In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability ${\displaystyle q=1-p}$). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.[1]

Quick facts: Notation, Parameters, Support, PMF, CDF...
Notation Probability mass function Cumulative distribution function ${\displaystyle B(n,p)}$ ${\displaystyle n\in \{0,1,2,\ldots \}}$ – number of trials${\displaystyle p\in [0,1]}$ – success probability for each trial${\displaystyle q=1-p}$ ${\displaystyle k\in \{0,1,\ldots ,n\}}$ – number of successes ${\displaystyle {\binom {n}{k}}p^{k}q^{n-k}}$ ${\displaystyle I_{q}(n-k,1+k)}$ (the regularized incomplete beta function) ${\displaystyle np}$ ${\displaystyle \lfloor np\rfloor }$ or ${\displaystyle \lceil np\rceil }$ ${\displaystyle \lfloor (n+1)p\rfloor }$ or ${\displaystyle \lceil (n+1)p\rceil -1}$ ${\displaystyle npq}$ ${\displaystyle {\frac {q-p}{\sqrt {npq}}}}$ ${\displaystyle {\frac {1-6pq}{npq}}}$ ${\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)}$ in shannons. For nats, use the natural log in the log. ${\displaystyle (q+pe^{t})^{n}}$ ${\displaystyle (q+pe^{it})^{n}}$ ${\displaystyle G(z)=[q+pz]^{n}}$ ${\displaystyle g_{n}(p)={\frac {n}{pq}}}$(for fixed ${\displaystyle n}$)
Close

The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.