Calculate Poisson distribution probabilities for modeling rare events. Find exact, cumulative, and range probabilities with mean rate λ. Includes distribution charts, probability tables, and step-by-step calculations.
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Calculate Poisson distribution probabilities instantly. Model the number of events occurring in a fixed interval given an average rate λ. Perfect for call centers, traffic analysis, and rare event modeling. Includes distribution charts, probability tables, and step-by-step calculations.
The Poisson distribution models the probability of a given number of events occurring in a fixed interval (time, space, or distance) when events occur independently at a constant average rate λ (lambda). It's ideal for modeling rare events like customer arrivals, phone calls, radioactive decay, or website visits. Unlike the binomial distribution, Poisson doesn't require a fixed number of trials.
Poisson Probability Formula
P(X = k) = (λᵏ × e⁻λ) / k!Model incoming call rates to optimize staffing levels and minimize wait times.
Predict vehicle arrivals at intersections or toll booths for signal timing optimization.
Model patient arrivals in emergency rooms or disease occurrence rates in epidemiology.
Predict defect rates in manufacturing when defects are rare and independent.
Use Poisson when: events occur continuously at a rate λ, you don't know the total number of trials, and events are rare. Use Binomial when: you have a fixed number of trials n, each with known success probability p. Poisson is actually the limit of Binomial as n→∞ and p→0 while np=λ remains constant.