Poisson distribution calculator
Enter the average rate and the count, get the probabilities.
What is the Poisson distribution?
It gives the probability of a number of events in a fixed interval when events happen independently at a steady average rate λ: P(X = k) = λᵏ e^−λ ÷ k!. OpenStax (Introductory Statistics 2e, section 4.6) states the conditions as events counted over a fixed interval of time or space, a known average rate, and each event occurring independently of when the previous one happened.
Example: calls to a help desk
If a desk averages 3 calls an hour, the chance of exactly 2 calls in an hour is 3² e^−3 ÷ 2 = 0.2240 (22.4%); the chance of none is 0.0498.
How do I find “at least” probabilities?
Use 1 − P(fewer than k). With an average of 3, the chance of 5 or more is 0.1847.
What are the mean and variance?
Both equal λ, so the standard deviation is √λ — 1.732 for λ = 3. A mean equal to the variance is a quick check on real data: counts that vary much more than their average suggest that events cluster and the Poisson model is too tidy.
Worked example: emails per day
OpenStax Example 4.30 describes a user who receives 147 emails a day on average. The chance of exactly 160 on a given day is 0.0180, about 1.8%. Even though 160 is only 13 above the average, a single exact value is rare because the probability is spread over many counts; the standard deviation is √147 = 12.1, so 160 is about one standard deviation above the mean. Enter λ = 147 and k = 160 to see the whole table.
How do I change the interval?
Scale λ to the interval you are asking about. At 3 calls an hour, the average over half an hour is λ = 1.5, and the chance of no call in those 30 minutes is e^−1.5 = 0.2231, not the 0.0498 for a full hour. The rate and the interval must use the same unit.
When does Poisson not fit?
When events cluster or the rate changes over the interval — for example, rush-hour arrivals versus late-night ones. It also fails when one event makes another more likely (aftershocks, a viral post) or when there is a hard ceiling on the count, as with seats on a bus. Then the independence and steady-rate conditions are broken.
Is it a substitute for the binomial?
Yes, when there are many trials and each has a small chance of success: take λ = np (OpenStax section 4.6). With 1,000 trials at 0.2% each, the chance of exactly 3 successes is 0.1806 exactly and 0.1804 with λ = 2. The waiting times between Poisson events follow the exponential distribution, which has its own calculator.
Common mistakes with the Poisson distribution
First, using a rate for one interval and asking about another without scaling λ. Second, entering a count that is not a whole number: events are counted in whole units, and the calculator rounds k. Third, treating λ as a certainty: it is an average estimated from data, and a short record gives an uncertain λ. Fourth, mixing “exactly k” with “k or more”; the table shows both, and the two can differ a lot for small λ.
Sources: OpenStax, Introductory Statistics 2e, section 4.6 (Poisson Distribution, including Example 4.30 and the binomial approximation). Related: binomial probability calculator, exponential distribution calculator, geometric distribution calculator, negative binomial calculator and normal distribution calculator.