Binomial Distribution Formula

Binomial Distribution Formula (Table of Contents)

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What is the Binomial Distribution Formula?

The binomial distribution is the probability distribution formula that summarizes the likelihood of an event occurs either A win, B loses or vice-versa under given set parameters or assumptions. However, there is an underlying assumption of the binomial distribution where there is only one outcome is possible for each trial, either success or loss. And each trial in itself is mutually exclusive from another one.

Suppose, if we have defined one outcome out of two is defined as a success, then the probability of x successes out of N trials can be computed as:

P(X) = _nC_x * p^x * (1 – p)^(n-x)

P(X) = (n! / (x! * (n – x)!)) * p^x * (1 – p)^(n-x)

Where p is the probability of success on one trial.

Examples of Binomial Distribution Formula

Let’s take an example to understand the calculation of Binomial Distribution in a better manner.

You can download this Binomial Distribution Formula Excel Template here – Binomial Distribution Formula Excel Template

Binomial Distribution Formula – Example #1

A coin is flipped 10 times. Calculate the probability of getting 5 heads using a Binomial distribution formula.

Solution:

Probability is calculated using the binomial distribution formula as given below

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P(X) = (n! / (x! * (n – x)!)) * p^x * (1 – p)^(n-x)

P(x=5) = (10! / (5! * (10 – 5)!)) * (0.5)^5 * (1 – 0.5)^(10 – 5)
P(x=5) = (10! / (5! * 5!)) * (0.5)^5 * (0.5)^5
P(x=5) = 0.2461

The probability of getting exactly 5 successes is 0.2461

Binomial Distribution Formula – Example #2

In a study, it is found that 70% of people who purchase pet insurance are mostly women. If we randomly select 9 pet insurance owners. What is the probability, out of them 7 will be women?

Solution:

Probability is calculated using the binomial distribution formula as given below

P(X) = (n! / (x! * (n – x)!)) * p^x * (1 – p)^(n-x)

P(x=7) = (9! / (7! * (9 – 7)!)) * (0.7)^7 * (1 – 0.7)^(9 – 7)
P(x=7) = (9! / (7! * 2!)) * (0.7)^7 * (0.3)^2
P(x=7) = 0.2668

Binomial Distribution Formula – Example #3

Last year in the survey of Autocar India, it was found that 70% of buyers of sports cars are men. If 10 sports car owners are randomly selected. What is the probability, out of them 6 will be men?

Solution:

Probability is calculated using the binomial distribution formula as given below

P(X) = (n! / (x! * (n – x)!)) * p^x * (1 – p)^(n-x)

P(x=5) = (10! / (6! * (10 – 6)!)) * (0.7)^6 * (1 – 0.7)^(10 – 6)
P(x=5) = (10! / (6! * 4!)) * (0.7)^6 * (0.3)^4
P(x=5) = 0.2001

Explanation

A binomial distribution basically depends much more on the number of trials or observations are done. While each trial defines its own probability of outcome value or in other words. A binomial random variable defines as a successful outcome of x in n number of the repeated trial of a binomial experiment. While a binomial random variable’s probability distribution is also known as a binomial distribution.

If we take an example, when we toss a coin, the probability of obtaining a head is 0.5 of 50% out of 100%. If we perform 100 trials. The expected value of obtaining heads is 50(100 x 0.5). The binomial distribution is a statistical term to predict the outcome of an event to occur like what is the probability of a sportsman to win in the competition.

There are certain steps and rules to meet the specific criteria of Binomial Distribution models in order to use the formula.

Step 1: Fixed Trials

In this course of action, there is a certain set of a fixed number of trials which can’t be altered in the course of the whole process. The number of trials in the binomial probability formula is represented by the letter “n”. In our case, flips a coin, free throws, wheel spins are the fixed number of trials.

Step 2: Independent Trials

Independent Trial is another condition of a binomial probability in which trials are independent of each other where the outcome of one trial doesn’t impact much more on the subsequent trials.

If we take an example where independent trials may be tossing a coin or rolling dice is independent of the subsequent events.

Step 3: Fixed Probability of success

In this type of distribution, the probability of getting success remain the same for all trials. For example, if we toss a coin the probability of an outcome of every event either head or tail is 0.5. Since there are two possible outcomes.

Step 4: Two Mutually exclusive Outcomes

In this distribution, there are only two types of mutually exclusive outcomes exist either success or failure. Where success has been defined in a positive term. The purpose of the trial is to validate what we have defined as a success. Either it is positive or negative.

Relevance and Uses of Binomial Distribution Formula

The binomial distribution model is the most important probability model it is required when there are two possible outcomes expected. It comes into existence when there were more than two distinct outcomes. In that case, a multinomial probability is more appropriate. But here our major concern is more on the situation where the outcome is dichotomous.

The use of the binomial distribution requires three models:

Each outcome of the process results in the one or two outcomes either success or failure.
The outcome of each process results in the same probability.
Each outcome is mutually exclusive to each other of the process.