Binomial Distribution

Successes in n independent trials: the PMF, probabilities, and the mean.

The idea

The binomial distribution describes the count of successes in a fixed number of independent yes-or-no trials: heads among $n$ coin flips, defectives among $n$ items, recoveries among $n$ patients. Suppose each trial succeeds with the same probability $p$, and let $X$ count the successes. We write $X \sim \text{Bin}(n, p)$.

To find $\mathbb{P}(X = k)$, first fix which $k$ trials succeed. Any one pattern of $k$ successes and $n - k$ failures has probability $p^{k}(1-p)^{n-k}$, because independent trials multiply and reordering the factors changes nothing. There are $\binom{n}{k}$ such patterns, so adding their probabilities gives $\mathbb{P}(X = k) = \binom{n}{k}\,p^{k}(1-p)^{n-k}, \qquad k = 0, 1, \ldots, n.$ The binomial coefficient counts the arrangements, and the powers give the probability of any one of them.

Plotted against $k$, these probabilities rise and then fall, peaking near the mean number of successes $np$ — symmetrically when $p = \tfrac{1}{2}$, skewed toward small $k$ when $p$ is small.

Ways to work on it

• Walkthrough. The PMF and why the binomial coefficient appears.
• Practice. Evaluate the binomial PMF at a value.
• Hardest. A cumulative probability via the complement, plus the mean.

Not sure where to start? Take the ten-question placement test.