Poisson Distribution Calculator: P(X = k), At Least & Table

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Formula at a glance

Poisson Distribution Calculator: Poisson probability — P(X = k) = λᵏ·e^(−λ) / k!. Example: 5 calls when the average is 3 → P(X = 5): 0.100819 = 10.0819% — How the Poisson Distribution Calculator works: Poisson probability — P(X = k) = λᵏ·e^(−λ) / k!

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📘 About the Poisson Distribution Calculator

The Poisson distribution models the number of events in a fixed interval when they happen independently at a constant average rate: customers per hour, calls per minute, accidents per month, typing errors per page, goals in a football match.

Enter the average rate λ and the number of events k. The calculator gives the probability of exactly k, at most, at least or between, the chance of no events at all, the mean and variance (both λ) and the distribution as a table and chart.

🧮 Formula

Poisson probability

P(X = k) = λᵏ·e^(−λ) / k!

Mean = variance

μ = σ² = λ

✏️ Worked examples

5 calls when the average is 3

You enter
Average rate λ	3
Number of events k	5
Probability of	Exactly k
k₂ (for between)	7

You get
P(X = 5)	0.100819 = 10.0819%
P(X = 5)	0.100819
P(X ≤ 5)	0.916082
P(X ≥ 5)	0.184737
P(X = 0) — none at all	0.0497871
Mean = variance = λ	3 (SD 1.73205)
Formula	P(X = k) = λᵏe^(−λ)/k! = 3^5·e^(−3)/5!

0.100819 = 10.0819%.

At least 2 goals (average 1.4)

You enter
Average rate λ	1.4
Number of events k	2
Probability of	At least k (≥)
k₂ (for between)	4

You get
P(X ≥ 2)	0.408167 = 40.8167%
P(X = 2)	0.241665
P(X ≤ 2)	0.833498
P(X ≥ 2)	0.408167
P(X = 0) — none at all	0.246597
Mean = variance = λ	1.4 (SD 1.18322)
Formula	P(X = k) = λᵏe^(−λ)/k! = 1.4^2·e^(−1.4)/2!

0.408167 = 40.8167%.

🧭 How to use this calculator

What you enter

Average rate λevents per interval
Number of events k
Probability of
k₂ (for between)

What you get

P(X = 5)
P(X = 5)
P(X ≤ 5)
P(X ≥ 5)
P(X = 0) — none at all
Mean = variance = λ
Formula

❓ Frequently asked questions

When should I use the Poisson distribution?

For counts of rare, independent events over time or space with a known average rate.

What is the probability of zero events?

e^(−λ). With an average of 3 events, the chance of none is e⁻³ ≈ 5%.

How is Poisson related to the binomial?

It approximates the binomial when n is large and p is small, with λ = np.

λ Poisson Distribution Calculator