Poisson Distribution Calculator - P(X = k), Cumulative & Range Probabilities
Poisson Distribution Calculator
Probabilities for counts of random events
Every Probability at Once
Exactly k, fewer, at most, more and at least are all shown, plus any range a to b.
Precise Tails
Tail probabilities come from the incomplete gamma function, so very small values such as 1e-12 stay accurate.
Visual Distribution
The chart highlights the outcomes counted in your chosen probability; point at a bar for its value.
Private
Calculated in your browser; nothing is sent.
When to Use the Poisson Distribution
The Poisson distribution models how many times an event happens in a fixed interval of time or space when events occur independently at a constant average rate λ: calls to a help desk per hour, typos per page, or meteors per night. The probability of exactly k events is P(X = k) = λke−λ / k!, and both the mean and the variance equal λ.
Example: a shop gets on average 4 customers every 10 minutes. The chance of exactly 6 in the next 10 minutes is P(X = 6) ≈ 0.1042, and the chance of at most 6 is about 0.8893. If the variance in your real data is much larger than the mean, the events are probably clustered and a negative binomial model may fit better.
Key Takeaways
- One parameter: Only the average rate λ is needed; the variance equals the mean.
- Inclusive or not: Mind the difference between "at most k" (≤) and "fewer than k" (<).
- Large λ: Works for large rates too, where the shape approaches a normal curve.