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WAMath MethodsSyllabus dot point

What is the expected value of a discrete random variable, and how do we use it to make decisions?

Calculate and interpret the expected value (mean) of a discrete random variable and apply it to decision contexts such as fair games

WACE Year 12 Mathematics Methods Unit 3 expected value: the mean as a probability-weighted average, its interpretation as a long-run average, expected value of a linear function, and fair-game decisions, with worked SCSA-style examples.

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  1. What this dot point is asking
  2. Definition and interpretation
  3. Expected value of a function
  4. Fair games

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 XX; it is the average you would approach over a large number of repetitions. The mean of a fair die is 3.53.5, even though no face shows 3.53.5.

Expected value of a function

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

E(g(X))=∑xg(x) P(X=x).E(g(X))=\sum_x g(x)\,P(X=x).

For a linear function this simplifies to E(aX+b)=aE(X)+bE(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.