# Poisson Distribution Calculator

URL: https://speedycalculator.com/calculators/statistics/poisson-probability/
Last updated: 2026-10-25

> The Poisson calculator gives the probability of exactly k events in an interval when events occur independently at an average rate λ, such as calls per hour or defects per batch.

## Inputs

- Average rate (λ)
- Number of events (k)

## Results

- P(X = k)
- P(X = 0)
- Standard deviation

## Formula

```
P(X = k) = e^(−λ) λᵏ ÷ k!
```

## Explanation

The Poisson distribution models counts of rare, independent events over time or space.

### How to use the Poisson Distribution Calculator

- **Average rate (λ)**
- **Number of events (k)**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Raise λ to the power k, multiply by e^(−λ) and divide by k factorial.

### Inputs explained

FieldWhat to enterAverage rate (λ)A numberNumber of events (k)A number

### What each result means

- **P(X = k)** = exp(-[Average rate (λ)])×[Average rate (λ)]^[Number of events (k)]÷fact([Number of events (k)])
- **P(X = 0)** = exp(-[Average rate (λ)])
- **Standard deviation** = sqrt([Average rate (λ)])

### Example values

How the main result changes when **Average rate** changes and everything else stays at the example values above:

Average rate (λ)P(X = k)20.270730.22440.146550.084260.044680.0107

## Worked example

Inputs — Average rate (λ): 4; Number of events (k): 2.

Results — **P(X = k) = 0.146525**; P(X = 0) = 0.018316; Standard deviation = 2.

## FAQ

**What is special about Poisson?**
Its mean and variance are both λ.

**When does it approximate the binomial?**
When n is large and p is small, with λ = np.

**What formula does the Poisson Distribution Calculator use?**
P(X = k) = e^(−λ) λᵏ ÷ k!.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)