Poisson Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The Poisson distribution calculates the probability of observing exactly k events with P(X=k) = λᵏ × e⁻λ ÷ k!, where λ is the average number of expected events. For an average of 3 events, the probability of observing exactly 2 is about 22.4%.
Explanation
The Poisson distribution models the number of events occurring in a fixed interval (of time, area, or volume), when these events occur independently of each other at a known average rate (λ, lambda). It is particularly suited to rare but possible events: the number of calls received by a call center in an hour, the number of manufacturing defects in a production batch, the number of customers arriving in a queue over a given period, or the number of radioactive decays detected by a Geiger counter. Unlike the binomial distribution (see our binomial distribution calculator), which counts successes over a fixed number of trials, the Poisson distribution does not require knowing a total number of trials: it only needs the average occurrence rate over the interval considered. In fact, the Poisson distribution can be seen as a limiting case of the binomial distribution, when the number of trials becomes very large and the probability of success of each very small, while keeping a constant product (n×p) equal to λ.
Example: an average of 3 events, probability of observing exactly 2
Inputs
Expected average (λ): 3. Occurrences sought (k): 2.
Calculation
P(X=2) = 3² × e⁻³ ÷ 2! = 9 × 0.049787 ÷ 2 ≈ 0.22404, that is 22.404%.
Result
The probability of observing exactly 2 events is about 22.4%.
Frequently asked questions
When should you use the Poisson distribution rather than the binomial?
Use the Poisson distribution when you know an average occurrence rate over an interval (for example, 3 calls per hour on average), with no identifiable discrete number of trials. Use the binomial distribution when you have a fixed number of independent trials, each with a known probability of success (for example, 10 coin flips).
Where does the constant e (about 2.71828) in this formula come from?
It is the base of natural logarithms, a mathematical constant that appears naturally in deriving the Poisson distribution from the binomial distribution, when the number of trials tends to infinity and the probability of success of each toward zero, while keeping their product constant.
Does this formula work for a very high λ?
Mathematically yes, but for large values of λ, the Poisson distribution approaches a normal distribution (bell curve) centered on λ, which makes its interpretation in terms of rare events less relevant. The Poisson distribution is most useful and intuitive for moderate values of λ, typically below a few dozen.