial distribution when trials are many and each success is rare, so it fits counts like calls per hour, typos per page, or arrivals per minute. A useful quirk is that its mean and variance are both equal to λ.
Worked example
A help desk gets 3 calls per hour on average (λ = 3). The chance of exactly 2 calls in an hour is (3² · e⁻³) ÷ 2! = (9 · 0.0498) ÷ 2 ≈ 0.224 — about 22.4%.
Sources: Poisson distribution — Wikipedia
Frequently asked questions
When is the Poisson distribution the right model?
Use it for counts of independent events over a fixed interval with a stable average rate, where two events cannot happen at the exact same instant. Examples include emails per hour, machine breakdowns per month, or defects per square metre.
What is λ and where do I get it?
λ (lambda) is the mean number of events expected in your interval — usually estimated from historical data, for example a long-run average of 3 arrivals per hour. If your interval changes, scale λ accordingly: half an hour would use λ = 1.5.
How does Poisson relate to the binomial distribution?
When n is large and p is small, Binomial(n, p) is closely approximated by Poisson with λ = n·p. That is why rare-event counts, where each of many opportunities has a tiny probability, follow a Poisson shape.
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For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.