Poisson Distribution Probability Calculator
A Poisson model answers one question: if events happen at a steady average rate, how likely is a particular count in the next interval? Give this calculator the average rate λ and the target count k, and it returns the probability as a decimal and a percentage, for exactly k, at most k, at least k, fewer than k, or more than k. It also shows the mean, variance, standard deviation and mode of the distribution, plus a probability table around your target, computed in log-space so large inputs never overflow the way direct factorials do.
Probability
Distribution properties
Probability table
The PMF is computed in log-space with the Lanczos log-gamma function, so large λ and k never overflow. For cumulative types with λ above 1000, the tool switches to the normal approximation with continuity correction and says so on the formula line.
Quick Answer
For a call center averaging 3 calls per hour, the chance of exactly 2 calls in the next hour is 0.22404181 (22.4042%). At most 2 calls: 0.42319008. At least 2 calls: 0.80085173. The distribution has mean 3, variance 3 and standard deviation 1.7321. Enter your own λ and k above and every one of those numbers recomputes instantly.
Key takeaways
- The Poisson model fits events that arrive at a steady average rate with no memory: calls, arrivals, typos, defects, decays. It does not fit fixed-trial questions such as “7 heads in 10 flips” (that is binomial).
- Mean and variance are both λ. That single number sets the whole shape of the distribution.
- Large k breaks naive calculators because 171! overflows a double. This one works in log-space, so k = 100,000 computes without drama.
- For very large rates (λ above 1000), cumulative probabilities switch to the normal approximation with continuity correction, and the tool says so on the formula line. Nothing is hidden.
- Quick Answer
- How to use the Poisson probability calculator
- The Poisson formula, and how this calculator evaluates it
- Worked example: a call center averaging 3 calls per hour
- Spot checks against published tables
- When the Poisson model fits, and when it does not
- What competing Poisson calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the Poisson probability calculator
Three inputs, one click, and the results stay hidden until you calculate.
- Enter the average rate λ: the expected number of events in one interval. The call-center example (3 calls per hour) is pre-filled. Decimals are allowed, and zero is valid.
- Enter k, a whole number: the count you are asking about. Fractional k is rejected with an explanation, because a Poisson variable counts whole events.
- Pick the probability type: exactly k for a single count; at most k and at least k include k itself; fewer than k and more than k exclude it. The comparison table below shows all five side by side for the worked example.
- Click Calculate and read the probability as a decimal and a percentage, the distribution properties, a probability table around your target, and a plain-English verdict. Copy my results and Share results sit under the verdict.
The Poisson formula, and how this calculator evaluates it
The probability of exactly k events is P(X = k) = λk × e−λ ÷ k!. Cumulative types sum those terms: P(X ≤ k) adds the PMF from 0 to k, and the rest are complements of that sum.
P(X ≤ k) = Σi=0..k P(X = i) | P(X ≥ k) = 1 − P(X ≤ k−1)
Mean μ = λ | Variance σ² = λ | Std dev σ = √λ
Two implementation details matter. First, the PMF is evaluated as k·ln(λ) − λ − ln(k!) using the Lanczos approximation of the log-gamma function, so k = 100,000 computes where a direct factorial would have overflowed at k = 171. Second, for cumulative probabilities with λ above 1000, summing thousands of terms is wasteful, so the tool switches to the normal approximation with continuity correction, Φ((k + 0.5 − λ) / √λ), and appends “[normal approximation, λ > 1000]” to the formula line. The exact probability type always uses the log-space PMF, whatever λ is. These are the standard definitions given in the NIST/SEMATECH e-Handbook of Statistical Methods and in OpenStax Introductory Statistics.
Worked example: a call center averaging 3 calls per hour
With λ = 3 and k = 2, exactly as pre-filled in the calculator: P(X = 2) = 3² × e−3 ÷ 2! = 9 × 0.0497871 ÷ 2 = 0.22404181 (22.4042%). The tool agrees to all eight decimals.
The five probability types for the same λ = 3, k = 2, all real runs of this calculator:
| Question | Type | Result |
|---|---|---|
| Exactly 2 calls | P(X = 2) | 0.22404181 (22.4042%) |
| 2 or fewer calls | P(X ≤ 2) | 0.42319008 (42.319%) |
| 2 or more calls | P(X ≥ 2) | 0.80085173 (80.0852%) |
| Fewer than 2 calls | P(X < 2) | 0.19914827 (19.9148%) |
| More than 2 calls | P(X > 2) | 0.57680992 (57.681%) |
Notice that “at most” plus “more than” equals 1 (0.42319008 + 0.57680992), and “at least” plus “fewer than” equals 1 (0.80085173 + 0.19914827). If those pairs ever fail to sum to 1, something is wrong with the arithmetic, so they make a good sanity check on any Poisson tool. The probability table under the results shows the PMF and running cumulative probability for k values around your target, with your k marked, which is the fastest way to see where the probability mass sits.
