Binomial distribution

The story

Flip the same biased coin $n$ times and count the heads. Each flip is a $\text{Bernoulli}(p)$ trial, the flips do not influence one another, and the total can be anything from 0 to $n$ . The probability of exactly $k$ successes is $\binom{n}{k} p^k (1 - p)^{n-k}$ : the chance of one particular arrangement, multiplied by the number of ways to arrange $k$ successes among $n$ slots.

On average you get $np$ successes, with variance $np(1 - p)$ . When $p = 0.5$ the distribution is perfectly symmetric; push $p$ towards 0 or 1 and it leans against the nearest wall. As $n$ grows the bars settle into a familiar bell shape, which is why the Normal approximation works so well for large $n$ .

Formulas

Support

k \in \{0, 1, \ldots, n\}

PMF

P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}

CDF

F(k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1 - p)^{n-i}

Expectation

\mathbb{E}[X] = np

Variance

\operatorname{Var}(X) = np(1 - p)

Mode

\lfloor (n + 1)p \rfloor

Parameters

Parameter	What it controls
$n$	The number of independent trials; $X$ can take any whole value from 0 to $n$ .
$p$	The probability of success on each trial. It sets the centre ( $np$ ) and, together with $n$ , the spread $np(1 - p)$ .

Where it shows up

The number of heads in 20 coin tosses.
How many of 50 surveyed people say they will vote for a given party.
The number of defective items in a batch of 30, if each is faulty independently with probability $p$ .
How many of $n$ emails in an A/B test are opened.

Try this

Set $n = 50$ and $p = 0.5$ , then slowly lower $p$ to 0.05. Watch the symmetric bell become a right-skewed pile squashed against zero.
Compare $n = 50$ , $p = 0.06$ with the Poisson page at $\lambda = 3$ : the two sets of bars are almost identical, because $np = 3$ in both.
With $n = 50$ and $p = 0.5$ , shade the interval from 22 to 28 (roughly the mean $\pm 1\sigma$ , since $\sigma \approx 3.54$ ), then press Draw 1000 and check that about that share of the histogram falls inside it.

Related distributions

A Binomial with $n = 1$ is the Bernoulli, and $\text{Binomial}(n, p)$ is the sum of $n$ independent $\text{Bernoulli}(p)$ variables.
The Poisson is the limit of the Binomial as $n \to \infty$ and $p \to 0$ with $np = \lambda$ held fixed.
For large $n$ (a common rule of thumb is $np$ and $n(1 - p)$ both at least 10) it is well approximated by a Normal with mean $np$ and variance $np(1 - p)$ .