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DEEPCHECKS GLOSSARY

# Binomial Distribution

## What is Binomial Distribution?

The binomial distribution is a probability distribution that specifies the number of successes in a certain number of independent trials, all with the same chance of success. It is a discrete distribution, meaning it can only be described for integer values of the random variable.

The binomial distribution is defined by two parameters: the probability of success (p) and the number of trials (n). The binomial distribution’s probability mass function (PMF) indicates the likelihood of detecting k successes in n trials.

Binomial distribution formula:

• P(k) = (n choose k) * p^k * (1-p)^(n-k)

where:

• P(X = k) is the probability of getting exactly k
• n is the total trials
• k is the frequency of successes
• (n choose k) is the binomial coefficient, which is the number of ways to choose k items from a set of n items and is defined as (n choose k) = n! / (k! * (n-k)!)
• p is the probability of success in each trial
• where (n choose k) is the binomial coefficient indicating the number of ways to choose k items from a collection of n different things

The binomial distribution’s mean is provided by np, while its variance is supplied by np(1-p). When p= 0.5, the distribution is symmetric, but it gets increasingly skewed as p approaches 0 or 1.

The binomial distribution notation is utilized in various statistics and probability applications, including quality control, genetics, and election predictions.

## What is Binomial Probability?

The possibility of receiving a given number of successes in a defined number of independent trials, each with the same probability of success, is known as binomial probability. The binomial distribution is used to compute it, which is a probability distribution that specifies the number of successes in a set number of independent trials.

Consider the case of flipping a fair coin ten times. Using the binomial distribution, the binomial probability formula of receiving precisely 5 heads is computed as follows:

• P(5 heads) = (10 choose 5) * (0.5)^5 * (0.5)^(10-5) = 0.24

This indicates that provided the coin is fair, there is a 24% binomial probability calculator of receiving precisely 5 heads in 10 coin flips.

Probability for binomial distribution is utilized in a variety of statistics and probability applications, including quality control, genomics, and election predictions.

## Importance of Binomial Distribution

The binomial distribution is significant for several reasons:

• It is a basic probability distribution commonly used in statistics and probability.
• Calculates the likelihood of attaining a particular number of successes in a certain number of independent trials, each having the same chance of success.
• It can be utilized to simulate a variety of events, including the number of faulty objects in a manufacturing process, the number of individuals who react to a marketing campaign, and the number of right answers on a multiple-choice exam.
• It serves as a foundation for many other distributions, including the Poisson and normal distributions, which are utilized in more complex statistical models.
• It has a wide range of practical applications, including finance, engineering, biology, and social sciences.

In general, the binomial distribution is a useful tool in probability theory and statistics for understanding and modeling real-world occurrences.

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## Applications of Binomial Distribution

• Sports– The binomial distribution could be utilized to determine a team’s chances of winning a game or a championship depending on the number of games they need to win and their chances of winning each game.
• Forecasting elections– Pollsters use the binomial distribution to predict the likelihood of a candidate winning an election based on the number of people who plan to vote for them and the margin of error in the survey.
• Finance– The binomial distribution can describe the likelihood of various outcomes in financial investments, such as the likelihood of a stock price increasing or decreasing.
• Genetics– It is utilized to represent the likelihood of kids acquiring a certain characteristic from their parents, given the likelihood of the trait being handed down through each generation.
• Quality control– When estimating the percentage of flawed goods in a batch based on production totals and individual defect rates, binomial distribution is a useful tool for quality control in manufacturing.
• Biology– Given the overall length of the sequence and the likelihood of a mutation happening at each place, the binomial distribution may be used to simulate the probability of a specified number of mutations occurring in a DNA sequence.