Poisson Distribution Calculator
Enter an average rate and a number of events to get the Poisson probability P(k; λ) = (λ^k × e^(−λ)) / k! — the chance of seeing exactly that many rare, independent events in a fixed interval.
What is the Poisson distribution?
Probability of rare, independent events
This Poisson distribution calculator returns the probability of observing exactly a given number of events in a fixed interval of time, space, or area, when those events happen independently at a constant average rate. You supply two numbers: the average rate λ (the mean count you expect per interval) and the count k (the exact number of events you want the probability for). The Poisson distribution is the model behind questions like "what are the chances of exactly 2 support calls this hour if we average 3?" and it underpins queueing, reliability, and rare-event analysis across science and business.
Enter an average rate λ and a whole number of events k to get the Poisson probability instantly.
The Poisson probability mass function raises the rate to the power of the count, multiplies by the natural exponential of the negative rate, and divides by the factorial of the count.
P(k; λ) = (λ^k × e^(−λ)) / k!Here e is Euler's number (about 2.71828) and k! is the factorial of k (with 0! defined as 1). Because λ sits inside both a power and the exponential, the probability peaks near λ and tapers off for counts far above or below it. The result is always a number between 0 and 1.
Suppose a help desk averages λ = 3 calls per hour and you want the probability of exactly k = 2 calls in the next hour.
Raise the rate to the power of k
3² = 9 — the rate raised to the number of events.
Multiply by e to the minus rate
9 × e^(−3) = 9 × 0.049787 = 0.448084 — weighting by the exponential decay.
Divide by k factorial
0.448084 ÷ 2! = 0.448084 ÷ 2 = 0.224042 — about a 22.4% chance of exactly 2 calls.
The Poisson distribution models the count of rare, independent events in a fixed interval, and your result is the probability of one specific count. A defining feature is that the mean and the variance are both equal to λ: the average outcome and the spread of outcomes are governed by the same single number. So with λ = 3 the most likely counts cluster around 2 and 3, and the probability for exactly 2 (0.224042) is one slice of that whole distribution. Probabilities for every possible count sum to 1, so a single value like 0.2240 tells you how much of the total likelihood lands on that exact outcome. Raising λ shifts the peak rightward and widens the spread; lowering it pushes mass toward 0. This is why the Poisson distribution describes things like calls to a switchboard, decay events from a radioactive sample, typos per page, or goals in a match — all situations where each event is independent and the rate is roughly constant.
The formula is exact, but the model only fits when its assumptions hold.
Independent events, a constant rate, and a whole-number count
The Poisson distribution assumes events occur independently and at a constant average rate over the interval. If events cluster, influence one another, or the rate drifts over time, the Poisson model understates or overstates the true probability. The count k must be a non-negative whole number — there is no such thing as 2.5 events — and the rate λ must be greater than zero. For a fixed number of trials with a fixed success probability, use the binomial distribution instead.