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Binomial Distribution

In statistics and probability theory, the binomial distribution is the probability distribution that is discrete and applicable to events having only two possible results in an experiment, either success or failure. (the prefix “bi” means two, or twice). A few circumstances where we have binomial experiments are tossing a coin: head or tail, the result of a test: pass or fail, selected in an interview: yes/ no, or nature of the product: defective/non-defective. Such a distribution of a binomial random variable is called a binomial probability distribution.

Binomial Distribution is a commonly used discrete distribution in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. Let us learn the formula to calculate the Binomial distribution considering many experiments and a few solved examples for a better understanding.

1.	What Is Binomial Distribution?
2.	Negative Binomial Distribution
3.	Binomial Distribution Formula
4.	Application of Binomial Distribution
5.	Binomial Distribution Mean and Variance
6.	Binomial Distribution Vs Normal Distribution
7.	Properties of Binomial Distribution
8.	FAQs on Binomial Distribution

What Is Binomial Distribution?

The binomial distribution is the probability distribution of a binomial random variable. A random variable is a real-valued function whose domain is the sample space of a random experiment. Let us consider an example to understand this better.

Toss a fair coin twice. This is a binomial experiment. There are 4 possible outcomes of this experiment. {HH, HT, TH, TT}. Consider getting one head as the success. Count the number of successes in each possible outcome. Here n(getting heads) is the success in n repeated trials of a binomial experiment.. n(X) = 0, 1, or 2 is the binomial random variable. The distribution of probability is of a binomial random variable, and this is known as a binomial distribution.

No. of heads(n(X))	Probability of getting a head(P(X))
0	P(x = 0) = 1/4 = 0.25
1	P(x = 1) = P(HT) = 1/4 + 1/4 = 0.50
2	P(x = 2) = P(HH) = 1/4 = 0.25

This table shows that getting one head in a single flip is 0.50. Now if a coin is flipped 3 times, consider we are intended to find the binomial distribution of getting two heads. Tossing 3 coins result in 8 outcomes. {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting two heads [P(HH)] is 3/8. Similarly, we can calculate the probability of getting one head, 2 heads, and 3 heads and 0 heads. The binomial probability distribution is given in terms of a random variable as:

P(X = 0) = 1/8

P(X = 1) = 3/8

P(X = 2) = 3/8

P(X = 3)= 1/8

Binomial Distribution in Statistics

The binomial distribution forms the base for the famous binomial test of statistical importance. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment. Two parameters n and p are used here in the binomial distribution. The variable ‘n’ represents the number of trials and the variable ‘p’ states the probability of any one(each) outcome. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process.

Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the boolean-valued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution.

If a die is thrown randomly 10 times, then the probability of getting a 3 for any throw is 1/6. Similarly, if we throw the dice 10 times, we have n = 10 and p = 1/6, q = 5/6

Negative Binomial Distribution

Let's understand with an example when can a binomial distribution be negative. Suppose we throw a die and determine that the occurrence of 2 will be a failure and all non-2’s will be successes. Let the failures be denoted by ‘r’. Now, if the die is thrown frequently until 2 appears the third time, i.e., r = three failures, then the binomial distribution of the number of non-2's that arrived would be the negative binomial distribution.

Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by; P(x:n,p) = ⁿC_x p^x(1-p)^n-xOr P(x:n,p) = ⁿC_x p^x (q)^n-x

Where p is the probability of success, q is the probability of failure, and n = number of trials. The binomial distribution formula is also written in the form of n-Bernoulli trials.

where ⁿC_x = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].p^x.(q)^n-x

Binomial Distribution Calculation

The image given below shows the formula used for binomial distribution calculation:

Binomial distribution formula

Application of Binomial Distribution

We now already know that binomial distribution gives the probability of a different set of outcomes. In real life, the concept of the binomial distribution 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
To find the number of male and female students in a university.
The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.

Consider a card selected at a random and replaced. If this experiment is repeated 5 times, let us find the probability of selecting exactly 3 hearts. Let us determine the number of trials, success, and the failure. The trial is the drawing a card 5 times. Thus n = 5.

success: card drawn is a heart = p = 1/4 = 0.25

failure: card drawn is not a heart = q = 1-0.25 = 0.75

Using the binomial distribution formula, we get ⁵C \(_3\) (0,25)³(0.75)²= 0.088

Binomial Distribution Mean and Variance

For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas

Mean, μ = np
Variance, σ²= npq
Standard Deviation σ= √(npq)

Where p is the probability of success q is the probability of failure, where q = 1-p

Binomial Distribution Vs Normal Distribution

The main difference between the binomial distribution and the normal distribution is that the binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.

