Poisson Distribution Probability Guide
Poisson probability models event counts in a fixed interval from one average rate parameter λ. The arithmetic is compact, but the model only makes sense when its constant-rate and independence assumptions are reasonable.
Method
- Identify λ, the expected average number of events in the interval.
- Identify k, the whole-number event count in the question.
- Use P(X = k) for exactly k events, P(X ≤ k) for at most k, or P(X ≥ k) for at least k.
- Compare the result with the model assumptions before interpreting it as a real-world probability.
Worked example
With λ = 4 and k = 2, P(X = 2) ≈ 0.1465 and P(X ≤ 2) ≈ 0.2381.
The distribution has mean 4, variance 4 and standard deviation 2.
Notes
- λ can be any nonnegative rate, while k must be a nonnegative whole-number count.
- A changing event rate, dependence between events or strong overdispersion can make a basic Poisson model inappropriate.
- Poisson and binomial can be related in some approximations, but they are not interchangeable models.
Use Poisson for counts tied to an average event rate
The Poisson model describes nonnegative event counts in a fixed interval when events occur with a stable average rate under its assumptions. The rate parameter lambda is both the model mean and variance.
Changing the interval requires changing lambda. If a process averages four events per hour, a half-hour interval would use an expected count of about two only when the rate is reasonably stable over time.
Watch for overdispersion, clustering and changing rates
Real event counts can vary more than a Poisson model allows because events cluster or the underlying rate changes. When the observed variance is much larger than the mean, the simple model may underestimate tail probabilities.
Use the calculation as a model-based estimate and inspect the process assumptions before treating the result as an empirical forecast.
Worked arrival-rate example: convert the hour to the question's interval
Assume a fictional service desk has a stable average of four arrivals per hour and that the simple Poisson assumptions are acceptable for an illustration. For a 30-minute interval, the model parameter is λ = 4 × 30/60 = 2 arrivals. The exact-three probability is P(X = 3) = e^(−2) × 2³/3! ≈ 0.18045, or 18.05%. Entering λ = 4 for this half-hour question would instead describe a different expected number of events.
At most one arrival includes both zero and one: P(X ≤ 1) = e^(−2) × (1 + 2) ≈ 0.40601, or 40.60%. Exactly one is only 2 × e^(−2) ≈ 0.27067. Use the linked calculator with λ = 2 and k = 1 to contrast exact and cumulative results. These numbers are probabilities under the stated model, not observed proportions or a service-level guarantee.
A change of time window changes the zero-event probability too
Under the same artificial four-per-hour constant-rate model, a 7.5-minute interval has λ = 4 × 7.5/60 = 0.5. Its probability of zero arrivals is e^(−0.5) ≈ 0.60653. A 30-minute interval instead uses λ = 2 and has P(X = 0) = e^(−2) ≈ 0.13534. A single rate cannot be copied between windows without converting the units.
A real help desk might be quiet in the morning and busy after lunch, with clustered contacts and changing rates. In that case, one constant λ for the whole day can understate busy-period risk. Separate the observed rate by an appropriate interval, check variability against the mean and choose a better process model when the simple Poisson assumptions do not fit.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Binomial Probability Calculator — Calculate exact, at-most and at-least binomial probabilities from trials, successes and a fixed success probability.
- Normal Distribution Calculator — Calculate normal-distribution left-tail, right-tail and between-values probabilities, density, z-scores and inverse percentiles.
- Permutation & Combination Calculator — Calculate exact nPr permutations and nCr combinations with or without repetition, using integer arithmetic instead of rounded factorial math.