Poisson distribution calculator
The Poisson distribution describes rare independent events: customers arriving, decays detected, typos per page.
Probability of exactly k events
10.081 881%
- Probability of no events
- 4.978 707 %
- Probability of at least one
- 95.021 293 %
- Standard deviation
- 1.732 05
- Z-score of k
- 1.1547
Show the working
- Average events per interval = 3
- Number of events = 5
- Probability of exactly k events = exp(-3) * 3^5 / fact(5) * 100 = 10.0819
- Probability of no events = exp(-3) * 100 = 4.978 71
- Probability of at least one = 100 - 4.978 71 = 95.0213
- Standard deviation = sqrt(3) = 1.732 05
- Z-score of k = 3 > 0 ? (5 - 3)/sqrt(3) : 0 = 1.1547
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
Scenarios & local history
Compare a saved scenario
Recent calculations
The formula
- P(X = k) = e^-lambda lambda^k / k!
- mean = variance = lambda
A Poisson distribution has its variance equal to its mean, which is a useful test of whether a count really is Poisson.