Bernoulli Distribution Calculator

A Bernoulli distribution models a single experiment (trial) that has exactly two possible outcomes: success (X=1) or failure (X=0). It is defined entirely by a single parameter p, the probability of success. It is the simplest discrete probability distribution and forms the building block of the binomial distribution. See also our use the Discrete Uniform Distribution Calculator.

The probability of success (p) is the likelihood that a single Bernoulli trial results in a success. It must be a value between 0 and 1, where 0 means the event never occurs and 1 means it always occurs. For example, a fair coin toss has p = 0.5.

The Bernoulli distribution is a special case of the binomial distribution where the number of trials n = 1. The binomial distribution generalizes Bernoulli to n independent trials, each with the same probability of success p, and counts the total number of successes across those trials.

For a Bernoulli random variable X with success probability p: P(X=1) = p and P(X=0) = 1 − p. These two probabilities always sum to 1. The mean is μ = p, the variance is σ² = p(1−p), and the standard deviation is σ = √(p(1−p)).

The mean (expected value) of a Bernoulli distribution is simply μ = p. This represents the average outcome you would expect if the trial were repeated many times. For instance, with p = 0.4, the long-run average value of X is 0.4.

The standard deviation of a Bernoulli distribution is σ = √(p × (1 − p)). It reaches its maximum value of 0.5 when p = 0.5 (maximum uncertainty) and approaches 0 as p approaches 0 or 1 (near-certain outcomes).

Any experiment with a binary outcome qualifies as a Bernoulli trial. Common examples include: flipping a coin (heads or tails), a free-throw attempt in basketball (make or miss), a quality control check (defective or not defective), and a medical test result (positive or negative).

Technically yes, but these are degenerate cases. If p = 0, the outcome is always failure (X always equals 0). If p = 1, the outcome is always success (X always equals 1). In both cases, the standard deviation is 0, meaning there is no variability in the outcome.