# Binomial Distribution: Definition, Formula, Analysis, and Example

## What Is Binomial Distribution?

Binomial distribution is a probability distribution used in statistics that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.

The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive, or independent of one another.

### Key Takeaways

• Binomial distribution is a probability distribution in statistics that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.
• The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive or independent of one another.
• Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution.

## Understanding Binomial Distribution

To start, the “binomial” in binomial distribution means two terms. We’re interested not just in the number of successes, nor just the number of attempts, but in both. Each is useless to us without the other.

Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution. This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure) given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial.

Binomial distribution summarizes the number of trials, or observations when each trial has the same probability of attaining one particular value. Binomial distribution determines the probability of observing a specified number of successful outcomes in a specified number of trials.

Binomial distribution is often used in social science statistics as a building block for models for dichotomous outcome variables, such as whether a Republican or Democrat will win an upcoming election, whether an individual will die within a specified period of time, etc. It also has applications in finance, banking, and insurance, among other industries.

## Analyzing Binomial Distribution

The expected value, or mean, of a binomial distribution is calculated by multiplying the number of trials (n) by the probability of successes (p), or n × p.

For example, the expected value of the number of heads in 100 trials of heads or tales is 50, or (100 × 0.5). Another common example of binomial distribution is by estimating the chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss.

The binomial distribution formula is calculated as:

P(x:n,p)nCx x px(1-p)n-x

where:

• n is the number of trials (occurrences)
• x is the number of successful trials
• p is probability of success in a single trial
• nCx is the combination of n and x. A combination is the number of ways to choose a sample of x elements from a set of n distinct objects where order does not matter and replacements are not allowed. Note that nCx=n!/(r!(n−r)!), where ! is factorial (so, 4! = 4 × 3 × 2 × 1).

The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean. When p > 0.5, the distribution is skewed to the left. When p < 0.5, the distribution is skewed to the right.

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure.

For instance, flipping a coin is considered to be a Bernoulli trial; each trial can only take one of two values (heads or tails), each success has the same probability (the probability of flipping a head is 0.5), and the results of one trial do not influence the results of another. Bernoulli distribution is a special case of binomial distribution where the number of trials n = 1.

## Example of Binomial Distribution

Binomial distribution is calculated by multiplying the probability of success raised to the power of the number of successes and the probability of failure raised to the power of the difference between the number of successes and the number of trials. Then, multiply the product by the combination between the number of trials and the number of successes.

For example, assume that a casino created a new game in which participants are able to place bets on the number of heads or tails in a specified number of coin flips. Assume a participant wants to place a \$10 bet that there will be exactly six heads in 20 coin flips. The participant wants to calculate the probability of this occurring, and therefore, they use the calculation for binomial distribution.

The probability was calculated as (20! / (6! × (20 - 6)!)) × (0.50)^(6) × (1 - 0.50) ^ (20 - 6). Consequently, the probability of exactly six heads occurring in 20 coin flips is 0.037, or 3.7%. The expected value was 10 heads in this case, so the participant made a poor bet.

So how can this be used in finance? One example: Let’s say you’re a bank, a lender, that wants to know within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they would render the bank insolvent? Once you use the binomial distribution function to calculate that number, you have a better idea of how to price insurance, and ultimately how much money to lend out and how much to keep in reserve.

## What is binomial distribution?

Binomial distribution is a probability distribution used in statistics that states the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.

## How is binomial distribution used?

This distribution pattern is used in statistics but has implications in finance and other fields. Banks may use it to estimate the likelihood of a particular borrower defaulting or how much money to lend and the amount to keep in reserve. It’s also used in the insurance industry to determine policy pricing and to assess risk.

## Why is binomial distribution important?

Binomial distribution is used to figure the likelihood of a pass or fail outcome in a survey or experiment replicated numerous times. There are only two potential outcomes for this type of distribution. More broadly, distribution is an important part of analyzing data sets to estimate all the potential outcomes of the data and how frequently they occur. Forecasting and understanding the success or failure of outcomes is essential to business development.

## The Bottom Line

The binomial distribution is an important statistical distribution that describes binary outcomes (such as the flip of a coin, a yes/no answer, or an on/off condition). Understanding its characteristics and functions is important for data analysis in various contexts that involve an outcome taking one of two independent values. It has applications in social science, finance, banking, insurance, and other areas. For instance, whether a borrower will default on a loan or not, whether an options contract will finish either in-the-money or out-of-the-money, or whether a company miss or beat earnings estimates.

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1. Cuemath. “Binomial Distribution Formula.”

2. Cuemath. “Bernoulli Trials.”

3. Research Optimus. “Difference Between Normal, Binomial, and Poisson Distribution.”