How to Use the Poisson Distribution Calculator
This calculator takes the average number of events per interval (λ, lambda) and a target count (k), then instantly returns two probabilities. Exact probability P(X=k) is the chance of the event occurring exactly k times, calculated as e^(-λ)×λ^k÷k!, while cumulative probability P(X≤k) is the sum of the chances of it occurring k times or fewer.
The Poisson distribution is widely used to model the count of independent, relatively rare events within a fixed interval — calls per hour at a call center, visitors per minute on a website, or defects per unit on a production line.
To avoid overflow when calculating k! for large values, this calculator uses a log-gamma-based approximation, so it stays stable even for large k. The average rate (λ) must be greater than 0, and the target count (k) must be an integer between 0 and 1000 — outside these conditions, the calculator immediately shows a guidance message.
Frequently Asked Questions
The Poisson distribution models the number of independent events happening within a fixed interval of time or space, such as calls per hour at a call center, daily website visitors, or defects per unit in a production line. Knowing just the average rate (lambda) lets you calculate the probability of any specific count.
Exact probability is the chance of the event occurring exactly k times, while cumulative probability is the sum of the chances of it occurring k times or fewer. For example, 'exactly 3 times' and '3 times or fewer' are different values.