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Binomial Distribution

A discrete distribution counting the number of successes in a fixed number of independent trials, each with the same success probability, the direct sum of independent Bernoulli trials.

Discrete Probability DistributionsBeginnerFree

In Plain English

The Binomial distribution answers the question: if you flip a coin (or run any fixed number of identical, independent yes/no trials) a fixed number of times, how many total successes should you expect, and how likely is each possible count? It's one of the most widely used discrete distributions in all of statistics, underlying everything from quality-control defect counts to clinical trial success rates to basic hypothesis tests about proportions.

Definition

A discrete random variable $X$ follows a Binomial distribution with $n$ independent trials, each with success probability $p$ , if it counts the total number of successes, $P (X = k) = (k n) p^{k} (1 - p)^{n - k}$ for $k = 0, 1, \dots, n$ , with mean $n p$ and variance $n p (1 - p)$ . It is the sum of $n$ independent, identically distributed Bernoulli( $p$ ) random variables, and for large $n$ it is well approximated by a Normal distribution with the same mean and variance (the De Moivre-Laplace theorem, an early special case of the Central Limit Theorem).

Formula

P (X = k) = (k n) p^{k} (1 - p)^{n - k}

Notation

n

—Number of independent trials

p

—Success probability per trial

Worked Example

If a factory's defect rate is $p = 0.05$ and 20 items are inspected ( $n = 20$ ), the number of defective items follows Binomial( $n = 20, p = 0.05$ ), with expected value $n p = 1$ defect and variance $n p (1 - p) = 0.95$ ; the probability of finding exactly zero defects is $(0.95)^{20} \approx 0.358$ .

Common Misconceptions

The Binomial distribution requires the trials to be independent with a constant success probability throughout, sampling without replacement from a finite population (where the probability shifts after each draw) instead follows the Hypergeometric distribution, not the Binomial.
As n grows large, the Normal approximation to the Binomial is a convenient approximation, not an exact equivalence, for small n or p near 0 or 1, the Normal approximation can be noticeably inaccurate and an exact Binomial calculation (or a continuity correction) is preferred.

Properties

The Binomial distribution assumes independent trials with a constant success probability, when sampling without replacement from a finite population causes that probability to shift between draws, the Hypergeometric distribution is the exactly correct model instead, though the Binomial remains a good approximation when the population is much larger than the sample.
For large $n$ , the Binomial distribution is well approximated by a Normal distribution with the same mean and variance, a special case of the Central Limit Theorem historically proven first (by De Moivre and Laplace) for exactly this distribution, well before the general theorem was established.
The Binomial distribution underlies the exact binomial test and the Wald and other approximate confidence interval methods for a single proportion, making it one of the most practically important discrete distributions in applied statistics.

At a Glance

Support{0, 1, ..., n}

Meann * p

Variancen * p * (1-p)

Constructed asthe sum of n independent Bernoulli(p) trials

Used in These Tests

Exact Binomial Test

Last updated August 9, 2026← Back to the Dictionary