The Binomial Distribution

Many random experiments consist of $n$ identical, independent trials, where each trial results in either a success or a failure. The probability of success $p$ is the same for every trial. Examples include flipping a coin $n$ times, rolling a die $n$ times and counting sixes, or inspecting $n$ parts for defects.

If $X$ counts the number of successes in $n$ such trials, then $X$ follows a binomial distribution with parameters $n$ and $p,$ written

$$X \sim B(n, p).$$

Its probability mass function is

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

The binomial coefficient $\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$ counts the number of ways to choose which $k$ of the $n$ trials are the successes. The factor $p^k(1-p)^{n-k}$ is the probability of any one specific sequence with $k$ successes and $n-k$ failures.

For example, suppose a fair coin is flipped $3$ times and $X$ is the number of heads. Then $X \sim B(3, 0.5),$ and

$$P(X = 2) = \binom{3}{2}(0.5)^2(0.5)^1 = 3 \cdot \tfrac{1}{4} \cdot \tfrac{1}{2} = \tfrac{3}{8}.$$