Definition
The binomial distribution describes the number of successes X in n independent trials, where each trial has the same probability p of success. The probability of exactly k successes is P(X = k) = C(n, k) × p^k × (1 - p)^(n-k), where C(n, k) is the binomial coefficient n choose k.
Its mean is np, its variance is np(1 - p). The Bernoulli distribution is the special case n = 1.
Why it matters
How it works
To use the binomial, identify n (the number of trials) and p (the per-trial success probability). Then any probability question — P(X = k), P(X ≥ k), expected number of successes — can be computed from the formula or read off a binomial calculator.
A classic example: 10 flips of a fair coin produce a binomial(10, 0.5) random variable. The probability of exactly 7 heads is C(10, 7) × 0.5^7 × 0.5^3 ≈ 0.117. The expected number of heads is 5, and the standard deviation is about 1.58. This kind of calculation generalises to any independent-trials experiment.