Binomial Distribution

Many real situations consist of a fixed number of independent, identical trials, each of which ends in only one of two outcomes — usually called "success" and "failure" — with the same probability of success on every trial. Such a single trial is called a Bernoulli trial. Examples: tossing a coin a fixed number of times (success = head); inspecting a fixed sample of items from a production line (success = defective item); a fixed number of shots at a target (success = hit).

A discrete random variable $X$ = number of successes in $n$ independent Bernoulli trials, each with probability of success $p$ (and probability of failure $q = 1 - p$), is said to follow a Binomial distribution, written $X \sim B(n, p)$, provided:

1. The number of trials $n$ is fixed in advance.
2. Each trial is independent of the others.
3. Each trial results in exactly two possible outcomes (success/failure).
4. The probability of success $p$ is constant across all trials.

Under these conditions, the probability of getting exactly $r$ successes in the $n$ trials is given by the binomial probability mass function:

where $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$ is the number of ways of choosing which $r$ of the $n$ trials are the successes.

Mean and variance of the Binomial distribution (stated results, derived from $E(X) = \sum r\,p(r)$):

$E(X) = np, \qquad Var(X) = npq, \qquad SD(X) = \sqrt{npq}$ …