# Binomial Probability Formula

The binomial distribution is a type of distribution in statistics that has two possible outcomes. For instance, if you toss a coin and there are only two possible outcomes: heads or tails. In the same way, taking a test could have two possible outcomes: pass or fail.

The binomial probability formula can calculate the probability of success for binomial distributions. The Binomial Probability distribution is an experiment that posses the following properties:

- There are fixed number of trials which is denoted by n
- All the trials are independent.
- The outcome of each trial can either be a “success” or “failure”.
- The probability of success remains constant and is denoted by p.

The Binomial Probability distribution of exactly x successes from n number of trials is given by the below formula-

\[\LARGE P(X) = C_{x}^{n} P^{x} q^{n-x}\]

n = Total number of trials

x = Total number of successful trials

p = probability of success in a single trial

q = probability of failure in a single trial = 1-p

**Binomial Probability : Few solved examples**

**Example 1:**A coin is flipped 6 times. What is the probability of getting exactly 2 tails.

**Solution:**

P(X) = $C_{x}^{n}$ $P^{x}$ $q^{n-x}$

P(x) = $\frac{6!}{2!\times 4!}$ $\times$ ($\frac{1}{2})^{3}$ $\times$ ($\frac{1}{2})^{6-3}$

= $\frac{6!}{2!\times 4!}$ $\times$ ($\frac{1}{2})^{3}$ $\times$ ($\frac{1}{2})^{3}$

= $\frac{15}{64}$

= 0.234