# Binomial Distribution in Probability

Binomial Distribution in Probability gives information about only two types of possible outcomes i.e. Success or Failure. Binomial Probability Distribution is a discrete probability distribution used for the events that give results in ‘Yes or No’ or ‘Success or Failure’. It’s particularly useful in scenarios where these outcomes are mutually exclusive, and the probability of success (usually denoted as “p”) and the probability of failure (usually denoted as “q”) is constant for each trial. This distribution helps calculate the probability of getting a specific number of successes in a fixed number of trials, making it valuable in fields such as statistics, economics, and quality control.

In this article, we will learn about binomial probability distributions, binomial distribution formulas, meaning, and properties of binomial distribution.

## What is Binomial Distribution in Probability?

Binomial Probability Distribution talks about the probability of success or failure of an outcome in a series of events. The Binomial Distribution in Probability maps the outcome obtained in the form of success or failure, yes or no, true or false, etc. Each trial done to obtain the outcome in success or failure is called Bernoulli Trial and the probability distribution for each Bernoulli Trial is called the Bernoulli Distribution. Let’s learn the definition and meaning of Binomial Distribution.

### Binomial Distribution Definition

Binomial Distribution for a Random Variable X = 0, 1, 2, …., n is defined as the probability distribution of two outcomes success or failure in a series of events. Binomial Distribution in statistics uses one of the two independent variables in each trial where the outcome of each trial is independent of the outcome of other trials.

### Binomial Distribution Meaning

The meaning of Binomial Distribution can be understood with the help of an example. Let us assume an event A. If we conduct a random experiment that gives us ‘A’ then we call it success ‘S’ and if not then we call it failure ‘F’. The probability of success is given as P(S) = p and the probability of failure is given as P(F) = 1. We know that in a random experiment either we will succeed in getting A or fail in getting A i.e. we have only two sets of probability which is either of failure or of success. Hence, the Binomial Distribution gives the idea of the likelihood of an event for each trial where there are possibilities of only resulting in the event or not resulting in the event.

## Binomial Distribution Formula

Binomial Distribution Formula which is used to calculate the probability, for random variable X = 0, 1, 2, 3,….,n is given as

P(X = r) = nCrprqn-r, r = 0, 1, 2, 3….

Where,

• p is success
• q is failure and q = 1 – p
• p, q > 0 such that p + q = 1.

## Binomial Distribution Calculation

Binomial Distribution in statistics is used to compute the probability of likelihood of an event using the above formula. To calculate the probability using binomial distribution we need to follow the following steps:

Step 1: Find the number of trials and assign it as ‘n’

Step 2: Find the probability of success in each trial and assign it as ‘p’

Step 3: Find the probability of failure and assign it as q where q = 1-p

Step 4: Find the random variable X = r for which we have to calculate the binomial distribution

Step 5: Calculate the probability of Binomial Distribution for X = r using the Binomial Distribution Formula.

The use of the above steps has been illustrated using an example below:

## Binomial Distribution Examples

Let’s say we toss a coin twice, and getting head is a success we have to calculate the probability of success and failure then, in this case, we will calculate the probability distribution as follows:

In each trial getting a head that is a success, its probability is given as p = 1/2

n = 2 as we throw a coin twice

r = 0 for no success, r = 1 for getting head once and r = 2 for getting head twice

Probability of failure q = 1 – p = 1 – 1/2 = 1/2.

P(Getting 1 head) = P(X = 1) = ncrprqn-r = 2c1(1/2)1(1/2)1 = 2 ⨯ 1/2 ⨯ 1/2 = 1/2

P(Getting 2 heads) = P(X = 2) = 2c2(1/2)2(1/2)0 = 1/4

P(Getting 0 heads) = P(X = 0) = 2c0(1/2)0(1/2)2 = 1/4

Random Variable X = r

P(X = r)

X = 0 (Getting 0 Head)

1/4

X = 1 (Getting 1 Head)

1/2

X = 2 (Getting 2 Head)

1/4

As of now, we know that Binomial Distribution is calculated for the Random Variables obtained in Bernoulli Trials. Hence, we should understand these terms.

## Bernoulli Trial

Bernoulli Trial is a trial that gives results of dichotomous nature i.e. result in yes or no, head or tail, even or odd. It means it gives two types of outcomes out of which one favors the event while the other doesn’t. A random experiment is called Bernoulli Trial if it satisfies the following conditions:

• Trials are finite in number
• Trials are independent of each other
• Each trial has only two possible outcomes
• The probability of success and failure in each trial is the same.

## Binomial Random Variable

A Binomial Random Variable can be defined by two possible outcomes such as “success” and a “failure”. For instance, consider rolling a fair six-sided die and recording the value of the face. The binomial distribution formula can be put into use to calculate the probability of success for binomial distributions. Often it states “plugin” the numbers to the formula and calculates the requisite values.

The binomial distribution is based on the following characteristics:

• The experiment contains n identical trials.
• Each trial results in one of the two outcomes either success or failure.
• The probability of success, denoted p, remains the same from trial to trial.
• All the n trials are independent.

