In probability theory and statistics, the **binomial distribution** is the discrete probability distribution which gives only two possible results in an experiment, either **Success or Failure**. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.

There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙.

Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.

**Also, read:**

## Binomial Probability Distribution

In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the boolean-valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a **Bernoulli process**. For n = 1, i.e. a single experiment, the binomial distribution is a **Bernoulli distribution**. The binomial distribution is the base for the famous binomial test of statistical importance.

### Negative Binomial Distribution

In probability theory and statistics, the number of successes in a series of independent and identically distributed Bernoulli trials before a particularised number of failures happens. It is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e., r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.

### Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In real life, the concept is used for:

- Finding the quantity of raw and used materials while making a product.
- Taking a survey of positive and negative reviews from the public for any specific product or place.
- By using the YES/ NO survey, we can check whether the number of persons views the particular channel.
- To find the number of male and female employees in an organisation.
- The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.

## Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by;

P(x:n,p) = ^{n}C_{x} p^{x} (1-p)^{n-x}
Or P(x:n,p) = |

Where,

n = the number of experiments

x = 0, 1, 2, 3, 4, …

p = Probability of Success in a single experiment

q = Probability of Failure in a single experiment = 1 – p

The binomial distribution formula can also be written in the form of n-Bernoulli trials, where ^{n}C_{x} = n!/x!(n-x)!. Hence,

**P(x:n,p) = n!/[x!(n-x)!].p ^{x}.(q)^{n-x}**

## Properties of Binomial Distribution

The properties of the binomial distribution are:

- There are two possible outcomes: true or false, success or failure, yes or no.
- There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
- The probability of success or failure varies for each trial.
- Only the number of success is calculated out of n independent trials.
- Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.

## Binomial Distribution Examples And Solutions

**Example 1: If a coin is tossed 5 times, find the probability of:**

**(a) Exactly 2 heads**

**(b) At least 4 heads.**

**Solution:**

**(a) **The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

x=2

P(x=2) = ^{5}C2 p^{2} q^{5-2 }= 5! / 2! 3! × (½)^{2}× (½)^{3}

P(x=2) = 5/16

**(b) **For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

Hence,

P(x = 4) = ^{5}C4 p^{4} q^{5-4} = 5!/4! 1! × (½)^{4}× (½)^{1} = 5/32

P(x = 5) = ^{5}C5 p^{5} q^{5-5} = (½)^{5} = 1/32

Therefore,

P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

**Example 2: For the same question given above, find the probability of:**

a) **Getting at least 2 heads**

Solution: P (at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)

P(X = 0) = (½)^{5} = 1/32

P(X=1) = ^{5}C_{1} (½)^{5.}= 5/32

Therefore,

P(X ≤ 2) = 1/32 + 5/32 = 3/16

Probability is a wide and very important topic for class 11 and class 12 students. By capturing the concepts here at BYJU’S, students can excel in the exams.