*Binomial Distribution Calculator* is an online statistics tool for probability data analysis programmed to find out the discrete probability distribution of the number of successes in a sequence of n independent experiments in which there are only two possible outcomes success and failure

### Definition

A random variable follows a **binomial distribution** when each trial has exactly two possible outcomes. Binomial Distribution is defined as the number of success can be expected to follow a particular sample. A probability distribution indicates the distribution of expected values. A histogram is used to represent the distribution of values that actually occur in a given sample. The binomial distribution is the origin for the popular binomial test of statistical significance. The success failure experiment also called as Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The Binomial distribution is n times repeated Bernoulli trial. In common a Binomial distribution arises when we have the following 4 conditions

1. n identical trials

2. Two possible outcomes for each trial are success and failure

3. Trials are independent, e.g. each coin toss doesn’t affect the others

4. P(success) = p is the same for each trial

### The Formula for Binomial Distribution

If all the above four conditions are set then the Binomial Distribution can be calculated from the formula

n and p are called the parameters of the distribution. We say X follows a binomial distribution with parameters n and p

X is a binomially distributed random variable, then the expected value of the **Mean** can be derived from the formula

Mean **μ = np**

The **Variance** can be derived from the formula

**Variance of X = np(1 - p)**

The **Standard Deviation** can be derived from the formula

Standard Deviation **σ = Root of (npq)**

where q = 1 - p

The **Skewness** can be derived from the formula

### Shapes of Binomial distributions

The skewness of a Binomial distribution will also depend upon the values of n and p, In general

if p < 0.5 the distribution will exhibit Positive Skew

if p = 0.5 the distribution will be Symmertric

if p > 0.5 the distribution will exhibit Negative Skew

The above illustrations can guide you to understand how to find out the Binomial Distribution in probability and statistics. In practice to make your calculations easy, this online Binomial Distribution calculator allows you to find out the discrete probability distribution of the number of successes in a sequence of n independent experiments in which there are only two possible outcomes success and failure