For Example, consider the following instance

A fair coin is flipped 20 times; X represents the number of heads.

X is a binomial random variable with n = 20 which is the total number of trials and p = 1/2 is the probability of getting head in each trial. The value of X represents the number of trials in which you succeed in getting head.

## Binomial Distribution Table

The binomial distribution for a situation when getting 6 is a success on throwing two dies is discussed in this section. First of all, we see that it is a Bernoulli Trial as getting 6 is the only success, and getting any different is a failure. Now we can get six on both die in a trial or six on only one of the die in a trial and getting no six on both die. Hence, the random variable for which we have to find the probability takes the value X = r = 0, 1, 2. The Binomial Distribution Table for getting 6 as success is plotted below:

Random Variable X = r

P(X = r)

X = 0 (Getting no 6)

25/36

X = 1 (Getting one 6)

10/36

X = 2 (Getting two 6)

1/36

We see that sum of all the probabilities 25/36 + 10/36 + 1/36 = 1.

## Binomial Distribution Graph

The Binomial Distribution Graph is plotted for X and P(X). We will plot a Binomial Distribution Graph for tossing a coin twice where getting the head is a success. If we toss a coin twice, the possible outcomes are {HH, HT, TH, TT}. The binomial Distribution Table for this is given below:

X (Random Variable)

P(X)

X = 0 (Getting no head)

P(X = 0) = 1/4 = 0.25

X = 1 (Getting 1 head)

P(X = 1) = 2/4 = 1/2 = 0.5

X = 2 (Getting two heads)

P(X = 2) = 1/4 = 0.25

The Binomial Distribution Graph for the above table is given below: ## Measure of Central Tendency for Binomial Distribution

In this, we will learn the formulas for Mean, Variance, and Standard Deviation of Binomial Distribution.

### Binomial Distribution Mean

The Mean of Binomial Distribution is the measurement of average success that would be obtained in ‘n’ number of trials. The Mean of Binomial Distribution is also called Binomial Distribution Expectation. The formula for Binomial Distribution Expectation is given as:

μ = n.p

Where,

• μ is the Mean or Expectation,
• n is the total number of trials, and
• p is the probability of success in each trial.

Example: If we toss a coin 20 times and getting head is the success then what is the mean of success?

Total Number of Trials n = 20

Probability of getting head in each trial, p = 1/2 = 0.5

Mean = n.p = 20 ⨯ 0.5

It means on average we would head 10 times on tossing a coin 20 times.

### Binomial Distribution Variance

The variance of Binomial Distribution tells about the dispersion or spread of the distribution. It is given by the product of the number of trials, probability of success, and probability of failure. The formula for Variance is given as follows:

σ2 = n.p.q

Where

• σ2 is the variance,
• n is the total number of trials,
• p is the probability of success in each trial, and
• q is the probability of failure in each trial.

Example: If we toss a coin 20 times and getting head is the success then what is the variance of the distribution?

We have, n = 20

Probability of Success in each trial (p) = 0.5

Probability of Failure in each trial (q) = 0.5

Variance of the Binomial Distribution, σ = n.p.q = (20 ⨯ 0.5 ⨯ 0.5) = 5

### Binomial Distribution Standard Deviation

Standard Deviation of Binomial Distribution tells about the deviation of the data from the mean. Mathematically, Standard Deviation is the square root of the variance. The formula for the Standard Deviation of Binomial Distribution is given as

σ = √n.p.q

Where

• σ is the standard deviation,
• n is the total number of trials,
• p is the probability of success in each trial, and
• q is the probability of failure in each trial.

Example: If we toss a coin 20 times and getting head is the success then what is the standard deviation?

We have, n = 20

Probability of Success in each trial (p) = 0.5

Probability of Failure in each trial (q) = 0.5

Standard Deviation of the Binomial Distribution, σ = √n.p.q

⇒ σ = √(20 ⨯ 0.5 ⨯ 0.5)

⇒ σ = √5 = 2.23

## Binomial Distribution Properties

The properties of Binomial Distribution are mentioned below:

• There are only two possible outcomes: success or failure, yes or no, true or false.
• There is a finite number of trials given as ‘n’
• The probability of success and failure in each trial is the same
• Only Success is calculated out of all trials
• Each trial is independent of any other trial

## Binomial Distribution Applications

Binomial Distribution is used where we have only two possible outcomes. Let’s see some of the areas where Binomial Distribution can be used

• To find the number of male and female students in an institute
• To find the likeability of something in Yes or No
• To find defective or good products manufactured in a factor
• To find positive and negative reviews on a product
• Votes collected in the form of 0 or 1

## Negative Binomial Distribution

Consider a situation where getting 6 is the success on throwing a die. Now if we throw the die and not get 6 then it is a failure. Now we throw again and do not get 6. Let’s say we don’t get 6 for three successive attempts and six is obtained in the fourth attempt and onwards then the binomial distribution of the number of getting 6 is called the Negative Binomial Distribution.

### Negative Binomial Distribution Formula

The formula for Negative Binomial Distribution is given as

P(x) = n+r-1Cr-1prqn

Where

• n is the total number of trial,
• r is the number of trials in which we get the first success,
• p is the probability of success in each trial, and
• q is the probability of failure in each trial.

