Expected value of a discrete random variable: WACE Year 12 Mathematics Methods Unit 3

What this dot point is asking

SCSA Unit 3 defines the expected value of a discrete random variable as its mean. This dot point asks you to compute it, interpret it as a long-run average, transform it under a linear change, and apply it to decisions such as whether a game is fair. It is examined in both sections.

Definition and interpretation

Each value is weighted by its probability, so likely outcomes pull the mean toward them. The expected value is not necessarily an attainable value of $X$; it is the average you would approach over a large number of repetitions. The mean of a fair die is $3.5$, even though no face shows $3.5$.

Expected value of a function

If you earn a payoff that is a function of $X$, the expected payoff weights each function value by the same probabilities:

For a linear function this simplifies to $E(aX+b)=aE(X)+b$, because expectation is linear.

Fair games

A game is fair if the expected net gain to the player is zero, meaning the expected winnings equal the cost to play. If the expected gain is negative, the game favours the operator; if positive, it favours the player. Commercial games are designed with a small negative expected gain to the player (the "house edge"), which guarantees the operator a profit over many plays even though individual players sometimes win. SCSA decision questions often ask you to set the expected gain to zero and solve for a fair entry price or a fair prize.

Why linearity works

The rule $E(aX+b)=aE(X)+b$ follows directly from the definition. Each value $x$ becomes $ax+b$ with the same probability, so

using $\sum P(X=x)=1$. This is why a constant shift $b$ moves the mean by exactly $b$ and a scale factor $a$ multiplies it by $a$. The same reasoning does not extend to nonlinear functions, which is why $E(X^{2})$ must be computed term by term rather than as $[E(X)]^{2}$.

Long-run interpretation in context

The phrase "expected value" can mislead: $E(X)$ is the average over very many repetitions, not the value to expect on a single trial. If a raffle ticket has expected winnings of $\$0.80$ but costs $\$1$, no single ticket returns $80$ cents, yet buying thousands of tickets loses about $20$ cents each on average. SCSA decision questions hinge on this long-run reading, so always interpret $E(X)$ as a per-trial average across repeated play.