Poisson Probability Calculator
Find the probability that exactly k events occur in a fixed interval, given a known average rate λ. Use it for modeling rare events like customer arrivals, server requests, manufacturing defects, or radioactive decay counts.
About this calculator
The Poisson distribution describes the probability of a given number of independent events occurring in a fixed time or space interval, given a constant average rate. The formula is P(X = k) = (λᵏ × e⁻λ) / k!, where λ (lambda) is the average number of events per interval, k is the exact number of events you want the probability for, e is Euler's number (≈ 2.71828), and k! is the factorial of k. The distribution assumes events occur independently of one another, at a constant average rate, and that two events cannot occur at exactly the same instant. It is widely used in telecommunications, epidemiology, insurance, and operations research to model counts of rare or random events over continuous intervals of time or space.
How to use
A call center receives an average of λ = 4 calls per minute. What is the probability that exactly k = 6 calls arrive in a given minute? First compute λᵏ = 4⁶ = 4096. Then e⁻λ = e⁻⁴ ≈ 0.018316. Multiply: 4096 × 0.018316 ≈ 75.04. Divide by k! = 6! = 720. P(X = 6) = 75.04 / 720 ≈ 0.1042. There is about a 10.4% probability of receiving exactly 6 calls in any given minute.
Frequently asked questions
What does the lambda (λ) parameter mean in the Poisson distribution?
Lambda (λ) is the expected number of events in the defined interval — it is simultaneously the mean and the variance of the Poisson distribution. For example, if a website receives 200 visits per hour on average, then λ = 200 for a one-hour window or λ = 200/60 ≈ 3.33 for a one-minute window. You can rescale λ to match any interval length by multiplying the per-unit rate by the size of your interval. The higher the value of λ, the more the distribution resembles a bell-shaped normal curve.
When is the Poisson distribution a good model for real-world events?
The Poisson distribution is appropriate when events are rare relative to the number of opportunities, occur independently, and happen at a roughly constant average rate. Classic examples include the number of accidents at an intersection per month, the number of typos per page, incoming network packets per second, or bacterial colonies on a petri dish. It is not a good model when events cluster together (overdispersion), when the rate changes over time, or when there is a hard maximum number of possible occurrences in the interval.
How is the Poisson distribution different from the binomial distribution?
The binomial distribution counts successes in a fixed number of discrete trials, each with probability p. The Poisson distribution counts events in a continuous interval with no fixed upper bound on the number of events. The Poisson distribution can be derived as a limiting case of the binomial when n is very large and p is very small, with np = λ remaining constant. In practice, use binomial when you have a defined number of trials and use Poisson when you are counting events over time or space with no clear upper ceiling.