Properties of Binomial Distribution

The properties of the binomial distribution are:

There are only two distinct possible outcomes: true/false, success/failure, yes/no.
There is a fixed number of 'n' times repeated trials in a given experiment.
The probability of success or failure remains constant for each attempt/trial.
Only the successful attempts are calculated out of 'n' independent trials.
Every trial is an independent trial on its own, this means that the outcome of one trial has no effect on the outcome of another trial.

Important Notes on Binomial Distribution

For using the binomial distribution, the number of observations or trials in an experiment is fixed or finite.
Each observation/attempt/trial is independent on its own. This means none of the trials have an effect on the probability of the next trial.
Each trial has an equal probability of occurrence. The probability of success is exactly the same from one trial to another.

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Binomial Distribution Examples

Example 1: If a coin is tossed 5 times, using binomial distribution 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

Using binomial distribution formula,

P(x=2) = ⁵C₂ p² 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, using binomial distribution formula,

P(x = 4) = ⁵C₄ p⁴ q^5-4 = 5!/4! 1! × (½)⁴× (½)¹ = 5/32

P(x = 5) = ⁵C₅ p⁵ q^5-5 = (½)⁵ = 1/32

Answer:Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16
Example 2: For the same question given above, using the binomial distribution find the probability of getting at most 2 heads.

Solution:

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

P(X = 0) = (½)5 = 1/32

Using binomial distribution formula, we get

P(X=1) = ⁵C₁ (½)⁵= 5/32

Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16
Example 3: A random variable X has the following binomial distribution. Determine P(X>6) and P(0<X<3)

X 0 1 2 3 4 5 6 7

P(X) 0 k 2k 2k 3k k² 2k² 7k²+ k

Solution:

This is a binomial distribution.

To find k. The sum of all the probabilities = 1

0 + k + 2k +2k + 3k + k²+ 2k²+ 7k²+ k = 1

10k²+ 8 k = 1

Solving for k , we get k = 0.1 and -1, We consider k = 0.1 as k = -1 makes the probability negative which is not possible.

i) P(X>6)= 7k²+ k

7(0.1)²+ 0.1

= 0.17

ii) P(0<X<3)

= k + 2k = 3k

= 0.3

Answer: P(X>6)= 0.17 and P(0<X<3) = 0.3

X	0	1	2	3	4	5	6	7
P(X)	0	k	2k	2k	3k	k²	2k²	7k²+ k

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Binomial Distribution Questions

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FAQs on Binomial Distribution

What Is a Binomial Distribution?

The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution. The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial, and is applicable to events having only two possible results in an experiment.

What Is the Purpose of Binomial Distribution?

The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

What Is the Formula for Binomial Distribution?

The formula for binomial distribution is:

P(x: n,p) = ⁿC_x p^x (q)^n-x

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)

What Is the Formula for the Mean and Variance of the Binomial Distribution?

The mean and variance of the binomial distribution are:

Mean = np
Variance = npq

Where p is the probability of success, q is the probability of failure, and n = number of trials.

What Are the Criteria for the Binomial Distribution?

The criteria for using the binomial distribution are:

The number of trials should be fixed.
Each trial should be independent.
The probability of success is exactly the same from one trial to the other trial.

What Is the Difference Between a Binomial Distribution and Normal Distribution?

The main difference between the binomial distribution and the normal distribution is that the binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.

How Do you Identify a Binomial Distribution?

For a variable to be a binomial random variable, all of the following conditions must be met:

There are a fixed number of trials (a fixed sample size).
On each trial, the event of interest either occurs or does not.
The probability of occurrence (or not) is the same on each trial.
Trials are independent of one another.

Is Binomial Distribution Discrete or Continuous?

A binomial distribution is a discrete distribution with parameters n and p, where n is the number of trials and p is the probability of success.

For the Binomial Distribution, Which Formula Finds the Standard Deviation?

The standard deviation formula for a binomial distribution is given by, σ = √(npq), where n = number of trials, p = probability of success, q = probability of failure = 1 - p.

What is Negative Binomial Distribution?

Negative binomial distribution is a discrete probability distribution in statistics. It helps in finding r success in x trials. Here we consider the n + r trials needed to get r successes.

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