# Binomial Distribution Definition, Formula, Analysis, and Example

## What Is the Binomial Distribution?

The binomial distribution is a statistical probability distribution that represents the chance of a variable taking one of two independent values under a particular set of factors or assumptions.

The binomial distribution’s basic assumptions are that each trial has just one result, that each trial has the same chance of success, and that each trial is mutually exclusive, or independent of one another.

### Key Takeaways

- In statistics, the binomial distribution quantifies the chance that a variable will take one of two independent values under a particular set of factors or assumptions.
- The binomial distribution’s basic assumptions are that each trial has just one result, that each trial has the same chance of success, and that each trial is mutually exclusive or independent of the others.
- The binomial distribution, as opposed to a continuous distribution like the normal distribution, is a frequent discrete distribution used in statistics.

## Understanding Binomial Distribution

To begin, the word “binomial” refers to two concepts. We’re interested not just in the number of successes, but also in the number of attempts. Without the other, neither is useful to us.

The binomial distribution, as opposed to a continuous distribution like the normal distribution, is a frequent discrete distribution used in statistics. This is due to the fact that the binomial distribution only counts two states, which are commonly expressed as 1 (for success) or 0 (for failure) given a number of trials in the data. Given a success probability p for each trial, the binomial distribution indicates the likelihood of x successes in n trials.

The binomial distribution describes the number of trials or observations where each trial has the equal chance of achieving a certain value. The binomial distribution calculates the likelihood of witnessing a certain number of successful outcomes in a given number of trials.

In social science statistics, the binomial distribution is often used as a building block for models for dichotomous outcome variables, such as whether a Republican or Democrat will win an upcoming election or if a person will die within a certain time period, etc. It is also used in finance, banking, and insurance, among other areas.

## Analyzing Binomial Distribution

A binomial distribution’s expected value, or mean, is derived by multiplying the number of trials (n) by the chance of success (p), or n x p.

The predicted value of the number of heads in 100 trials of heads or stories, for example, is 50, or (100 * 0.5). Another typical use of the binomial distribution is evaluating the probability of success for a basketball free-throw shooter, where 1 Equals a basket made and 0 Means a miss.

The binomial distribution formula is calculated as:

P_{(x:n,p)}=_{n}C_{x}x p(1-p)

where:

- n denotes the number of tries (occurrences)
- X represents the number of successful trials.
- The chance of success in a single trial is given by p.
- nCx is the product of n and x. A combination is the number of possible methods to choose a sample of x items from a collection of n different objects, where order is irrelevant and replacements are not permitted. It is worth noting that nCx=n!/(r!(nr)!), where! is factorial (thus, 4! = 4 x 3 x 2 x 1).

The mean of the binomial distribution is np, and the variance is np (1 p). The distribution is symmetric around the mean for p = 0.5. When p is greater than 0.5, the distribution is skewed to the left. The distribution is biased to the right when p = 0.5.

The binomial distribution is the sum of many independently dispersed and identically distributed Bernoulli trials. The experiment in a Bernoulli trial is considered to be random, with just two potential outcomes: success or failure.

For example, flipping a coin is a Bernoulli trial because each trial can only have one of two outcomes (heads or tails), each success has the same probability (the chance of flipping a head is 0.5), and the results of one trial have no impact on the results of another. The Bernoulli distribution is a subset of the binomial distribution with n = 1 trials.

## Example of Binomial Distribution

The binomial distribution is computed by multiplying the chance of success multiplied by the number of successes by the likelihood of failure multiplied by the difference between the number of successes and the number of tries. The product is then multiplied by the sum of the number of tries and the number of successes.

Assume a casino develops a new game in which players may wager on the number of heads or tails in a set number of coin flips. Assume a person wishes to stake $10 that precisely six heads will come up in 20 coin flips. The participant wants to determine the likelihood of this happening, so they utilize the binomial distribution computation.

The probability was computed as follows: (20! / (6! * (20 – 6)!)) * (0.50)(6) * (1 – 0.50) (20 – 6).As a result, the likelihood of precisely six heads happening in 20 coin flips is 0.037, or 3.7%. In this situation, the anticipated value was 10 heads, hence the player placed a bad bet.

So, how does this apply to finance? Assume you’re a bank or a lender that wants to know the chance of a certain borrower failing to three decimal places. What are the possibilities of so many borrowers failing that the bank goes bankrupt? Once you’ve calculated that figure using the binomial distribution function, you’ll have a better sense of how to price insurance and, ultimately, how much money to give out and how much to hold in reserve.

## What Is Binomial Distribution?

The binomial distribution is a statistical probability distribution that describes the chance that a variable will take one of two independent values given a set of factors or assumptions.

## How Is Binomial Distribution Used?

This distribution pattern is employed in statistics, but it also has applications in finance and other areas. Banks may use it to predict the chance of a certain borrower failing or to determine how much money to lend and how much to maintain in reserve. In the insurance sector, it is also used to calculate policy price and risk.

## Why Is Binomial Distribution Important?

The binomial distribution is used to calculate the chance of a pass or fail result in a survey or experiment that is repeated several times. This sort of distribution has just two possible outcomes. More generally, distribution is a crucial aspect of studying data sets since it allows us to estimate all of the possible outcomes and how often they occur. Forecasting and comprehending the success or failure of outcomes is critical to corporate growth.

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