Binomial distribution

Definition

Let $n \ge 0$ be an integer and let $p \in [0, 1]$. The binomial distribution with parameters $n$ and $p$ is the distribution on $\{0, 1, \dots, n\}$ with the probability mass function $$p(k) = \binom{n}{k} p^k (1 - p)^{n-k} \quad \text{for } k = 0, 1, \dots, n.$$ Here $\binom{n}{k}$ is the binomial coefficient. One writes $X \sim \operatorname{B}(n, p)$ or $X \sim \operatorname{Bin}(n, p)$.

Remarks

The values add up to $1$ by the binomial theorem: $\sum_{k} \binom{n}{k} p^k (1 - p)^{n-k} = (p + 1 - p)^n = 1$. The number of successes in $n$ independent Bernoulli trials with success probability $p$ has this distribution. The factor $\binom{n}{k}$ counts the positions of the $k$ successes. The mean is $np$ and the variance is $np(1 - p)$. The probability of exactly $2$ heads in $4$ tosses of a fair coin is $\binom{4}{2} / 2^4 = 6/16$.

Explanations

L1 human

Repeat a yes-or-no experiment $n$ times, independently, and count the yes answers. The count has the binomial distribution. Each sequence with $k$ yes answers has probability $p^k (1 - p)^{n-k}$, and there are $\binom{n}{k}$ such sequences.