Poisson Distribution Calculator
Calculate exact, cumulative, tail, range and no-event Poisson probabilities from an average event rate, with interval scaling and step-by-step work.
Poisson probability calculator
Calculate exact, cumulative, tail, and range probabilities for event counts.
Probability distribution
Probability of each event count in the displayed range.
| k | Probability | Bar |
|---|---|---|
| 0 | 1.8316% | |
| 1 | 7.3263% | |
| 2 | 14.6525% | |
| 3 | 19.5367% | |
| 4 | 19.5367% | |
| 5 | 15.6293% | |
| 6 | 10.4196% | |
| 7 | 5.9540% | |
| 8 | 2.9770% | |
| 9 | 1.3231% | |
| 10 | 0.5292% | |
| 11 | 0.1925% | |
| 12 | 0.0642% | |
| 13 | 0.0197% | |
| 14 | 0.0056% | |
| 15 | 0.0015% |
Step-by-step calculation
- Set the effective rate: λ = 4.0000.
- Use the Poisson formula: P(X = k) = e⁻λ λᵏ / k!.
- Calculate the selected probability: P(X = 3) = 19.5367%.
- For cumulative and range questions: add the probabilities for all included integer counts.
When should I use Poisson?
Use it for counts of independent events in a fixed interval with a stable average rate. If observed variance is much larger than the mean, a Poisson model may not describe the data well.
How to use the Poisson distribution calculator
- Enter the average rate λ for one interval.
- Enter the number of intervals if your question covers a different duration or exposure.
- Enter the event count k and optional upper value b.
- Read exact, cumulative, tail, range and no-event probabilities.
Poisson distribution formula
The mean is λ, the variance is λ, and the standard deviation is √λ.
Worked example
If a help desk receives an average of 4 calls per hour, the probability of exactly 3 calls in one hour is P(X = 3) = e⁻⁴ × 4³ / 3! ≈ 19.5367%.
Common Poisson applications
- Calls or arrivals per hour
- Defects per unit of material
- Emails or messages per minute
- Accidents per month
- Other independent event counts over a fixed exposure
Frequently asked questions
What is a Poisson distribution?
A Poisson distribution models the number of events in a fixed interval when events occur independently at a stable average rate.
What does λ mean?
λ is the expected number of events in the interval. It is also both the mean and variance.
How do I calculate P(X = k)?
Use P(X = k) = e⁻λ λᵏ / k!.
What is the difference between at most and at least?
At most k means X ≤ k; at least k means X ≥ k. Less than k excludes k, while greater than k excludes k.
Can λ be a decimal?
Yes. λ is an average rate and does not need to be an integer.
When should I use Poisson?
Use it for counts of independent events in a fixed interval when the average rate is reasonably stable.