Assumed knowledgeThe content of the modules:
This module focusses on the binomial distribution. The module Discrete probability distributions includes many examples of discrete random variables. But the binomial distribution is such an important example of a discrete distribution that it gets a module in its own right.
The importance of the binomial distribution is that it has very wide application. This is because at its heart is a binary situation: one with two possible outcomes. Many random phenomena worth studying have two outcomes. Most notably, this occurs when we examine a sample from a large population of 'units' for the presence of a characteristic; each unit either has the characteristic or it doesn't. The generic term 'unit' is used precisely because the situation is so general. The population is often people, in which case a unit is a person; but a unit might be a school, an insect, a bank loan, a company, a DNA sequence, or any of a number of other possibilities.
This module starts by introducing a Bernoulli random variable as a model for a binary situation. Then we introduce a binomial random variable as the number of 'successes' in \(n\) independent Bernoulli trials, each with the same probability of success \(p\). We show how to calculate probabilities associated with a binomial distribution, and illustrate the use of binomial distributions to solve practical problems. The last section covers the mean and variance of a binomial distribution.