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Statistics

Poisson probability

Probability of exactly k events in a fixed interval when events arrive independently at an average rate λ.

Ref: STA-09
Rev: v1
Field: Statistics
Result: P(k)

P (k) = \frac{λ^{k} e^{- λ}}{k!}

λ is the average number of events per interval and k the exact count asked about (k = 0, 1, 2, …). The mean and the variance of a Poisson count are both λ. Events are assumed independent, random in time or space and at a steady average rate.

Probability of exactly k events in a fixed interval when events arrive independently at an average rate λ.

Calculator

Solve for

P(k)ProbabilityComputed

λAverage events per interval

0.150

kEvents counted

050

Probability of no events, e^(−λ)0.04979
Probability of at least one event0.9502

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Probability () current value 0.224

λAverage events per intervalKey driver

2.41e-190.267

kEvents counted

1.18e-42

With an average of 3 events per hour the chance of exactly 2 is about 22 %; the chance of none is only about 5 %.
The standard deviation is √λ, so counts of 100 typically vary by ±10.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
λ Average events per interval	0.1 to 50	2.411e-19	0.2666	0.2666	+10%	−10.4 %
k Events counted	0 to 50	1.175e-42	0.224	0.224	+1	+8.88e-14 %

Sweep of Average events per interval

Average events per interval	Probability
0.1	0.004524
6.337	0.03552
12.57	2.734e-4
18.81	1.196e-6
25.05	4.145e-9
31.29	1.264e-11
37.52	3.554e-14
43.76	9.448e-17
50	2.411e-19

Sweep one variable

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Probability ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

P(k)

Probability

Result ·

What the equation gives you. Everything else feeds it.

Average events per interval

· typical 0.1 to 50

The average number of events in the interval. Moves the peak of the distribution towards k = λ. For k = 2 the chance rises up to λ = 2 and falls beyond it, so at the default a 10 % rise in λ lowers P by about 10 %.

Events counted

· typical 0 to 50

The exact number of events whose probability is wanted. Probability is highest for k close to λ and falls off on either side.

Worked example with the values above

Numerator: λ^k e^−λ = 3² × 0.049787 = 0.44808.
Factorial: k! = 2.
P(k) = 0.44808 / 2 = 0.22404.

Assumptions and limits

Events occur independently at a constant average rate.
Two events cannot occur at exactly the same instant.

Common mistakes

Using it when events cluster, such as bursts of faults from one cause.
Forgetting to rescale λ when the interval length changes.

Statistics and uncertainty

The Poisson distribution has mean and variance both equal to λ. If observed counts vary much more than that, events are clustered and the model understates the spread.

Where it is used

Arrivals of contacts or faults per hour.
Defects per metre of weld or per batch.

Maths you need

Factorials: k!
Exponentials: e^(−λ)

Related formulas

Practise this formula