Poisson Distribution Calculator
StatisticsCalculate Poisson probability P(X = k) for any event count in seconds. Enter the average rate (lambda) to get exact and cumulative probabilities.
Reviewed by the thecalcu.com team · Last updated July 18, 2026
P(X = k)
What is a Poisson Distribution?
The Poisson Distribution Calculator computes the probability of a specific number of events occurring within a fixed interval, given a known average event rate (λ). Enter your average rate and the target number of events, and the calculator returns P(X = k), the cumulative P(X ≤ k) and P(X ≥ k), and the distribution's standard deviation.
The Poisson distribution is the standard tool for modeling rare, independent events occurring at a known average rate, customer arrivals, website errors, equipment failures, or accidents per time period. Unlike the binomial distribution, it requires only a single parameter (the average rate) rather than a fixed number of discrete trials.
For scenarios with a known, fixed number of trials and a per-trial success probability instead of a continuous rate, use the Binomial Distribution Calculator instead, the Poisson distribution is actually the limiting case of the binomial as trials become numerous and individual success probability becomes small.
Why Use a Poisson Distribution Calculator?
Computing Poisson probabilities by hand requires evaluating e raised to a negative power, computing a factorial in the denominator, and combining these terms precisely, calculations that become unwieldy and numerically unstable for larger k values using standard calculators or spreadsheet functions.
This calculator uses a numerically stable logarithmic computation internally, producing accurate results even for large event counts, and returns both exact and cumulative probabilities together, giving you the complete picture needed for threshold-based decisions like staffing or alerting.
Who Should Use This Calculator?
Operations and capacity planning analysts estimating the probability of exceeding service capacity (call volume, support tickets, order volume) in a given time period based on historical average rates.
Site reliability and DevOps engineers setting data-driven alerting thresholds for error rates, incidents, or system events based on historical baseline rates.
Actuaries and insurance analysts modeling the probability of rare events (claims, accidents, defaults) occurring within a policy period, based on historical average incidence rates.
Students in statistics, probability, and operations research courses learning discrete probability distributions for modeling count-based, rate-driven events.
Quality control and reliability engineers estimating the probability of a specific number of defects or failures occurring within a batch or time period, given a known historical defect rate.
What Insights Does This Calculator Give You?
P(X = k), Exact Probability, the headline result: the probability of exactly the specified number of events occurring.
P(X ≤ k), Cumulative Probability, the probability of k or fewer events, useful for capacity or threshold-based planning ("what's the chance we stay under X events?").
P(X ≥ k), Upper Cumulative Probability, the probability of k or more events, useful for risk assessment ("what's the chance we exceed X events?").
Standard Deviation (σ), the natural variability around the average rate, calculated simply as the square root of lambda, a distinctive property of the Poisson distribution.
How to use this Poisson Distribution calculator
Enter the average rate (λ), your historical average number of events per interval (per hour, per day, per batch), estimated from past data.
Enter the target number of events (k), the specific count you want to evaluate the probability of.
Read P(X = k), the probability of exactly that many events occurring.
Check the cumulative probabilities, P(X ≤ k) for "at most" style capacity planning questions, or P(X ≥ k) for "at least" style risk questions.
Compare to the standard deviation, √λ gives you a sense of how much natural variation to expect around your average rate.
Re-estimate λ periodically, since the model assumes a constant rate, refresh your λ estimate periodically using recent historical data to keep predictions accurate as underlying conditions change.
Show formula & methodology ↓Show less ↑
Formula & Methodology
Poisson probability mass function: P(X = k) = (e^−λ × λᵏ) / k! Standard deviation: σ = √λ Variable definitions: - λ (lambda), the average rate of events per interval - k, the target number of events being evaluated - e, Euler's number (≈ 2.71828) Worked example: A customer support inbox receives an average of λ = 4 tickets per hour. What's the probability of receiving exactly 6 tickets in a given hour? Step 1, e^−4 ≈ 0.01832 Step 2, 4⁶ = 4096, and 6! = 720 Step 3, P(X = 6) = (0.01832 × 4096) / 720 ≈ 0.1042 (10.42%) Step 4, Standard deviation: σ = √4 = 2 A count of 6 tickets (2 above the mean of 4) is well within one standard deviation, so this outcome would not be considered unusual. Assumption: This calculation assumes events occur independently at a constant average rate throughout the interval studied. If your actual event rate varies significantly within the interval (e.g., much higher during business hours), consider modeling separate intervals with their own estimated λ values instead of one combined average.
Frequently Asked Questions