Poisson Distribution Calculator
Modeling Events That Arrive at a Steady Average Rate
Some things don't have a natural "number of trials" — calls arriving at a support line, customers entering a store, typos on a page. What they do have is an average rate over a given interval, and that's all the Poisson distribution needs to work out the probability of seeing exactly, at most, or at least a given count in that same interval.
The Formula
λ (lambda) is the average number of occurrences in the interval, and k is the specific count you're evaluating. Both the mean and the variance of a Poisson distribution equal λ.
Worked Example: Call Center Volume
A support line averages 4 calls per hour. The probability of receiving exactly k calls in the next hour:
| Calls (k) | Probability |
|---|---|
| 0 | 1.83% |
| 1 | 7.33% |
| 2 | 14.65% |
| 3 | 19.54% |
| 4 | 19.54% |
| 5 | 15.63% |
| 6 | 10.42% |
The probability of 2 or fewer calls in the hour is 23.81%, while the probability of 6 or more is 21.49% — useful for staffing decisions even though 4 is the "expected" figure.
Where This Matters
- Staffing and operations — call centers, emergency rooms, and service counters use Poisson models to plan for demand variability, not just the average.
- Reliability engineering — modeling the number of equipment failures expected in a given time period.
- Epidemiology — modeling rare disease case counts across a population and time period.
- Network and queueing theory — packet arrivals, server requests, and similar arrival processes are classic Poisson applications.
How to Use This Calculator
- Choose Exactly, At Most, or At Least K Occurrences from the dropdown.
- Enter the average rate (λ) for the interval you're analyzing.
- Enter the number of occurrences (k) you're interested in.
- Select Calculate to get the probability.
Related Calculations
For the time between Poisson-distributed events rather than the count itself, see the Exponential Distribution Calculator. For fixed-trial success counts instead of rate-based events, use the Binomial Distribution Calculator.