The **binomial distribution** is a discrete probability distribution which describes the number of successes in a sequence of *n* independent experiments, each of which yielding success with probability *p*. Such a success/failure experiment is also called a Bernoulli experiment.

A typical example is the following: 5% of the population are HIV-positive. You pick 500 people randomly. How likely is it that you get 30 or more HIV-positives?
The number of HIV-positives you pick is a random variable *X* which follows a binomial distribution with *n* = 500 and *p* = .05. We are interested in the probability Pr[*X* ≥ 30].

In general, if the random variable *X* follows the binomial distribution with parameters *n* and *p*, we write *X* ~ B(*n*, *p*). The probability of getting exactly *k* successes is given by

- Pr[
*X*=*k*] = C(*n*,*k*)*p*^{k}(1-*p*)^{n-k}for*k*= 0, 1, 2, ...,*n*

If *X* ~ B(*n*, *p*), then the expected value of *X* is

- E[
*X*] =*np*

- Var(
*X*) =*np*(1-*p*).

If *X* ~ B(*n*, *p*) and *Y* ~ B(*m*, *p*) are independent binomial variables, then *X* + *Y* is again a binomial variable; its distribution is B(*n*+*m*, *p*).

Two other important distributions arise as approximations of binomial distributions:

- If both
*np*and*n*(1-*p*) are greater than 5 or so, then an excellent approximation to B(*n*,*p*) is given by the normal distribution N(*np*,*np*(1-*p*)). This approximation is a huge time saver; historically, it was the first use of the normal distribution. Nowadays, it can be seen as a consequence of the central limit theorem since B(*n*,*p*) is a sum of*n*independent, identically distributed 0-1 indicator variables.- For example, suppose you randomly sample
*n*people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If you sampled groups of*n*people repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion*p*of agreement in the population and with standard deviation σ = (*p*(1 -*p*)/*n*)^{1/2}. Large sample sizes*n*are good because the standard deviation gets smaller, which allows a more precise estimate of the unknown parameter*p*.

- For example, suppose you randomly sample
- If
*n*is large and*p*is small, so that*np*is of moderate size, then the Poisson distribution with parameter λ =*np*is a good approximation to B(*n*,*p*).

The formula for Bézier curves was inspired by the binomial distribution.