This Poisson distribution calculator finds exact and cumulative probabilities for counted events arriving at an average rate λ — calls per hour, defects per batch, arrivals per day — where there is no fixed number of trials, only a rate.
Variance equals the mean for a true Poisson process (σ² = λ) — if real data shows much bigger spread (overdispersion), the negative binomial usually fits better. For a different time window, scale λ proportionally: 5 per hour is 0.0833 per minute.
When Events Have a Rate, Not a Roster
P(X = k) = λᵏ e^(−λ) / k! — one parameter does everything: mean = variance = λ. The Poisson arises as the binomial's limit when trials multiply and success gets rare (n→∞, p→0, np = λ), which is why both live side by side here: fixed trials belong to the Binomial Distribution Calculator; open-ended counts in time or space belong to Poisson. Its assumptions: events independent, rate constant, two events can't share an instant. Part of our Statistics Calculators collection.
Worked Example (Verified)
A support desk averaging λ = 5 calls per hour (the widget's defaults):
- P(exactly 3 calls) = 5³ × e^(−5) ÷ 3! = 0.1404 — about one hour in seven
- P(3 or fewer) = 0.2650 — a quiet-ish hour is a one-in-four event, not a mystery
- P(0 calls) = e^(−5) = 0.67% — a completely silent hour should happen about twice a year of round-the-clock operation; if it happens monthly, λ isn't 5 anymore
- SD = √5 = 2.24 — hours with 9–10 calls (+2σ) are built into the model, no surge explanation required
Spotting Poisson in the Wild
- Website hits, call centers, arrivals, radioactive counts (the original Geiger-counter math), typos per page, defects per roll of fabric.
- The clumping illusion: Poisson events cluster by pure chance — random "hot streaks" of calls or accidents are what λ looks like, which is why apparent clusters need statistics before stories.
- Rate changes break it: lunch-hour vs midnight λ differ — model separate windows or scale λ by the window (an hour at λ 5 → 15 minutes at λ 1.25).
Related Tools
- Binomial Distribution Calculator — fixed-trial counts
- Expected Value Calculator — weighting outcomes by these probabilities
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate a Poisson probability?
P(X = k) = λᵏe^(−λ)/k!. At λ = 5, k = 3: 125 × 0.00674 ÷ 6 = 0.1404. Cumulative values sum the individual terms — slow by hand, instant here.
What does λ (lambda) mean?
The average number of events in the interval you're modeling — per hour, per page, per square meter. It must match your interval: λ = 5/hour becomes 40 per 8-hour shift. Mean and variance both equal λ.
Poisson or binomial — which do I need?
Fixed number of trials with a success chance → binomial. Events counted over time/space with only an average rate → Poisson. They're linked: binomial with huge n and tiny p converges to Poisson with λ = np.
What is overdispersion?
Real count data whose variance clearly exceeds its mean — the Poisson's σ² = λ fails. Causes: rates varying (day/night mixtures), events triggering events. The negative binomial distribution is the standard upgrade.
How likely are zero events?
P(X = 0) = e^(−λ): λ = 1 → 36.8%, λ = 3 → 5.0%, λ = 5 → 0.67%. It's the single most-used Poisson value in practice (reliability, safety cases, "why was it so quiet?").
Is this Poisson calculator free?
Yes — free, instant, no sign-up.