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Calculator Answers

Poisson Distribution Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A Poisson distribution calculator is a statistics utility that calculates the probability of a given number of events occurring within a fixed interval of time or space. The events must occur independently and at a constant average rate. The calculator uses the Poisson formula, requiring the average rate of occurrence (lambda) and the target number of successes to compute the exact probability of occurrence. It also provides cumulative probabilities, helping researchers, operations managers, and analysts predict customer arrivals, equipment failures, or call center volumes.

Enter the average event rate λ and the event count k. Pick a probability mode (exactly k, at most k, at least k) and the calculator returns the Poisson probability as a decimal and a percent.

Quick Answer

Estimate the probability of occurrences over time or space. Enter the average rate of occurrence and the target count to compute exact and cumulative Poisson probabilities.

Probability mode

Average events per interval. Must be greater than 0. · e.g. 4

Non-negative whole number. · e.g. 3

Formula

P(X = k) = e^(−λ) × λ^k ÷ k!

The Poisson distribution models the count of events in a fixed interval when events occur independently and at a constant average rate λ. Common uses: calls per hour, visitors per minute, defects per unit.

Poisson probability

Exactly k · λ = 4, k = 3

0.195367

19.5367% chance

P(X = k) (exact)0.195367
As percent19.5367%
Mean and variance both equal λ4
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Examples

λ = 4, k = 3, exactly

P(X = 3) ≈ 0.195367

λ = 2.5, k = 5, at most

P(X ≤ 5) ≈ 0.957979

λ = 10, k = 15, at least

P(X ≥ 15) ≈ 0.083458

How it works

Start with P(X = 0) = e^(−λ). Each subsequent probability is the previous one times λ divided by the new k. Summing those in order gives the cumulative P(X ≤ k). The at-least case uses the complement rule for numerical stability.

P(X = k) · e^(−λ) × λ^k ÷ k!

Recursive form · P(k) = P(k − 1) × λ ÷ k

Mean and variance both equal λ. For large λ, the Poisson distribution is well approximated by a normal distribution with mean λ and standard deviation √λ.

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Frequently asked questions

The Poisson distribution models the number of times an event occurs in a fixed interval (time, area, or volume) when those events are independent and the average rate is constant. It is parameterized by a single number, lambda (λ), which is both the mean and the variance. Real-world examples: calls arriving at a help desk per hour, typos per page, photons reaching a sensor per second.

P(X = k) = e^(−λ) × λ^k ÷ k!. e is Euler's number (≈ 2.71828). λ is the average rate. k is the integer count of events. The calculator uses an iterative form for stability: P(0) = e^(−λ), then P(i) = P(i − 1) × λ ÷ i. This avoids computing large factorials directly.

Three conditions. (1) Events occur independently of each other. (2) The average rate λ is roughly constant over the interval. (3) Two events cannot occur at the exact same instant. Common violations: rush-hour traffic with bursty arrivals, contagious infections where one case raises the probability of the next, or any process with a hard cap on events.

λ (lambda) is the expected number of events in the interval you care about. If a help desk averages 12 calls per hour and you want the probability for a 30-minute window, use λ = 6 for that window. Make sure λ and k refer to the same fixed interval. The mean equals λ; the variance also equals λ, which is a distinctive property of the Poisson distribution.

P(X = k) is the chance of exactly k events. P(X ≤ k) is the chance of k or fewer events (cumulative up through k). P(X ≥ k) is the chance of k or more events. The calculator computes the cumulative form by summing the per-i probabilities iteratively, and uses 1 − P(X ≤ k − 1) for the at-least case to keep the result numerically stable.

The Poisson distribution is the limiting case of the binomial when n is large, p is small, and n × p stays finite. Setting λ = n × p and using Poisson gives a good approximation when np stays roughly constant as n grows. In practice, for n above 50 and p below 0.05, Poisson with λ = np is often used instead of binomial because the math is easier.