Discrete probability distribution
Binomial distribution
X ~ Binomial(n, p)
The number of successes in n independent Bernoulli(p) trials. As n grows large it is well approximated by the Normal distribution — the simplest illustration of the Central Limit Theorem.
Explore it interactivelyKey facts
- Notation
- X ~ Binomial(n, p)
- Type
- discrete
- Parameters
- n — number of independent trials; p — success probability per trial
- Support
- {0, 1, …, n}
- PMF
- P(X = k) = C(n, k) · pᵏ (1 − p)ⁿ⁻ᵏ
- Mean
- np
- Variance
- np(1 − p)
When to use it
- Defective items in a batch of n
- Correct answers on an n-question test
- Conversions out of n visitors
- Heads in n coin flips
Related distributions
Frequently asked questions
What is the Binomial distribution?
The number of successes in n independent Bernoulli(p) trials. As n grows large it is well approximated by the Normal distribution — the simplest illustration of the Central Limit Theorem.
What are the mean and variance of the Binomial distribution?
For X ~ Binomial(n, p), the mean is np and the variance is np(1 − p).
When is the Binomial distribution used?
It is commonly used for: Defective items in a batch of n; Correct answers on an n-question test; Conversions out of n visitors; Heads in n coin flips.