# Binomial Distribution

#### What is the Binomial Distribution?

The binomial distribution is a distribution function for discrete processes, where each independently generated value has a fixed probability.

Mathematically, the binomial distribution is the discrete probability distribution of success events in a sequence of n independent experiments, each with two possible outcomes (success /failure) and with probability p of success (and probability 1-p of failure). This probability distribution is denoted as B(n,p), and the probability of getting k successes is given by the following probability density function:

Where k = 0, 1, … n, and

The coefficient is called the binomial coefficient and gives its name to the distribution, with the prefix “bi” meaning two or twice. Each single success/failure experiment is called a Bernoulli trial or Bernoulli experiment, and the sequence of n experiments is named a Bernoulli process. Thus, when n equals 1, the distribution is called Bernoulli distribution.

Historically, French mathematician Blaise Pascal first derived it for p = 0.5, and later the Swiss mathematician Jacob Bernoulli obtained the general expression.

#### Why is the Binomial Distribution Important?

The binomial distribution is one of the fundamental distributions in probability theory and statistics. It can be used in experiments or surveys that are repeated several times and that either have a characteristic or don’t. For example, whether an individual will die or won’t within a period of time or whether party A will win the presidential election or won’t.

In statistics, it is the basis of the binomial test of statistical significance and used to model the extraction with replacement of a sample of size n from a population of size N. It is also used as an approximation for extraction without replacement when n is very small when compared to N.

#### Binomial Distribution + LogicPlum

Although easy to understand, applying the binomial distribution correctly requires a sound knowledge of its mathematical foundations. LogicPlum’s platform eliminates this need from its users by automating the calculation process. In addition, the company has a team of dedicated data analysts, who constantly update the platform with the latest technologies available. As a result, users can focus on using the model and interpreting the results obtained, knowing that they are working with the most efficient tool for the problem under study.

##### Guide to Further Reading

NIST/SEMATECH. *e-Handbook of Statistical Methods.* Available at https://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm