This is Chapter 7 – Discrete Random Variables for the Year 12 Mathematical Methods course.

This chapter develops a more formal approach to probability by introducing random variables and probability distributions. Instead of examining isolated events, students build mathematical models that describe all possible numerical outcomes of a random process and the probability associated with each outcome.

We begin by defining a random variable and distinguishing between the underlying experiment and the numerical values assigned to its outcomes. Students then construct and interpret discrete probability distributions, ensuring that all probabilities are valid and combine to give a total of one.

Expected value is introduced as the long-run average outcome of a random variable. This idea is especially useful for analysing games, financial decisions and repeated processes, although students must understand that the expected value is not necessarily an outcome that can occur in a single trial. Variance and standard deviation are then used to measure how widely the possible outcomes are spread around the mean.

The properties of aX+b allow students to determine how the mean and variability change when a random variable is transformed. This strengthens the connection between algebra and probability and becomes important in later statistical work.

The chapter finishes with Bernoulli and binomial distributions. Students learn to recognise situations involving repeated independent trials with two possible outcomes and apply an appropriate binomial model. Interpretation remains essential: a correct numerical result must be explained within the context of the original problem.

Discrete probability models are used in finance, insurance, actuarial science, engineering reliability, quality control and data analysis.

Subchapters:

• 7A: Random variables

• 7B: Discrete probability distributions
• 7C: Expected value
• 7D: Variance and standard deviation
• 7E: Properties of aX + b
• 7F: The Bernoulli distribution
• 7G: The binomial distribution

By the end of this chapter, students will be able to construct and interpret discrete probability models, calculate their key measures and apply binomial distributions to contextual problems.