Poisson Probability Calculator
Computes the probability of observing exactly k events in a fixed interval when events occur at a known average rate. Use it for call-centre arrivals, equipment failures, or any count of rare, independent events.
About this calculator
The Poisson distribution models the number of times a random event occurs in a fixed interval of time or space, given that events happen independently at a constant average rate λ. The probability mass function is P(X = k) = (e^−μ × μᵏ) / k!, where μ = λ × t is the expected number of events in interval t, k is the observed count, and e ≈ 2.71828. Both the mean and variance of a Poisson distribution equal μ, which makes it easy to check whether Poisson is a good fit: if the sample mean and variance are approximately equal, the distribution is plausible. The cumulative probability P(X ≤ k) sums the PMF from 0 to k. The Poisson distribution can also approximate the binomial when n is large and p is small (μ = np). Common applications include network packet arrivals, radioactive decay counts, and insurance claim modelling.
How to use
A call centre receives an average of λ = 3 calls per minute. What is the probability of exactly k = 5 calls in a t = 1 minute interval? Step 1 — compute μ = λ × t = 3 × 1 = 3. Step 2 — apply the formula: P(X = 5) = (e^−3 × 3⁵) / 5! = (0.04979 × 243) / 120 = 12.099 / 120 ≈ 0.1008. Step 3 — enter λ = 3, t = 1, k = 5 and select PMF; the calculator returns approximately 10.08%.
Frequently asked questions
What is the difference between Poisson PMF and Poisson CDF probability?
The PMF (probability mass function) gives the probability of observing exactly k events: P(X = k). The CDF (cumulative distribution function) gives the probability of observing at most k events: P(X ≤ k), summing the PMF from 0 up to k. Use the PMF when you need a precise count (e.g., 'exactly 3 defects') and the CDF when you need an upper bound (e.g., 'no more than 3 defects'). Many real-world decisions — like setting service-level thresholds — rely on the CDF.
When is the Poisson distribution not an appropriate model for count data?
The Poisson distribution assumes events are independent and occur at a constant rate. It becomes inappropriate when events cluster (positive correlation), when the rate varies over time, or when counts are overdispersed (variance substantially exceeds the mean). In those cases, a negative binomial distribution is often a better fit. It also should not be used for counts with a natural upper bound, where the binomial or hypergeometric distribution is more suitable.
How do you choose the right time interval when using the Poisson calculator?
The rate λ and the time interval t must use consistent units. If λ is given per hour and your interval is 30 minutes, convert: t = 0.5 hours so μ = λ × 0.5. Alternatively, convert λ to per-minute before entering. The calculator uses μ = λ × t internally, so as long as units match, any interval length is valid. Choosing a meaningful interval — such as one shift or one day — makes the resulting probabilities directly actionable for operational decisions.