How to use it
- Type the Average rate λ: the average number of events in the interval, for example 3 arrivals per minute.
- Type the Number of events k you want the probability for.
- Read P(X = k): exactly k events, P(X ≤ k) and P(X ≥ k).
- Check the Distribution table and How it is worked out.
The Poisson formula
The Poisson distribution counts events that happen independently at a constant average rate λ in a fixed interval of time, length or area. The probability of exactly k events is e^(−λ) λᵏ ÷ k!. Its mean and its variance are both λ, so a count whose spread is much larger or smaller than its mean does not fit this model.
The chance of no event in the interval is e^(−λ). Each further event multiplies the probability by λ and divides by the count reached, which is why the probabilities rise to a peak near λ and then fall.
P(X = k) = e^(−λ) × λᵏ ÷ k!
P(X ≤ k) = Σ P(X = j) for j from 0 to k
mean = variance = λWhen the Poisson distribution fits
It fits counts of rare, independent events, such as customers arriving in a minute, calls to a helpdesk in an hour, or typing errors on a page, when the average rate does not change during the interval. It is also the limit of the binomial distribution with many trials and a small success probability, where λ = np. The binomial probability calculator works with a fixed number of trials.
What the calculator does not cover
- Counts whose variance differs from the mean, such as clustered events. Those need the negative binomial distribution.
- Rates that change during the interval. Split the interval into parts with their own rates and add the results, or use a rate for each part.
- Average rates above 1000, which the calculator does not accept, and counts above 1000.
Worked example: 3 arrivals per minute
A shop gets an average of 3 customers a minute, so λ = 3. The probability of exactly 2 arrivals in a minute is e^(−3) × 3² ÷ 2! = 0.224042. The probability of at most 2 arrivals is 0.423190, and of at least 2 arrivals 0.800852.
The mean is 3 and the variance is also 3.
| Events j | P(X = j) | P(X ≤ j) |
|---|---|---|
| 0 | 0.049787 | 0.049787 |
| 1 | 0.149361 | 0.199148 |
| 2 | 0.224042 | 0.423190 |
| 3 | 0.224042 | 0.647232 |
| 4 | 0.168031 | 0.815263 |
Frequently asked questions
What is the difference between a Poisson and a binomial distribution?
The binomial counts successes in a fixed number of trials, each with a success probability. The Poisson counts events over a continuous interval at an average rate, with no fixed number of trials. For many trials and a small probability the two give nearly the same answer.
Why do the mean and the variance have the same value?
That is a property of the Poisson distribution. If counts are observed to vary much more than their average, the Poisson model is not appropriate, and a negative binomial model is needed.
How do I get the rate λ for a different interval?
Scale it. If the rate is 3 per minute, it is 15 per five minutes and 180 per hour. Use the same unit for the interval you ask about.
What is the probability of no event at all?
It is P(X = 0) = e^(−λ). With λ = 3, it is about 0.0498, so about 5% of minutes have no arrival.
What does P(X ≥ k) mean?
It is the probability of k events or more. It equals 1 minus the probability of at most k − 1 events, which is why the steps show that form.
Does the calculator send my numbers anywhere?
No. The calculation runs in your browser, and nothing you type is sent to our servers.
Sources
- Wikipedia: Poisson distribution
- Wikipedia: Poisson distribution, relation to the binomial and the incomplete gamma function
Figures and text last checked on Tuesday, November 10, 2026.