Poisson Distribution Calculator
Enter the average number of events per interval and a count k, and the calculator gives the Poisson probability of exactly k, at most k, at least k, fewer than k, more than k or a range of counts. You can also enter a rate and a length of time, such as 1.5 events per hour for 2 hours, and it works out the average for you. A table shows every probability around the average, with cumulative columns and a bar for each count. Everything runs in your browser.
| k | P(X = k) | P(X <= k) | P(X >= k) | Chance |
|---|---|---|---|---|
| 0 | 0.049787 | 0.049787 | 1 | |
| 1 | 0.14936 | 0.19915 | 0.95021 | |
| 2 | 0.22404 | 0.42319 | 0.80085 | |
| 3 | 0.22404 | 0.64723 | 0.57681 | |
| 4 | 0.16803 | 0.81526 | 0.35277 | |
| 5 | 0.10082 | 0.91608 | 0.18474 | |
| 6 | 0.050409 | 0.96649 | 0.083918 | |
| 7 | 0.021604 | 0.9881 | 0.033509 | |
| 8 | 0.0081015 | 0.9962 | 0.011905 | |
| 9 | 0.0027005 | 0.9989 | 0.003803 | |
| 10 | 0.00081015 | 0.99971 | 0.0011025 | |
| 11 | 0.00022095 | 0.99993 | 0.00029234 | |
| 12 | 5.5238e-5 | 0.99998 | 7.1387e-5 | |
| 13 | 1.2747e-5 | 1 | 1.6149e-5 | |
| 14 | 2.7315e-6 | 1 | 3.4019e-6 | |
| 15 | 5.4631e-7 | 1 | 6.7039e-7 |
The Poisson distribution counts how many times something happens in a fixed interval when events are independent and occur at a steady average rate lambda, such as calls per hour or typos per page. The chance of exactly k events is lambda to the power k times e to the power minus lambda, divided by k factorial, and both the mean and the variance equal lambda. If a rate is 1.5 per hour and you watch for 2 hours, lambda is 3. Cumulative probabilities come from the incomplete gamma function, so they stay accurate for large rates and tiny tails. Real counts that cluster or depend on each other are more spread out than a Poisson model predicts. Everything is worked out in your browser.
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Math & MoneyAbout the Poisson Distribution Calculator
The Poisson distribution models the number of times something happens in a fixed interval of time or space when events occur independently at a steady average rate, called lambda. Typical examples are calls arriving at a help desk, typos on a page or defects in a roll of fabric. The chance of exactly k events is lambda to the power k, times e to the power minus lambda, divided by k factorial. Both the mean and the variance equal lambda, so the standard deviation is its square root. With lambda = 3, the chance of no events is 0.0498, of exactly 2 events is 0.2240, and of 2 or fewer is 0.4232. The calculator computes exact counts from the logarithm of the mass function, so large rates do not overflow, and cumulative probabilities from the regularized incomplete gamma function, since the chance of k or fewer events equals the upper incomplete gamma Q of k plus 1 and lambda. This keeps tiny tail probabilities accurate. The chance of at least one event is 1 minus e to the power minus lambda. The Poisson model fits when events do not cluster and do not influence each other. If real counts vary more than the mean, a different model is needed.
When should I use the Poisson distribution?▾
Use it for counts of independent events in a fixed interval when the average rate is steady, such as arrivals per hour or errors per page. If the number of possible events has a clear upper limit, such as successes in a fixed number of trials, the binomial distribution is usually better.
What is lambda in the Poisson distribution?▾
Lambda is the average number of events in your interval. It is also the variance. If events happen at a rate of 1.5 per hour and you watch for 2 hours, lambda is 3.
How do I find the chance of at least one event?▾
Subtract the chance of none from 1. The chance of none is e to the power minus lambda, so at least one has a probability of 1 minus e to the power minus lambda. With lambda = 3 that is 0.9502.
What is the difference between Poisson and binomial?▾
A binomial count has a fixed number of trials with a set chance of success each. A Poisson count has no fixed maximum and is described by an average rate. The Poisson is the limit of the binomial for many trials with a small chance each.
Is my data uploaded?▾
No. The calculation runs in your browser and nothing you enter is sent anywhere.