Compute exact, cumulative, and interval Poisson probabilities from worksheets with flexible inputs, tables, and graphs. Export results and understand event frequency patterns more clearly.
| Scenario | Mean Rate (λ) | k | Meaning |
|---|---|---|---|
| Calls per minute | 3.2 | 4 | Chance of receiving four calls in one minute |
| Defects per batch | 1.8 | 0 | Chance of a defect-free batch |
| Website errors per hour | 5.0 | 7 | Likelihood of seven errors in one hour |
| Arrivals per interval | 2.4 | 1 | Probability of one arrival in the interval |
The Poisson probability mass function measures the chance of observing exactly k events when the average event rate is λ within a fixed interval.
P(X = k) = e-λ × λk / k!
Cumulative probability is found by summing exact probabilities from zero up to the selected k value. The model also has mean λ, variance λ, and standard deviation √λ.
The Poisson distribution is one of the most useful discrete probability models in mathematics, statistics, engineering, operations research, and data analysis. It helps estimate how often an event happens when the event count is whole, the interval is fixed, and the average rate stays reasonably stable. Typical examples include incoming calls, machine faults, service arrivals, claims, defects, traffic incidents, and website errors.
This calculator goes beyond one simple probability output. It returns exact probability for a chosen event count, cumulative probability up to that count, probability of at least that count, and inclusive interval probability between two bounds. It also builds a distribution table and converts each probability into expected frequency based on the number of observed intervals you provide.
The table helps compare several k values at once instead of checking one outcome in isolation. That is useful when you want to study tails, spot the most likely count, or understand how quickly cumulative probability rises. The graph gives a faster visual view of the distribution shape. With smaller λ values, the mass sits near zero. With larger λ values, the pattern spreads wider across event counts.
The model works best when events occur independently, counts happen within consistent intervals, and the mean rate is fairly constant. If the rate changes sharply over time or events cluster together, another model may fit better. Even then, the Poisson model remains a practical baseline for fast estimation, teaching, simulation, planning, and quality review.
λ is the average number of events expected in one fixed interval. It is the core input for every Poisson probability calculation.
k is the exact number of events you want to test. It must be a whole number equal to zero or more.
Use it for independent event counts across a fixed period, distance, area, or volume when the average rate is stable.
Cumulative probability is the chance of getting up to and including a selected number of events. It adds several exact probabilities together.
That is a defining property of the Poisson distribution. Both the expected value and the variance equal λ.
Expected frequency estimates how often a result should appear across many intervals. It equals probability multiplied by the number of observations.
Yes. If λ equals zero, zero events have probability one, and every positive event count has probability zero.
Exporting helps you document results, compare scenarios, share outputs, and keep a clean record for analysis or reporting.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.