Poisson Distribution Calculator
This Poisson distribution calculator gives the probability of k events for an average rate λ. Example: if you get 3 calls per minute on average, exactly 2 calls has a probability of 22.4%.
Example: Rate λ (expected value) 3 · Find the probability of …: exactly k events: P(X = k) · Count k 2 → Probability: P(X = 2) = 22.404%. Source: 1.3.6.6.19, NIST/SEMATECH e-Handbook – Poisson distribution. Updated: .
How it is calculated
The Poisson distribution describes how often a rare event occurs in a fixed period or space when it happens λ times on average and the events are independent. Typical cases: calls per minute, goals per match, typos per page, claims per year.
The formula
P(X = k) = λk · e−λ / k!. The mean and the variance are both equal to λ. For "at most k" add the individual probabilities from 0 to k; "at least k" is 1 minus "at most k − 1".
Example
A call centre gets 3 calls per minute on average (λ = 3). Exactly 2 calls: 3² · e⁻³ / 2! = 0.2240, i.e. 22.4%. At least 5 calls: 1 − P(X ≤ 4) = 0.1847, i.e. 18.5%. The table shows all individual and cumulative probabilities around the mean.
When does the Poisson distribution fit?
- Events occur one at a time and independently.
- The average rate is constant over the period.
- Several events at exactly the same moment are practically excluded.
- As an approximation of the binomial distribution for a large number of trials n and a small probability p, with λ = n · p.
For probabilities with a fixed n and p use the probability calculator (binomial); for bell-curve measurements use the normal distribution calculator.
Frequently asked questions
How do you calculate a Poisson probability?
With P(X = k) = λ^k · e^(−λ) / k!. You only need the average rate λ and the number k you are interested in. For "at most" or "at least" the individual probabilities are added.
What does λ mean in the Poisson distribution?
λ (lambda) is the average number of events in the period or area you look at. It is both the mean and the variance of the distribution.
How do I calculate "at least k"?
"At least k" is the opposite of "at most k − 1": P(X ≥ k) = 1 − P(X ≤ k − 1). The calculator adds up the individual probabilities without rounding loss.
When should I use Poisson instead of binomial?
When the number of trials is very large and the single probability very small (rule of thumb: n ≥ 50 and p ≤ 0.1), or when there is no fixed number of trials at all, only a rate per time.
Can I use the Poisson distribution for football goals?
As a simple model, yes: with the average goals per match as λ you get the probability of 0, 1, 2 … goals. It is an approximation, not a forecast.
Sources and legal basis
- 1.3.6.6.19: NIST/SEMATECH e-Handbook – Poisson distribution
- DLMF §8: NIST Digital Library of Mathematical Functions – incomplete gamma function
- Wikipedia – Poisson distribution
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