# Poisson Probability Calculator

> Find Poisson probabilities for k events when the average count is λ: exactly k, at most k, fewer than, at least and more than k, with mean and SD.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/poisson-probability-calculator

The Poisson distribution models how many times an event happens in a fixed interval when events occur independently at a steady average rate λ, such as calls per hour or defects per batch.

Also called: Poisson distribution calculator, Poisson CDF calculator.

## Inputs
- Average number of events (λ) (default 2)
- Number of events (k) (default 3)

## Results
- P(X = k)
- P(X = k), full precision
- P(X ≤ k), at most k
- P(X < k), fewer than k
- P(X ≥ k), at least k
- P(X > k), more than k
- Mean (λ)
- Standard deviation (√λ)

## Formula

```
P(X = k) = e^(−λ)·λᵏ ÷ k!;  P(X ≤ k) = Q(k + 1, λ);  P(X ≥ k) = P(k, λ)
```

Where:
- λ: average number of events in the interval
- k: number of events
- P, Q: regularised lower and upper incomplete gamma functions (exact cumulative sums)

## Worked example
1. P(X = 3) = e⁻² × 2³ ÷ 3! = 0.135335 × 8 ÷ 6 = 0.180447.
2. P(X ≤ 3) = e⁻² × (1 + 2 + 2 + 4/3) = 0.857123.
3. P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − e⁻² × 5 = 0.323324.

**Result:** If a help desk averages 2 calls an hour, the chance of exactly 3 calls in an hour is 18.0% and of 3 or more is 32.3%.

## What the result means

Use “at least” for capacity questions (how likely is a busy hour) and “at most” for service levels (how often can we handle everything). Variance equals the mean in a Poisson process; if your real counts vary much more than that, the data are over-dispersed and a negative binomial model fits better.

## Assumptions and limits
- Events occur independently, one at a time, at a constant average rate over the interval.
- Rates that change through the day (rush hours, seasonality) break the constant-rate assumption; split the interval instead.

## Questions

### How do I change the interval?

Scale λ. If you average 2 calls per hour, use λ = 1 for half an hour and λ = 16 for an 8-hour shift.

### How does Poisson relate to the binomial?

For many trials with a small success probability, binomial(n, p) is close to Poisson with λ = np.

## Related calculators
- [Binomial Probability Calculator](https://calculators.geekmidst.com/calculators/binomial-probability-calculator)
- [Exponential Distribution Calculator](https://calculators.geekmidst.com/calculators/exponential-distribution-probability-calculator)
- [Probability Calculator](https://calculators.geekmidst.com/calculators/probability-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/poisson-probability-calculator