Year 8 Mathematics · AC9M8P01

Complementary Events and Probability

An event and its complement divide the same sample space into “A” and “not A”. Because nothing is omitted or counted twice, their probabilities add to 1—even when…

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Key conceptTeach from the board

The complement of event A is every outcome in the same sample space that is not in A. The two events cannot occur together and together cover the entire sample space, so P(A)+P(not A)=1.

The complement rule does not require equally likely outcomes. If a supermarket promotion gives a particular novelty toy with probability 0.18, then the probability of receiving any other toy is 1−0.18=0.82.

Always define the sample space before naming the complement. “Not late” is the complement of “late” only when those two categories cover every possibility being considered. In binary artificial-intelligence classification such as spam/not spam or fraud/not fraud, the model outputs may be treated as complementary categories only if the classification system is genuinely binary.

Worked examplesWe do

Worked examples

AC9M8P01 - Complementary Events and Probability
Example 1

If P(bus is on time)=0.87, find P(bus is not on time).

Example 2

For a fair die, find P(not rolling an even number).

Example 3

Define the complement of “student chooses basketball” when the only choices are basketball or football.

Example 4

Explain why “blue” is not necessarily the complement of “red” if several colours are possible.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M8P01: recognise that complementary events have a combined probability of one; use this relationship to calculate probabilities in applied contexts.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8P01:Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-PRO-C-01; MAO-WM-01:Partial. Stage 4 probability covers simple chance experiments; the national descriptor gives complementary-event reasoning a more explicit Year 8 emphasis.

LessonAC v9VictoriaNSW
Complementary eventsAC9M8P01VC2M8P01 — ExactMA4-PRO-C-01; MAO-WM-01 — Partial
Questions and answersWith answers
  1. What is a complementary event? Every outcome in the same sample space that is not in the original event.
  2. Why do complementary probabilities sum to 1? Together they cover the whole sample space without overlap.
  3. Do outcomes need to be equally likely? No; the rule uses the probability of the event, not equal-outcome counting.
  4. What is the fastest complement calculation? P(not A)=1−P(A).
Practice and reviewReady for practice
  • “Opposite sounding” means complement. The two events must cover the whole stated sample space with no overlap.
  • Complement only works for fair dice or coins. It works for any event whose probability is known.
  • P(not A)=−P(A). Subtract from 1, not from zero.
  • Two labels are automatically complementary. Check that no third category or unclassified outcome exists.
Curriculum alignmentStart here
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