Two Events: Tables, Trees and Venn Diagrams — AC9M8P02

• Determine all four membership combinations for events A and B.
• Choose between two-way tables, tree diagrams and Venn diagrams.
• Interpret “and”, inclusive/exclusive “or” and “at least”.
• Distinguish mutually exclusive from overlapping events.

Recall sample spaces, event notation, complements, fractions/decimals/percentages and basic set ideas such as overlap and outside a set.

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.

• 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.

1. List the four A/B membership combinations.
2. Shade A and B, then A or B, on a Venn diagram.
3. Explain why Head/Tail are mutually exclusive on one toss.
4. Draw a tree for choosing red/blue then circle/square.

1. A class has 30 students: 17 play music, 14 play sport and 8 do both. How many do at least one?
2. Using the same data, how many do neither?
3. Create a two-way table for 50 people classified by two yes/no attributes.
4. Give an example of two mutually exclusive events and two non-mutually-exclusive events.
5. Explain the difference between exclusive and inclusive “or”.
6. Draw a tree for two fair coin tosses and find P(at least one head).
7. Describe how a Battendi-style two-event game context could be represented with a two-way table or Venn diagram without assuming cultural details not supplied.

A survey reports P(A)=0.55, P(B)=0.48 and P(A and B)=0.21. Find P(A or B), then determine the probabilities of all four A/B regions and check that they sum to 1. Explain why simply adding 0.55+0.48 is invalid.

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.

1. In 80 people, 46 are in A, 35 in B and 20 in both. Find P(A or B) and P(neither).
   Use union = A+B−intersection, then complement for neither.
2. Explain, with an example, why mutually exclusive and independent do not mean the same thing.
   Focus on whether events can occur together versus whether one changes the probability of the other.
3. Choose the most useful representation—tree, two-way table or Venn diagram—for a stated two-event situation and justify the choice.
   Match the representation to sequence, paired categories or overlap.

• I enumerate all two-event combinations.
• I interpret and/or/at least accurately.
• I avoid double-counting intersections.
• I choose a useful representation for the context.

Exit ticket: Why must the four disjoint A/B regions total probability 1?

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.

• AC9M8P01 — complementary events
• AC9M8P03 — simulations of compound events

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Determine all possible combinations for 2 events, using two-way tables...

Mapped skill: 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

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.