Discrete probability distribution
Bernoulli distribution
X ~ Bernoulli(p)
A single trial with two outcomes — success (1) or failure (0). It is the simplest distribution in probability and the building block of the Binomial: a Binomial is just the count of successes across many independent Bernoulli trials.
Explore it interactivelyKey facts
- Notation
- X ~ Bernoulli(p)
- Type
- discrete
- Parameters
- p — success probability (0–1)
- Support
- {0, 1}
- PMF
- P(X = k) = pᵏ (1 − p)¹⁻ᵏ, k ∈ {0, 1}
- Mean
- p
- Variance
- p(1 − p)
When to use it
- A single coin flip
- One yes/no or pass/fail event
- Whether one visitor converts
- Whether a single item is defective
Related distributions
Frequently asked questions
What is the Bernoulli distribution?
A single trial with two outcomes — success (1) or failure (0). It is the simplest distribution in probability and the building block of the Binomial: a Binomial is just the count of successes across many independent Bernoulli trials.
What are the mean and variance of the Bernoulli distribution?
For X ~ Bernoulli(p), the mean is p and the variance is p(1 − p).
When is the Bernoulli distribution used?
It is commonly used for: A single coin flip; One yes/no or pass/fail event; Whether one visitor converts; Whether a single item is defective.