Binomial distributions are used to represent situations can that can be thought as the result of \(n\) Bernoulli experiments (here the \(n\) is defined as the size of the experiment). The classical example is \(n\) independent coin flips, where each coin flip has probability p of success. In this case, the individual probability of flipping heads or tails is given by the Bernoulli(p) distribution, and the probability of having \(x\) equal results (\(x\) heads, for example), in \(n\) trials is given by the Binomial(n, p) distribution. The equation of the Binomial distribution is directly derived from the equation of the Bernoulli distribution.

Binomial(size, p = 0.5)



The number of trials. Must be an integer greater than or equal to one. When size = 1L, the Binomial distribution reduces to the bernoulli distribution. Often called n in textbooks.


The success probability for a given trial. p can be any value in [0, 1], and defaults to 0.5.


A Binomial object.


The Binomial distribution comes up when you are interested in the portion of people who do a thing. The Binomial distribution also comes up in the sign test, sometimes called the Binomial test (see stats::binom.test()), where you may need the Binomial C.D.F. to compute p-values.

We recommend reading this documentation on, where the math will render with additional detail.

In the following, let \(X\) be a Binomial random variable with parameter size = \(n\) and p = \(p\). Some textbooks define \(q = 1 - p\), or called \(\pi\) instead of \(p\).

Support: \(\{0, 1, 2, ..., n\}\)

Mean: \(np\)

Variance: \(np \cdot (1 - p) = np \cdot q\)

Probability mass function (p.m.f):

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

Cumulative distribution function (c.d.f):

$$ P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n \choose i} p^i (1 - p)^{n-i} $$

Moment generating function (m.g.f):

$$ E(e^{tX}) = (1 - p + p e^t)^n $$

See also


set.seed(27) X <- Binomial(10, 0.2) X
#> Binomial distribution (size = 10, p = 0.2)
random(X, 10)
#> [1] 5 0 3 1 1 2 0 0 1 1
pdf(X, 2L)
#> [1] 0.3019899
log_pdf(X, 2L)
#> [1] -1.197362
cdf(X, 4L)
#> [1] 0.9672065
quantile(X, 0.7)
#> [1] 3
cdf(X, quantile(X, 0.7))
#> [1] 0.8791261
quantile(X, cdf(X, 7))
#> [1] 7