Spot checks against published tables
Every row below is a real run of this calculator, cross-checked against independent published Poisson tables.
| Inputs | This calculator | Independent check |
|---|---|---|
| λ = 4, k = 2, exact | 0.14652511 (14.6525%) | Matches the PMF table on statistics reference tables (14.6525%) |
| λ = 1.5, k = 3, exact | 0.12551072 (12.5511%) | Matches the published Poisson table value 0.1255 |
| λ = 10, k = 10, exact | 0.12511004 (12.511%) | Hand check: e−10·1010/10! = 0.125110 |
| λ = 5, k = 0, exact | 0.00673795 (0.6738%) | e−5 = 0.0067379, the no-event probability |
| λ = 2000, k = 1990, exact | 0.00872147 (0.8721%) | Log-space evaluation; a direct factorial would overflow |
When the Poisson model fits, and when it does not
The Poisson distribution fits when events arrive independently at a constant average rate over the interval: calls per hour, website hits per minute, defects per batch, radioactive decays per second. If the rate drifts (rush hour vs midday) or events cluster (one failure triggering others), the model understates the extremes.
| Question | Right model |
|---|---|
| Calls in the next hour, averaging 3 | Poisson (this page): open-ended count in an interval |
| Heads in exactly 10 coin flips | Binomial: fixed number of trials. See the coin flip probability calculator. |
| Ways to choose k from n | Combinatorics, not a distribution. See the binomial coefficient calculator. |
| A probability stated as a percent of another number | Plain percentage arithmetic. See the probability percentage calculator. |
A related boundary: a general discrete probability distribution calculator handles arbitrary custom distributions you define point by point, while this page is the dedicated Poisson specialist with its log-space engine and five built-in probability types.
What competing Poisson calculators miss
Most Poisson tools compute the PMF with direct factorials, which overflow past k = 171 and either error out or silently return wrong numbers. Others cap k at 100, handle only the exact probability, or switch to approximations without telling you. This page keeps all five probability types on one picker, evaluates the PMF in log-space with the Lanczos log-gamma function for any k up to 100,000, and discloses its one approximation: cumulative types with λ above 1000 use the normal approximation with continuity correction, labelled on the formula line and in the verdict. The mean, variance, standard deviation and mode appear beside the probability instead of on a separate page, and the probability table shows the PMF and cumulative columns around your target so you can see the shape of the distribution, not just one number.
Frequently asked questions
A Poisson distribution models the number of events occurring in a fixed interval when events happen independently at a constant average rate λ. It is the right model for counts like calls per hour or defects per batch, and it is defined entirely by the single parameter λ, which is both the mean and the variance.
Divide the total events observed by the number of intervals: λ̂ = total events / number of intervals. If a call center logged 240 calls over 80 hours, λ = 3 calls per hour, which is the worked example pre-filled in the calculator.
P(X = k) is the probability of exactly k events. P(X ≤ k) adds up the probabilities for 0 through k, and P(X ≥ k) is 1 minus the probability of k − 1 or fewer. “Fewer than k” and “more than k” exclude k itself. With λ = 3 and k = 2, the calculator gives 0.22404181, 0.42319008 and 0.80085173 respectively.
They compute k! directly, and 171! overflows a standard double-precision number, so results past k = 170 are wrong or missing. This calculator evaluates the PMF in log-space as k·ln(λ) − λ − ln(k!) with the Lanczos log-gamma function, so k up to 100,000 computes correctly.
Exact-probability results still use the log-space PMF. Cumulative types switch to the normal approximation with continuity correction, Φ((k + 0.5 − λ) / √λ), because summing thousands of terms is wasteful and the approximation is excellent there. The tool adds “[normal approximation, λ > 1000]” to the formula line and the verdict so the switch is never hidden.
λ = 0 is valid and means nothing ever happens: P(X = 0) = 1 and every other count has probability 0. Fractional k is rejected, because a Poisson variable counts whole events; the tool asks for a whole number instead of silently rounding your input.
Use Poisson for an open-ended count in an interval at a steady average rate, such as calls per hour. Use binomial for a fixed number of trials with a per-trial success chance, such as heads in 10 flips. The binomial approaches the Poisson when the trial count is large and the per-trial chance is small, which is why rare-event counts are often modelled either way.
The mode of a Poisson distribution is floor(λ), the largest whole number not above λ. One subtlety the tool discloses: when λ is itself a whole number, λ − 1 is also a mode (for λ = 3, counts 2 and 3 tie at 0.224042). The tool reports floor(λ) and this FAQ notes the tie.
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References
- National Institute of Standards and Technology and SEMATECH. Poisson Distribution, NIST/SEMATECH e-Handbook of Statistical Methods. Accessed October 24, 2026.
- OpenStax. 4.6 Poisson Distribution, Introductory Statistics. Accessed October 24, 2026.
- Penn State Eberly College of Science. Lesson 12: The Poisson Distribution, STAT 414 Probability Theory. Accessed October 24, 2026.
- Weisstein, Eric W. Poisson Distribution, Wolfram MathWorld. Accessed October 24, 2026.
- Khan Academy. Poisson distribution, Statistics and Probability. Accessed October 24, 2026.