# 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.