What is a binomial distribution?

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials.

In probability theory and statistics, a binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials. The binomial distribution is used when there are only two possible outcomes for each trial, such as success or failure, heads or tails, or yes or no. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p) in each trial.

The probability of getting exactly k successes in n trials is given by the binomial probability formula:

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

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items. The binomial coefficient can be calculated using the formula:

(n choose k) = n! / (k! * (n-k)!)

where n! is the factorial of n, which is the product of all positive integers up to n.

The mean and variance of a binomial distribution are given by:

mean = np
variance = np(1-p)

The binomial distribution is widely used in many fields, including genetics, finance, and quality control. It is also a fundamental concept in hypothesis testing and statistical inference.