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Calculate exact and cumulative Poisson distribution probabilities instantly. Enter your known average rate (Lambda) and target occurrences to generate a complete probability breakout matrix.
The average number of expected occurrences (lambda).
The exact number of successes you want the probability for.
The Poisson distribution describes the probability of counting a given number of events in a fixed window—time, distance, area, or volume—when events are independent, occur at a steady average rate (λ), and are individually rare. It is the standard model for queues, arrivals, failures, clicks, and defects when you care about "how many" rather than a fixed number of trials.
Why it matters: λ turns historical averages into forward-looking risk. Teams use Poisson probabilities to size staffing, set safety stock, design SLAs, and explain variability to stakeholders with numbers instead of intuition.
Primary use cases
The calculator is powered by the Poisson probability mass function (PMF) for exact counts, and by sums of PMF terms for cumulative questions (at most, fewer than, at least, more than).
P(X ≤ k) = P(0) + P(1) + … + P(k)
"At most k" adds every exact probability from 0 through k. Strict inequalities use complements—for example, P(X < k) = P(X ≤ k − 1), and P(X ≥ k) = 1 − P(X ≤ k − 1).
Key property: For Poisson, mean and variance both equal λ. That identity is handy for quick sanity checks.
Three worked scenarios showing the PMF in action. Use our calculator to double-check each numeric result.
A midnight shift averages λ = 4 patients per hour. What is P(X = 6)?
(46 · e−4) / 6! = (4096 · 0.01832) / 720
≈ 0.104 → about 10.4% chance of exactly six arrivals that hour—useful for bed and triage planning.
A line averages λ = 1.2 defects per batch. What is P(X = 0)—a perfectly clean batch?
P(X = 0) = λ0e−λ / 0! = e−1.2
≈ 0.301 → about 30.1% of batches have no defects at this rate—baseline for Six Sigma or SPC follow-up.
You average λ = 3 signups per hour. How likely is a busy hour with exactly 5 signups?
(35 · e−3) / 5! = (243 · 0.0498) / 120
≈ 0.101 → about 10.1%. Compare to P(X ≤ 2) for a "slow" hour using cumulative sums.
Use these reference bands to interpret λ, choose the right tail vs cumulative question, and contrast Poisson with the binomial model.
| λ range (mean events) | Distribution shape | Typical questions | Modeling tip |
|---|---|---|---|
| λ < 1 | Strong spike at 0–1 | P(X = 0), P(X ≤ 1) | Great for very rare events; watch data sparsity. |
| 1 ≤ λ ≤ 10 | Skewed, single mode near λ | Staffing, queues, daily volumes | Most classroom and ops examples live here. |
| 10 < λ ≤ 25 | Smoother, mild skew | P(X ≥ k) tail risk, SLAs | Normal approximation may be reasonable for some sums. |
| λ > 25 | Nearly bell-shaped | High-volume call centers, web traffic | Check overdispersion; consider alternatives if variance ≫ λ. |
| Topic | Poisson | Binomial |
|---|---|---|
| What is fixed? | Interval + average rate λ | Number of trials n |
| Upper bound on count? | No hard cap (theoretically unbounded) | Cannot exceed n successes |
| Classic question | "How many arrivals this hour?" | "How many heads in 20 flips?" |
| Scenario | Lambda (avg rate) | Query type | Typical interpretation | | --- | ---: | --- | --- | | Website signups/hour | 6 | P(X = 10) | Chance of exactly 10 signups this hour | | Support tickets/day | 14 | P(X <= 10) | Workload stays at or below 10 tickets | | Defects per batch | 1.2 | P(X = 0) | Clean-batch probability (~30% at λ=1.2) | | ER arrivals/hour | 4 | P(X >= 7) | Surge staffing / overflow risk | | Fraud alerts/day | 0.8 | P(X < 2) | Low-alert day (0 or 1 alerts) | | λ range | Shape | Typical questions | | --- | --- | --- | | < 1 | Spike at 0-1 | Rare events, P(X=0) | | 1-10 | Skewed ops curve | Queues, staffing | | > 25 | Near symmetric | High volume; validate variance ≈ λ |
Share this Poisson Distribution calculator with your classmates to instantly verify complex probability equations without wrestling with factorial exponents by hand.
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