Poisson Distribution Calculator
Enter the average rate (λ) and a number of events (k) to find the Poisson probability.
The Poisson distribution gives the probability of a number of independent events in a fixed interval when they occur at a known average rate λ. The probability of exactly k events is λ^k times e^(−λ), divided by k factorial. For λ = 3 and k = 2, that is 3² × e^(−3) ÷ 2! ≈ 0.224.
How to Use This Calculator
Enter the average rate λ — the mean number of events expected in your interval — and the specific count k you are interested in. The calculator returns the probability of exactly k events and the cumulative probability of k or fewer.
The Poisson distribution models counts of independent events over time or space: calls to a call center in an hour, typos per page, or arrivals in a queue, when they happen at a steady average rate.
How the Calculation Works
The formula combines three pieces: λ raised to the power k reflects the chance of that many events, e^(−λ) is the normalising factor from the underlying process, and dividing by k factorial accounts for the ways the events can occur.
The cumulative probability of k or fewer events is the sum of the individual probabilities from 0 up to k. The mean and the variance of a Poisson distribution are both equal to λ.
P(X = k) = (λ^k × e^(−λ)) ÷ k!
Worked Example
Suppose a shop averages λ = 3 customers per five-minute window and you want the probability of exactly k = 2. Compute 3² × e^(−3) ÷ 2! = 9 × 0.0498 ÷ 2 = 0.224.
So there is about a 22.4% chance of exactly two customers, and summing from 0 to 2 gives the cumulative chance of two or fewer, about 42.3%.
Assumptions and Limitations
- Assumes events are independent and occur at a constant average rate
- λ is the mean per interval; k must be a whole number
- Mean and variance of a Poisson distribution both equal λ
- Not suitable when events cluster or the rate changes over the interval
When to Use a Poisson Calculator
Statistics students and operations analysts use this to find the probability of a number of events given an average rate. Run it for problems involving arrivals, defects, calls, or other rare, independent events over a fixed interval. Enter the rate and count to get the probability.
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Open calculator →Frequently Asked Questions
What is the Poisson distribution used for?
It models the count of independent events in a fixed interval at a steady average rate — such as arrivals, calls, defects or accidents — giving the probability of any specific number occurring.
What is the Poisson formula?
P(X = k) = (λ^k × e^(−λ)) ÷ k!, where λ is the average rate and k is the number of events. It gives the probability of exactly k events.
What is λ in the Poisson distribution?
Lambda (λ) is the average number of events expected in the interval. It is both the mean and the variance of the distribution, so a larger λ means both more events and more spread.
What is cumulative Poisson probability?
The probability of k or fewer events, found by adding the individual probabilities from 0 up to k. It answers questions like 'at most k events' rather than 'exactly k'.
When should I not use the Poisson distribution?
When events are not independent, cluster together, or the rate changes over the interval. In those cases the constant-rate assumption breaks and other models fit better.
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