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Poisson Distribution Calculator

Find exact, at-most and at-least Poisson probabilities from an average rate.

By Nitin Raturi · Last reviewed

Your worksheet

How to use the Poisson Distribution Calculator

Find the probability of a number of events when they happen at a known average rate.

  1. Enter the average number of events in the interval.
  2. Enter the count you are asking about.
  3. Choose exactly, at most, at least or another relation.
  4. Read the probability and standard deviation.

Good to know

  • The mean and the variance of a Poisson count are equal.

P(X = k) = μᵏ e^(−μ) ÷ k!. Only the average rate μ is needed.

Average rate (μ)

The average number of events in the interval you care about.

How to fill it in

  • Scale it to the interval: 41.5 a day is 1.7292 an hour.

Example

With μ = 147 the standard deviation is 12.1244.

Number of events (k)

The count you are asking about.

How to fill it in

  • Enter a whole number from 0.

Example

With an average of 2, exactly 3 has probability 0.1804.

Probability of

Exactly k, at most k, fewer than k, at least k or more than k.

How to fill it in

  • Choose =, ≤, <, ≥ or >.

Example

μ = 147: P(X = 160) = 0.0180 and P(X ≤ 160) = 0.8666; μ = 0.75: P(X > 1) = 0.1734.

P(X ≤ 160)

0.8666

With an average of 147 events per interval, the probability of at most 160 is 0.8666 (86.66%).

P(X = 160)
0.0180
Mean
147
Standard deviation (√μ)
12.1244

Events happen independently, one at a time, at a steady average rate over the interval.

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How this calculator works

  1. Step 1Rateμ per intervalThe average count.
  2. Step 2Each countμᵏ e^(−μ) ÷ k!Chance of exactly k.
  3. Step 3Rangesadd the termsFor at most, at least and so on.

The Poisson distribution describes counts of events that happen independently at a steady average rate: calls to a help line, emails, typos on a page.

Only the average matters. The standard deviation is the square root of the mean.

Worked example: OpenStax Example 4.30

147 emails a day on average: at most 160?
P(X ≤ 160)
Σ 147ᵏ e⁻¹⁴⁷ ÷ k! for k = 0 to 160
  1. P(X = 160)0.0180
  2. Standard deviation√14712.1244
  3. P(X ≤ 160)0.8666

OpenStax gives 0.0180, 0.8666 and 12.1244.

Read the result

  • μ = 2: exactly 30.1804
  • μ = 0.75: more than 10.1734
Two Poisson questions

Even with an average under one a day, more than one event happens on 0.1734 of days (OpenStax Example 4.29).

  • Poissonrate onlyNo fixed number of trials.
  • Binomialn and pA set number of yes-or-no trials.

Assumptions and limits

  • IndependentOne event does not trigger another.
  • Steady rateThe average is the same throughout.
  • Not clusteredBursts of events break the model.
  • Same intervalμ must match the interval asked about.

The Poisson models events at an average rate. A fixed number of tries uses the binomial.

For successes in a set number of trials, use the binomial distribution calculator. The probability calculator combines simpler independent events.

Common mistakes

  • Mistake: Using a daily rate for an hourly question.

    Instead: Scale μ to the interval first.

  • Mistake: Mixing up at least and more than.

    Instead: At least k includes k.

Checking the method

A separate script rebuilds every probability term by term with 50-digit decimals and reproduces OpenStax Examples 4.29 to 4.32.

Read how we test our calculators, or browse more maths calculators.

Frequently asked questions

If you get 147 emails a day on average, how likely is 160 or fewer?

0.8666, and exactly 160 is 0.0180 (OpenStax Example 4.30).

What is P(X = 3) when μ = 2?

0.1804, from 2³ e⁻² ÷ 3!.

What is P(X > 1) when μ = 0.75?

0.1734, as in OpenStax Example 4.29.

What is the standard deviation?

The square root of the mean: √147 = 12.1244.

When should I use Poisson instead of binomial?

When events come at a rate with no fixed number of trials, such as arrivals per hour.

Sources and methodology

The formulas are standard time-value-of-money results. We check every figure against golden values worked out independently, by closed form and by month-by-month simulation; see the methodology.

Written and maintained by Nitin Raturi, a software engineer (about the author). Last reviewed 4 Nov 2026; next review due by . Found an error? Tell us.

Changelog

  • : Initial release.