Exact probability
Calculates the probability of observing exactly a chosen number of events.
Advanced Math and Statistics
Poisson Distribution Calculator
Calculate exact, cumulative, at-least, interval, and zero-event probabilities for counts occurring at a known average rate.
- Exact and cumulative probabilities
- Mean, variance, and standard deviation
- Interactive probability distribution
Poisson distribution inputs
Enter the average event rate and choose the count probability you want to calculate.
Event rate
Event count
Optional rate conversion
Quick examples
Advanced Math and Statistics
Model event counts over time, distance, area, or volume
The Poisson distribution measures the probability of a specified number of events occurring within a fixed interval when events happen independently and at a constant average rate.
Cumulative probability
Finds the probability of no more than or fewer than a selected event count.
Upper-tail probability
Calculates the probability of at least or more than a selected number of events.
Range probability
Adds exact event-count probabilities across an inclusive interval.
Zero-event probability
Finds the chance that no events occur during the selected interval.
Distribution visualization
Displays event-count probabilities with selected bars highlighted.
Core formulas
Poisson probability and distribution statistics
Exact probability
Probability of exactly x events when the expected count is λ.
P(X = x) = e−λλˣ ÷ x!
Mean and variance
For a Poisson distribution, both equal the expected event count.
μ = λ and σ² = λ
Standard deviation
The typical spread of event counts around the expected value.
σ = √λ
How it works
Calculate Poisson probability in four steps
1
Enter the average rate
Add the expected number of events in the selected interval.
2
Choose the probability
Select exact, cumulative, at-least, zero-event, or range probability.
3
Enter event counts
Provide one event count or a lower and upper interval.
4
Review the distribution
See the probability, summary statistics, highlighted bars, and complete probability table.
Frequently asked questions
Poisson distribution calculator FAQs
A Poisson distribution gives probabilities for the number of events occurring in a fixed interval when events happen independently at a stable average rate.
Lambda, written λ, is the expected number of events during the selected interval. It is also the mean and variance of the Poisson distribution.
Use it for counts such as calls per hour, defects per batch, arrivals per minute, accidents per month, or particles per area when events are independent and the average rate is approximately constant.
Substitute x = 0 into the Poisson formula. Because zero factorial equals one and λ raised to zero equals one, the probability simplifies to e raised to negative λ.
At most x includes every event count from zero through x. At least x includes x and every larger possible count.
Yes. Multiply the base average rate by the relative interval length. For example, if the average is four events per hour, the expected count for three hours is λ = 12.