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Poisson Distribution Calculator
The Poisson distribution gives the probability of seeing a certain number of events in a fixed interval when you know the average rate. Enter the average λ and a count k to get the probability of exactly k events and the cumulative probabilities either side.
- P(X ≤ k) at most k
- 0.42319
- P(X ≥ k) at least k
- 0.800852
- P(X < k) fewer than k
- 0.199148
- P(X > k) more than k
- 0.57681
- Standard deviation (√λ)
- 1.7321
Distribution around the mean
| k | P(X = k) | P(X ≤ k) |
|---|
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Formula
How to use it
- Enter the average number of events per interval, λ.
- Enter the number of events you are asking about, k.
- Pick the line that matches your question: exactly k, at most k, at least k, fewer than k or more than k.
Worked examples
An average of 3 events, and exactly 2 happen
- P(X = k) exactly k
- 0.224042
- P(X ≤ k) at most k
- 0.42319
- P(X ≥ k) at least k
- 0.800852
- Standard deviation (√λ)
- 1.7321
A call center averaging 4.5 calls an hour gets exactly 6
- P(X = k) exactly k
- 0.12812
- P(X ≤ k) at most k
- 0.831051
- P(X > k) more than k
- 0.168949
No defects when the average is 0.8 per unit
- P(X = k) exactly k
- 0.449329
- P(X ≥ k) at least k
- 1
When to use it
Poisson fits counts of events that happen independently at a steady average rate: calls arriving per hour, typos per page, defects per roll of fabric, goals per match. There is no fixed number of trials — that is what separates it from the binomial distribution.
It is also a good approximation to the binomial when there are many trials and success is rare; use λ = n × p.
Match λ to the interval
λ must be the average for the same interval you are asking about. If a help desk averages 3 calls an hour, use λ = 1.5 for a half-hour window and λ = 24 for an eight-hour shift.
Questions people ask
If the average is 3 events, what is the probability of exactly 2?
22.40% (0.224042), from 3² × e⁻³ ÷ 2!.
With an average of 3, what is the probability of 2 or fewer events?
42.32% (0.423190).
What is the probability of zero events when the average is 0.8?
44.93%, which is simply e^(−0.8).