Poisson Distribution Calculator
Poisson Distribution Calculator
Enter the average rate of occurrence (λ, lambda) over your interval of interest, and the number of occurrences you want the probability for. This calculates both the exact probability of that many events happening, and the cumulative probability of that many or fewer.
The Poisson distribution models rare, independent events happening over a fixed interval of time or space — calls arriving at a support line per hour, typos per page of a manuscript, or customer arrivals per minute at a quiet counter are all classic Poisson scenarios. It's related to but distinct from the binomial distribution: binomial counts successes across a fixed number of discrete trials, while Poisson counts occurrences over a continuous interval where the "number of possible trials" isn't really a meaningful concept.
- Formula: P(X=k) = (λ^k × e^−λ) ÷ k!, where λ is the average rate and k is the target number of occurrences.
- Models rare, independent events over a fixed interval: like calls arriving at a call center per hour, typos per page, or customer arrivals per minute.
- Different from binomial: Poisson doesn't need a fixed number of "trials" — it models counts of events over continuous time or space, given only the average rate.
What kinds of events fit the Poisson model?
Events that occur independently of each other at a known constant average rate, within a fixed unit of time or space — such as website hits per minute, defects per meter of material, or emails per hour.
What if I want the probability of "more than" a number, not "at most"?
Subtract the cumulative probability from 1 — the probability of more than k occurrences equals 1 minus the probability of k or fewer.
Poisson Distribution Calculator
Enter the average rate of occurrence (λ, lambda) over your interval of interest, and the number of occurrences you want the probability for. This calculates both the exact probability of that many events happening, and the cumulative probability of that many or fewer.
The Poisson distribution models rare, independent events happening over a fixed interval of time or space — calls arriving at a support line per hour, typos per page of a manuscript, or customer arrivals per minute at a quiet counter are all classic Poisson scenarios. It's related to but distinct from the binomial distribution: binomial counts successes across a fixed number of discrete trials, while Poisson counts occurrences over a continuous interval where the "number of possible trials" isn't really a meaningful concept.
- Formula: P(X=k) = (λ^k × e^−λ) ÷ k!, where λ is the average rate and k is the target number of occurrences.
- Models rare, independent events over a fixed interval: like calls arriving at a call center per hour, typos per page, or customer arrivals per minute.
- Different from binomial: Poisson doesn't need a fixed number of "trials" — it models counts of events over continuous time or space, given only the average rate.
What kinds of events fit the Poisson model?
Events that occur independently of each other at a known constant average rate, within a fixed unit of time or space — such as website hits per minute, defects per meter of material, or emails per hour.
What if I want the probability of "more than" a number, not "at most"?
Subtract the cumulative probability from 1 — the probability of more than k occurrences equals 1 minus the probability of k or fewer.