# Situations which Give Rise to a Binomial Distribution

# Situations which Give Rise to a Binomial Distribution

## Definition and Key Features

**Binomial distribution**is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials each with the same probability of success.- It’s characterised by only two possible outcomes in every trial often termed as ‘success’ or ‘failure’.
- An experiment is termed a binomial experiment if it has the following four properties: fixed number of trials, each trial is independent, there are two possible outcomes, and the probability of each outcome remains constant from trial to trial.

## Typical Situations

- Tossing a coin: Here, the outcome can either be a ‘head’ or a ‘tail’ - two definitive outcomes. When you are interested in the number of ‘heads’ (successes) from a fixed number of tosses, then the situation is governed by a
**binomial distribution**. - An example in quality control could be testing a batch of items off a production line to determine if each is a ‘pass’ or ‘fail’. If the probability of a defective item remains constant, the situation follows a
**binomial distribution**. - Other common examples include predicting the number of students who will pass a certain exam (pass or fail), or predicting the number of voters who will vote for a particular candidate (vote or not vote) - both situations considering a fixed number of trials.

## Applications in Various Fields

- Binomial distributions are commonly used in
**statistics**,**business analytics**,**quality control**,**marketing**and many other areas that require making predictions based on data or making decisions in the presence of uncertainty. - In
**medical research**, it could be used to model the number of patients who respond positively to a treatment, given the probability of a positive response for any individual patient. - Binomial distribution is also used extensively in
**sports analytics**, for example, to model the number of successful free throws a basketball player might make in a certain number of attempts.

## Key Variables in Binomial Distribution

- In binomial distribution, we are interested in the number of successes (k), out of a fixed number of trials (n), given that each trial has a fixed probability of success (p). The binomial distribution is then represented as
**B(n,p)**.

## Note on Sample Space

- In a binomial setting, the sample space is sets of n ‘successes’ and ‘failures’ where order matters, where each outcome in the sample space can be labelled with the number of successes. Therefore, for n trials, there are exactly (n+1) possible outcomes, corresponding to 0, 1, 2, …, n successes.

## Limitations

- The binomial distribution remains appropriate only if the trials are independent and the success probability is constant. If this isn’t the case, then a different probability model could be a better fit.