# Binomial distribution

The binomial distribution counts successes in n independent trials, each with the same success chance p. P(X = k) = nCk x p^k x (1-p)^(n-k). The mean is np and the variance is np(1-p).

Statistics (university) · Part 03: Random variables and discrete distributions · Card 5 of 8 · Last updated 2026-10-19

## Analogy

Shooting five free throws with a fixed 60 percent chance each, and counting how many go in.

## Example

**Exactly 3 of 5**

n = 5, p = 0.6
P(X=3) = 10 x 0.6^3 x 0.4^2
= 10 x 0.216 x 0.16 = 0.3456
Mean = 5 x 0.6 = 3
Variance = 5 x 0.6 x 0.4 = 1.2

## In practice

Used for yes/no counts: defective items in a batch, voters who say yes, patients who recover.

## Practice

### Flashcards

- **Mean and variance of a binomial?** Mean np, variance np(1-p).

### Interview questions

**A shooter scores 60% of free throws. Find the chance of exactly 3 scores in 5 shots, and state the binomial conditions you assumed.**

- n = 5, p = 0.6
- P = 10 x 0.216 x 0.16 = 0.3456
- Fixed number of trials
- Independent trials, same p

## Questions people ask

### What conditions must hold for a binomial model?

There is a fixed number of trials n. Each trial has two outcomes, success or failure. The success chance p is the same every time, and trials are independent. Drawing cards without replacement breaks independence, though the model works well for a small sample from a huge population.

### Where does the nCk come from in the binomial formula?

It counts the different orders in which k successes can occur among n trials. Each particular order has probability p^k x (1-p)^(n-k). Multiplying by the number of orders gives the total. With 3 successes in 5 trials there are 10 orders.

## Related concepts

- [Poisson distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-6)
- [Geometric distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-7)
- [Cumulative distribution function](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-8)

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