Discrete Random Variables & Probability Distributions
Learn discrete random variables and probability distributions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-07). A discrete random variable takes separate values, such as the number of heads in four tosses, and each value comes with its own probability.
You will learn to set up a probability distribution as a table or graph, check that the probabilities total 1, find a missing probability or constant, and recognise a uniform distribution where every outcome is equally likely — the basis for games, quality checks and risk.
Theory
A discrete random variable takes separate values, each with a probability that all add to 1. This Year 12 Mathematics Advanced topic (MAV-12-07) works with probability distributions.
\(P(X=x)\) is the probability \(X\) takes the value \(x\); each \(0\le P(x)\le 1\) and \(\sum P(x)=1\). A uniform distribution over \(n\) values has each probability \(\dfrac1n\).
Method
- Use \(\sum P(x)=1\) to find a missing probability.
- Add probabilities for cumulative events like \(P(X\le k)\).
- Uniform: each value has probability \(\dfrac1n\).
| \(P(3)\) | \(=\) | \(1-0.6=0.4\) |
| \(P(X\le2)\) | \(=\) | \(0.2+0.3=0.5\) |
| \(P(X=4)\) | \(=\) | \(\dfrac16\) |
| \(10k\) | \(=\) | \(1\) |
| \(k\) | \(=\) | \(0.1\) |
Common pitfalls
Frequently asked questions
What is a discrete random variable?
A variable that takes separate values, each with a probability, such as the number rolled on a die.
What makes a valid probability distribution?
Every probability is between 0 and 1, and they all add up to 1.
What is a uniform distribution?
One where every value is equally likely, so each has probability 1 over n.
How do you find a missing probability?
Use the fact that all the probabilities add to 1.