Year 8 Mathematics · AC9M8P02

Two Events: Tables, Trees and Venn Diagrams

Represent every possible combination of two events systematically, then read “and”, inclusive/exclusive “or”, “at least” and mutually exclusive relationships without…

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Key conceptTeach from the board

For two yes/no events A and B, the complete sample space splits into four disjoint regions: A and B, A and not B, not A and B, and neither. Their probabilities must sum to 1.

A two-way table is useful for paired categories and totals. A Venn diagram makes overlap, union and exclusivity visible. A tree diagram is useful when outcomes occur in sequence. The representation changes, but the underlying complete set of outcomes must remain the same.

Language matters. “A and B” means intersection. Inclusive “A or B” includes the overlap. Exclusive “A or B” means one or the other but not both. “At least one” often becomes easier by using the complement “none”. Mutually exclusive events have no overlap, so P(A and B)=0.

The Australian Curriculum also uses instructive games such as Battendi, associated with Ngarrindjeri Peoples of the Lake Murray and Lake Albert region, as contexts for analysing possible combinations and relationships. Treat the cultural context respectfully and focus on the stated mathematical structure rather than turning it into decorative trivia.

Worked examplesWe do

Worked examples

AC9M8P02 - Two Events: Tables, Trees and Venn Diagrams
Example 1

List the four A/B membership combinations.

Example 2

Shade A and B, then A or B, on a Venn diagram.

Example 3

Explain why Head/Tail are mutually exclusive on one toss.

Example 4

Draw a tree for choosing red/blue then circle/square.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M8P02: determine all possible combinations for 2 events, using two-way tables, tree diagrams and Venn diagrams, and use these to determine probabilities of specific outcomes in practical situations.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8P02:Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-PRO-C-01; MAO-WM-01:Supporting. Current Stage 4 probability is framed around simple chance experiments; the national Year 8 descriptor explicitly develops two-event Venn/two-way/tree representations beyond that single Stage 4 outcome.

LessonAC v9VictoriaNSW
Representing two eventsAC9M8P02VC2M8P02 — ExactMA4-PRO-C-01; MAO-WM-01 — Supporting
Questions and answersWith answers
  1. What are the four regions for two events? A∩B, A∩not B, not A∩B and neither.
  2. What does inclusive “A or B” mean? A, B or both.
  3. What makes events mutually exclusive? Their intersection is empty, so they cannot happen together.
  4. When is a tree diagram especially useful? When outcomes occur in stages or ordered sequences.
Practice and reviewReady for practice
  • Inclusive “or” excludes the overlap. Inclusive “or” contains A, B and their intersection.
  • Row/column totals are additional outcomes. They summarise cells; do not include them again in the denominator.
  • Mutually exclusive means independent. It means the events cannot occur together.
  • A tree always means equal branch probabilities. Branch probabilities come from the context and may differ.
Curriculum alignmentStart here
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