Poisson Distribution Calculator
Probability of exactly k events, of none, and of at least one, when the average rate is known.
Results
The dashed line marks the value you entered.
What this tool does
The Poisson distribution answers the staffing question that averages cannot: a counter that serves three people an hour on average is empty for five minutes in twenty and faces five at once often enough to matter. It has the unusual property that its mean and its variance are the same number, so the rate alone fixes the whole shape — there is nothing else to estimate.
Formula
P(k) = lambda^k e^(-lambda) / k!
Variables
| Symbol | Meaning | Unit |
|---|---|---|
lm | Average events in the period | — |
kk | Events to test for | — |
PK | Probability | % |
P0 | Probability of none | % |
P1 | Probability of at least one | % |
PL | Probability of at most k | % |
SD | Standard deviation (sample) | — |
Worked example
- Average events in the period3
- Events to test for2
- Probability22.4042 %
- Probability of none4.9787 %
- Probability of at least one95.0213 %
- Probability of at most k42.3190 %
- Standard deviation (sample)1.7321
Limitations
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
Frequently asked questions
When does Poisson apply?
When events happen independently at a constant average rate over a fixed window, and two of them never coincide exactly: customers arriving at a counter, faults on a length of cable, typing errors per page, machine breakdowns per month. If the events cluster or influence each other, Poisson understates the extremes and a different model is needed.