Quick answer
The Poisson probability of k events is P(X = k) = e^(−λ) λ^k ÷ k!. With an average of 2 events, P(X = 3) = 0.1804, P(X ≤ 3) = 0.8571 and P(X ≥ 3) = 0.3233.
Enter the average rate λ and the number of events k. The calculator returns the exact, cumulative and upper-tail probabilities. Use this poisson distribution calculator to check your own figures.
Related tools: Binomial Distribution Calculator: Binomial probabilities for n trials. Normal Distribution Calculator: Probabilities for a normal distribution. Sampling Distribution of the Mean Calculator: Probability for a sample mean using the central limit theorem.
How to use the Poisson Distribution Calculator
- Enter average rate (λ) (between 0 and 1000).
- Enter events (k) (a whole number, between 0 and 1000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Average rate (λ) | A number | > 0, ≤ 1000 |
| Events (k) | A whole number | ≥ 0, ≤ 1000 |
Formula and method
Formula
P(X = k) = e^(−λ) λ^k ÷ k!
IEEE-754 double precision; shown to at most 6 decimal places.
Computed in log space (using ln k! summation) for numerical stability; P(X ≤ k) sums the terms.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Average rate (λ) = 2; Events (k) = 3.
| Result | Value |
|---|---|
| P(X = 3) | 0.180447 |
| P(X ≤ k) | 0.857123 |
| P(X ≥ k) | 0.323324 |
Example 2
Inputs: Average rate (λ) = 4; Events (k) = 0.
| Result | Value |
|---|---|
| P(X = 0) | 0.018316 |
| P(X ≤ k) | 0.018316 |
| P(X ≥ k) | 1 |
The first line is the headline result (p(X = 3)). Below it the poisson Distribution Calculator lists p(X ≤ k) and p(X ≥ k), which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
e^−2 = 0.13534; 2³ = 8; 3! = 6, so 0.13534 × 8 ÷ 6 = 0.1804. The mean and variance both equal λ.
When it applies
Use it for counts of rare, independent events in a fixed interval: calls per hour, defects per metre, arrivals per minute. λ and k are limited to 1,000 here.
How the result changes with average rate
Holding every other input at the first worked example, this table changes only average rate. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Average rate | P(X = 3) | P(X ≤ k) | P(X ≥ k) |
|---|---|---|---|
| 1 | 0.061313 | 0.981012 | 0.080301 |
| 2 | 0.180447 | 0.857123 | 0.323324 |
| 4 | 0.195367 | 0.43347 | 0.761897 |
| 8 | 0.028626 | 0.04238 | 0.986246 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Average rate (λ): entering "0" shows "Average rate (λ) must be greater than 0." beside that field and no result.
- Events (k): entering "1.5" shows "Events (k) must be a whole number." beside that field and no result.
Choosing between the Poisson Distribution Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Normal Distribution Calculator](/statistics/normal-distribution-calculator/) | mean, standard deviation and probability | P(X < x₁) |
| [Sampling Distribution of the Mean Calculator](/statistics/normal-probability-sampling-distributions/) | population mean, population SD and sample size | P(x̄ < value) |
| [IQ Percentile Calculator](/statistics/iq-percentile/) | iQ score, mean and standard deviation | Percentile |
| [Empirical Rule Calculator](/statistics/empirical-rule/) | mean and standard deviation | ≈68% (μ ± σ) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- λ up to 1000
- k up to 1000
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for average rate (λ) and events (k) to confirm this.
Frequently asked questions
What is the mean of a Poisson?
The mean of a Poisson distribution is λ, and so is its variance; for λ = 2 both equal 2.
What is P(X = 0)?
e^(−λ), 0.1353 here.
When is it like a binomial?
With large n and small p.
What does the Poisson Distribution Calculator calculate?
Poisson probabilities for a given mean rate. It returns P(X = 3), P(X ≤ k) and P(X ≥ k).
What formula does the Poisson Distribution Calculator use?
P(X = k) = e^(−λ) λ^k ÷ k! Computed in log space (using ln k! summation) for numerical stability; P(X ≤ k) sums the terms.
What do I need to enter in the poisson Distribution Calculator?
You enter Average rate (λ) and Events (k).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. λ up to 1000.
What are the limits of this calculator?
λ up to 1000. k up to 1000.
What is an example of a poisson Distribution result?
With average rate (λ) 2 and events (k) 3, the p(X = 3) is 0.180447.
Related calculators
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Normal Distribution Calculator — Probabilities for a normal distribution.
- Sampling Distribution of the Mean Calculator — Probability for a sample mean using the central limit theorem.
- IQ Percentile Calculator — Percentile of an IQ score on a normal scale.
- Empirical Rule Calculator — 68–95–99.7 intervals for a normal distribution.
- Geometric Probability Calculator — Probability the first success occurs on trial k.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.