# Binomial Probability Calculator

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

> The binomial calculator finds the probability of exactly k successes in n independent trials with success probability p, plus the cumulative probabilities of at most and at least k.

## Inputs

- Number of trials (n)
- Successes (k)
- Probability of success (%)

## Results

- P(X = k)
- Mean (np)
- Standard deviation

## Formula

```
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
```

## Explanation

The binomial distribution models yes/no outcomes such as coin flips, defective parts or survey responses.

### How to use the Binomial Probability Calculator

- **Number of trials (n)**
- **Successes (k)**
- **Probability of success**

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

### How it works

Multiply the number of ways to choose k successes by the probability of each arrangement.

### Inputs explained

FieldWhat to enterNumber of trials (n)A numberSuccesses (k)A numberProbability of successIn %

### What each result means

- **P(X = k)** = comb([Number of trials (n)], [Successes (k)])×[p as decimal]^[Successes (k)]×(1 - [p as decimal])^([Number of trials (n)] - [Successes (k)])
- **Mean (np)** = [Number of trials (n)]×[p as decimal]
- **Standard deviation** = sqrt([Number of trials (n)]×[p as decimal]×(1 - [p as decimal]))

### Example values

How the main result changes when **Number of trials** changes and everything else stays at the example values above:

Number of trials (n)P(X = k)50.31257.50.2486100.117212.50.0435150.0139200.0011

## Worked example

Inputs — Number of trials (n): 10; Successes (k): 3; Probability of success: 50 %.

Results — **P(X = k) = 0.117188**; Mean (np) = 5; Standard deviation = 1.5811.

## FAQ

**When can I use the binomial distribution?**
Fixed number of independent trials, each with the same success probability.

**What is P(3 heads in 10 flips)?**
About 0.117.

**What formula does the Binomial Probability Calculator use?**
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ.


## Sources

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