Binomial Distribution

The binomial distribution is one of the most important discrete probability distributions in mathematics and statistics. It models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes — success or failure — and the probability of success remains constant from trial to trial. Examples appear everywhere: counting how many free throws a basketball player makes out of 10 attempts, how many students pass an exam, or how many defective items appear in a batch from a factory. To use the binomial distribution, a situation must satisfy four specific conditions, collectively called the binomial setting: (1) there are a fixed number of trials n, (2) each trial results in only one of two outcomes (success or failure), (3) the probability of success p is the same on every trial, and (4) the trials are independent of each other. When all four conditions are met, the count X of successes follows a binomial distribution, written X ~ B(n, p). The binomial distribution connects combinatorics (counting arrangements) with probability theory. The famous binomial coefficient, which counts the number of ways to choose k successes from n trials, sits at the heart of its formula. Understanding this distribution equips you to model and analyse a huge variety of real-world random processes, and it serves as a foundation for more advanced topics such as the normal approximation, hypothesis testing, and inferential statistics.

Key takeaways

Identify the Binomial Setting
Define the Random Variable
Understand the Binomial Coefficient
Compute a Single Probability
Use the Mean and Standard Deviation

Identify the Binomial Setting

Check all four BINS conditions: fixed number of trials (n), independent trials, only two outcomes per trial (success/failure), and the same probability of success (p) each time. If any condition fails — for example, if trials affect each other or p changes — the binomial model does not apply.

Define the Random Variable

Let X be the count of successes in n trials. X can take integer values 0, 1, 2, …, n. We write X ~ B(n, p) to declare that X follows a binomial distribution with parameters n (number of trials) and p (probability of success on each trial).

Understand the Binomial Coefficient

The binomial coefficient C(n, k) = n! / (k!(n−k)!) counts the number of distinct sequences of n trials that contain exactly k successes. For example, with n = 3 trials and k = 2 successes, the possible sequences are {SSF, SFS, FSS}, giving C(3,2) = 3. The factorial n! = n × (n−1) × … × 2 × 1, with 0! = 1 by convention.

Compute a Single Probability

Each specific sequence with exactly k successes has probability p^k × (1−p)^(n−k): the probability of k successes multiplied by the probability of (n−k) failures. Multiplying by the number of such sequences gives the binomial probability formula: P(X = k) = C(n,k) × p^k × (1−p)^(n−k).

Use the Mean and Standard Deviation

The expected value (mean) of a binomial distribution is μ = np, and the variance is σ² = np(1−p), so the standard deviation is σ = √(np(1−p)). These formulas let you quickly describe the centre and spread of the distribution without computing every probability individually. For example, if n = 100 and p = 0.3, then μ = 30 and σ ≈ 4.58.

The Key Equation

P(X = k) = \binom{n}{k} p^{k}(1-p)^{n-k}, \quad k = 0, 1, 2, \ldots, n

n — Number of trials (dimensionless): The fixed total number of independent trials in the experiment.
k — Number of successes (dimensionless): The specific count of successes whose probability we are computing; k is an integer from 0 to n.
p — Probability of success (dimensionless): The constant probability of a success on any single trial; 0 ≤ p ≤ 1.
\binom{n}{k} — Binomial coefficient (dimensionless): Equals n! divided by k!(n−k)!; counts the number of ways to arrange k successes among n trials.
(1-p) — Probability of failure (dimensionless): The probability of a failure on a single trial, often denoted q.

Worked Example

Problem. A fair coin is tossed 8 times. Let X be the number of heads. (a) State the distribution of X. (b) Find P(X = 3). (c) Find the mean and standard deviation of X.

P(X=3)=\binom{8}{3}(0.5)^{8}=56\times0.00390625\approx0.2188

Answer. X ~ B(8, 0.5); P(X = 3) ≈ 0.2188; mean = 4; standard deviation ≈ 1.414

Figure

Diagram

Origins · 1 minJacob Bernoulli and how this was discovered

Jacob Bernoulli

1655–1705 · Swiss

Jacob Bernoulli (Jakob Bernoulli) was a pioneering Swiss mathematician who laid the groundwork for the binomial distribution in his landmark work Ars Conjectandi ('The Art of Conjecturing'), published posthumously in 1713. He formalised the study of repeated independent trials with two outcomes — now called Bernoulli trials in his honour — and proved the Law of Large Numbers, showing that the sample proportion of successes converges to the true probability p as the number of trials grows. Bernoulli was so fascinated by the logarithmic spiral that he requested one be carved on his gravestone with the inscription 'Eadem mutata resurgo' ('Though changed, I rise again the same') — though the stonemason accidentally carved an Archimedean spiral instead.

Geometric Distribution — The geometric distribution counts the number of Bernoulli trials needed to achieve the first success, using the same binomial setting conditions but asking a different question.
Combinations and Permutations — The binomial coefficient C(n, k) = n!/(k!(n−k)!) is the combinatorics formula at the heart of every binomial probability calculation.
Normal Approximation to the Binomial — When n is large and p is not too extreme, the binomial distribution can be approximated by a normal distribution with the same mean and variance, connecting discrete and continuous probability.
Expected Value — The mean of the binomial distribution, μ = np, is derived using the general definition of expected value for discrete random variables.