Poisson Distribution Calculator
Find the probability of exactly, fewer than, at most, more than, or at least a selected number of events when events follow a Poisson model.
Inputs
Required integer from 0 to 1,000. Example: 3.
Required mean event count from 0 to 500. Use a period for decimals.
Live results
Probability mass profile
The profile joins discrete Poisson probabilities to make the distribution shape easier to scan; each plotted x-position still represents an integer event count.
Probabilities around the selected count
| Event count k | P(X = k) | P(X ≤ k) | Relative to selected x |
|---|
How to use this Poisson distribution calculator
What this calculator does
This calculator estimates probabilities for a Poisson random variable: a count of events observed in a fixed interval of time, distance, area, volume, or another exposure. It reports the chance of exactly x events and four cumulative probabilities around x. It also shows the distribution mean, variance, standard deviation, a probability-mass profile, and a compact table around the selected count. The model is useful when a count can reasonably be treated as the result of independent events arriving at a stable average rate. It does not test whether your data actually satisfy those assumptions, and it is not a substitute for a fitted count model when the rate changes with predictors or over time.
Use it for practical questions such as the chance of receiving exactly three support tickets in an hour when the long-run average is five; the probability of no equipment faults in a shift; the chance that arrivals exceed a staffing threshold; or the likelihood that a defect count stays at or below a chosen limit. The NIST Poisson distribution reference gives the probability-mass and cumulative formulas used here.
How to calculate
- The calculator opens with a complete demonstration: x = 3 and λ = 5. Its outputs, chart, table, and XLSX workbook are ready immediately.
- Replace Number of occurrences (x) with the non-negative integer count you want to evaluate. Then enter Rate of success (λ), the average number of events expected in the same interval. Results update live after each valid change.
- Read Probability P(X = x) for the exact-count probability and the four cumulative cards for fewer than, at most, more than, or at least x events. Use Download Excel to export the current typed inputs and calculated values. Reset clears the demonstration values, results, chart, table, and workbook state; Excel export remains unavailable until both required inputs form a complete valid state again.
Input guide
Number of occurrences (x) is required and must be a whole number from 0 through 1,000. It has no physical unit; it is the count whose probability you want. Enter plain digits such as 3. Decimal values, negative numbers, grouped values such as “1,000,” scientific notation, and unit symbols are rejected rather than silently reinterpreted. Raising x moves the exact-count query along the same Poisson distribution; whether the probability rises or falls depends on how x compares with λ.
Rate of success (λ) is required and represents the mean count for the same observation interval as x. This calculator accepts a non-negative decimal from 0 through 500 using a period as the decimal separator, such as 5 or 2.5. A common mistake is mixing intervals – for example, using an hourly λ with a daily event count. Convert the rate first so both refer to the same exposure. Larger λ shifts the center of the distribution toward larger counts and also increases its spread. λ = 0 is treated as the degenerate case in which X = 0 with probability 1.
Output guide
Probability P(X = x) is the exact probability of observing the selected count. Cumulative probability P(X < x) is the probability of fewer than x events, while P(X ≤ x) includes x. P(X > x) excludes x on the upper side, and P(X ≥ x) includes it. All five are shown as percentages. Values near 0% indicate outcomes that are very unlikely under the stated λ; values near 100% indicate cumulative events that are very likely.
The live pills report Mean, Variance, and Std. dev. For a Poisson distribution, mean and variance both equal λ, while standard deviation is √λ. The Probability mass profile plots P(X = k) across integer counts k and highlights the selected x. The table columns Event count k, P(X = k), P(X ≤ k), and Relative to selected x show a local numerical view around x. These are exact identities of the specified Poisson model, not empirical confidence statements about whether the model is appropriate.
Worked example
With the startup values x = 3 and λ = 5, the exact-count formula is P(X = 3) = e – 5 × 53 / 3! = 0.1403738958, displayed as 14.037390%. Summing P(X = 0), P(X = 1), and P(X = 2) gives 12.465202% for P(X < 3). Adding the exact-count probability gives 26.502592% for P(X ≤ 3). Their complements produce 73.497408% for P(X > 3) and 87.534798% for P(X ≥ 3).
How the Poisson model works
P(X = x) = e – λ λx / x!, for x = 0, 1, 2, ...
The Poisson distribution is discrete: X is a count, so x takes integer values. Its single parameter λ sets both the expected count and the variance. That equality is a distinctive property of the theoretical distribution. In observed data, variance can be larger than the mean, a pattern called overdispersion; the Penn State discussion of Poisson regression explains why real count data often depart from the equal-mean-and-variance ideal.
For cumulative probabilities, the calculator sums the individual mass values through the relevant integer boundary. Thus P(X ≤ x) is the sum from k = 0 through x, P(X < x) stops at x – 1, and the upper-tail probabilities are computed as complements. This is why P(X < x) + P(X ≥ x) = 1 and P(X ≤ x) + P(X > x) = 1 apart from negligible floating-point rounding.
When should you be cautious?
Poisson probabilities are most meaningful when the counting process has a reasonably stable rate over the chosen exposure and events are sufficiently independent. Strong clustering, contagion, seasonality, capacity limits, or a changing underlying rate can make a simple Poisson model misleading. If you observe data rather than merely planning from a known rate, compare the sample mean and variance and inspect the context before interpreting model probabilities as operational forecasts.
Keep the interval consistent. If an average of 5 events applies per hour, λ = 5 answers one-hour questions. For a two-hour interval under a constant rate, λ would be 10. The count x and rate λ must describe the same exposure.