# The binomial distribution

## Understanding The Binomial Distribution

- The
**Binomial Distribution**is a specific type of discrete probability distribution. - It models the number of successes in a fixed number of
**Bernoulli Trials**or**independent experiments**with only two possible outcomes. - Each Bernoulli trial is independent of each other and has the same probability of success, usually denoted by
**p**. - The distribution is denoted as B(n, p) where
**n**is the number of trials and**p**is the probability of success on each trial.

## Properties of The Binomial Distribution

- The binomial distribution has two parameters:
**n**(the number of trials) and**p**(the probability of success on each trial). - The possible values of a Binomial random variable are the whole numbers from
**0 to n**. - The
**mean**of a binomial distribution B(n, p) is**np**, and the**variance**is**np(1-p)**.

## Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)

- The
**Probability Mass Function**of a binomial distribution gives the probability of getting**k successes**in**n trials**. It can be calculated using the combination formula C(n, k) * p^k * (1-p)^(n-k). - The
**Cumulative Distribution Function**for a binomial distribution is the sum of the probabilities of the outcomes up to and including the defined upper limit.

## Working with The Binomial Distribution

- The binomial distribution is commonly used in statistical modelling and experiments where the outcome can be classified as a success or failure.
- The
**binomial coefficient**or “n choose k”, denoted C(n,k) or nCk in many statistics textbooks, is an important concept when working with binomial distributions. - As an example, the binomial distribution can be used to answer questions such as “what’s the probability of getting exactly 6 heads in 10 coin tosses?”
- The binomial distribution simplifies when
**p = 0.5**to a**symmetrical distribution**, commonly known as a**bell curve**.

To cement these concepts, review your lecture notes and work through as many practice problems as possible.