## Binomial Distribution vs Normal Distribution

Binomial Distribution differs from the Normal Distribution in the manner that Binomial Distribution is discrete in nature while Normal Distribution is continuous in nature. This implies that the number of events in the case of Binomial Distribution is finite while the number of events in Normal Distribution is infinite. The Binomial Distribution Curve resembles the Normal Distribution Curve only when the Binomial Distribution curve is plotted for a large sample size.

Related Resources,

## Solved Examples on Binomial Distribution

Example 1: A die is thrown 6 times and if getting an even number is a success what is the probability of getting (i) 4 Successes (ii) No success

Given: n = 6, p = 3/6 = 1/2, and

q = 1 – 1/2 = 1/2

P(X = r) = nCrprqn-r

(i) P(X = 4) = 6C4(1/2)4(1/2)2 = 15/64

(ii) P(X = 0) = 6C0(1/2)0(1/2)6 = 1/64

Example 2: A coin is tossed 4 times what is the probability of getting at least 2 heads?

Given: n = 4

Probability of getting head in each trial, p = 1/2 ⇒ q = 1 – 1/2 = 1/2

P(X = r) = 4Cr(1/2)r(1/2)4-r

⇒ P(X = r) = 4Cr(1/2)4 {Using the laws of Exaponents}

And we know, Probability of getting at least 2 heads = P(X ≥ 2)

⇒ Probability of getting at least 2 heads = P(X = 2) + P(X = 3) + P(X = 4)

⇒ Probability of getting at least 2 heads = 4C2(1/2)4 + 4C3(1/2)4 + 4C4(1/2)4

⇒ Probability of getting at least 2 heads = (4C2 + 4C3 + 4C4)(1/2)4

⇒ Probability of getting at least 2 heads = 11(1/2)4 = 11/16

Example 3: A pair of die is thrown 6 times and getting sum 5 is a success then what is the probability of getting (i) no success (ii) two success (iii) at most two success

Given: n = 6

5 can be obtained in 4 ways (1,4) (4,1) (2,3) (3,2)

Probability of getting the sum 5 in each trial, p = 4/36 = 1/9

Probability of not getting sum 5 = 1 – 1/9 = 8/9

(i) Probability of getting no success, P(X = 0) = 6C0(1/9)0(8/9)6 = (8/9)6

(ii) Probability of getting two success, P(X = 2) = 6C2(1/9)2(8/9)4 = 15(84/96)

(iii) Probability of getting at most two successes, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

⇒ P(X ≤ 2) = (8/9)6 + 6(85/96) + 15(84/96)

## Practice Problems on Binomial Distribution

Q1: A box has 5 red, 7 black and 8 white balls. If three balls are drawn one by one with replacement what is the probability that all i) all are white ii) all are red iii) all are black

Q2: What is the probability distribution of the number of tails when three coins are tossed together?

Q3: A die is thrown three times what is the probability distribution of getting sixes.

Q4: A coin is tossed 4 times then what is the probability distribution of getting head.

## FAQs on Binomial Distribution

### 1. What is Binomial Distribution in Maths?

Binomial Distribution is the probability distribution of the success of obtained in a Bernoulli Trial

### 2. What is Binomial Distribution Formula?

The Binary Distribution Formula is given as P(X = r) = nCrprqn-r. Here r = 0, 1, 2, 3 . . . .

Where, p is success, q is failure and is given by q = 1 – p, and p, q > 0 such that p + q = 1.

### 3. What are the Binomial Distribution Conditions?

The conditions for Binomial Distribution are mentioned below

• Only two possible outcomes such as success or failure, yes or no, true or false.
• There is a finite number of trials given as ‘n’
• The probability of success and failure in each trial is the same
• Only Success is calculated out of all trials
• Each trial is independent of any other trial

### 4. What is Binomial Distribution Mean and Variance?

The Binomial Distribution Mean tells about the average success obtained in ‘n’ number of trials. Binomial Distribution Mean is also called Binomial Distribution Expectation. The formula for Binomial Distribution Expectation is given as μ = n.p.

Where μ is the Mean or Expectation, n is the total number of trials, and p is the probability of success in each trial.

Binomial Distribution Variance is the measurement of the spread of the distribution. The formula of the variance is given by σ2 = n.p.q.

Where σ2 is the variance, n is the total number of trials, p is the probability of success in each trial and q is the probability of failure in each trial.

### 5. What are the Binomial Distribution Characteristics?

The characteristics of Binomial Distribution are mentioned as follows:

• The possible outcomes such as success or failure, yes or no, true or false.
• The number of observations is finite ‘n’
• The probability of success and failure in each trial is the same
• Each trial is independent of any other trials

### 6. What Is the Purpose of the Binomial Distribution Formula?

The purpose of the Binomial Distribution formula is to calculate the probability of obtaining a specific number of successes (often denoted as “k”) in a fixed number of independent and identical trials (often denoted as “n”) when each trial has only two possible outcomes: success or failure.

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