Poisson Distribution Calculator

Find the probability of a given number of events in a fixed interval when events happen at a known average rate.

Your numbers

Average rate (λ)

The average number of events per interval, such as 3 calls per hour.

Number of events (k)

Mean = variance = 3; standard deviation = 1.7321.

Poisson distribution, λ = 3

Red bar: exactly k events. Lighter bars: fewer than k.

Probability	Value
P(X = 2)	0.224042
P(X ≤ 2)	0.42319
P(X < 2)	0.199148
P(X ≥ 2)	0.800852
P(X > 2)	0.57681

P(X = 2) = e^−λ λ² ÷ 2! = e⁻³ × 3² ÷ 2! = 0.224042

How to use this tool

Enter the average number of events per interval (λ), such as 3 calls per hour.
Enter k, the number of events you are asking about.
Read the probability of exactly k, at most k, fewer than k, at least k, and more than k.
Use the chart to see where k sits in the whole distribution.

Formula Reference

Poisson probability

P(X = k) = e^(−λ) λ^k ÷ k!

Mean and variance

E[X] = Var(X) = λ

At most k

P(X ≤ k) = Σ from i=0 to k of P(X = i)

At least k

P(X ≥ k) = 1 − P(X ≤ k − 1)

ArticleThe Poisson Distribution ArticleThe Exponential Distribution ArticleThe Binomial Distribution CalcBinomial Probability Calculator GlossaryGlossary: Poisson Distribution

Frequently Asked Questions

When can I use the Poisson distribution?+

When you count events in a fixed interval of time or space, events occur independently of each other, and they occur at a constant average rate. Typical examples are calls to a help line, typos on a page, or defects along a length of cable.

What do the mean and variance tell me?+

For a Poisson distribution, the mean and the variance are both equal to λ. If your data have a variance much larger than the mean, the counts are overdispersed and a Poisson model may understate the spread.

How is it related to the binomial distribution?+

A binomial with many trials and a small success probability is well approximated by a Poisson with λ = np. The Poisson is often easier when the number of trials is large or unknown.

What if my interval is different from the rate I know?+

Scale λ to match the interval. If calls arrive at 3 per hour on average, then for a 30-minute interval use λ = 1.5.

Probability

Value

P(X = 2)

0.224042

P(X ≤ 2)

0.42319

P(X < 2)

0.199148

P(X ≥ 2)

0.800852

P(X > 2)

0.57681