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 ). 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...**▼

Probability mass function | |||

Cumulative distribution function | |||

Notation | |||
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Parameters |
– number of trials – success probability for each trial | ||

Support | – number of successes | ||

PMF | |||

CDF | (the regularized incomplete beta function) | ||

Mean | |||

Median | or | ||

Mode | or | ||

Variance | |||

Skewness | |||

Ex. kurtosis | |||

Entropy |
in shannons. For nats, use the natural log in the log. | ||

MGF | |||

CF | |||

PGF | |||

Fisher information |
(for fixed ) |

Part of a series on statistics |

Probability theory |
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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.