Year 9 Mathematics · AC9M9P01

Two-Step Compound Events and Sample Spaces

Two-step chance experiments require a complete sample space. Replacement changes second-step probabilities; tree diagrams, tables and arrays help list outcomes without…

Ready to project and teach

Learning goalsSay it simply

Two-step chance experiments require a complete sample space. Replacement changes second-step probabilities; tree diagrams, tables and arrays help list outcomes without omissions or duplicates.

By the end of this lesson, you should be able to:

  • list all outcomes for two-step chance experiments
  • distinguish experiments with and without replacement
  • represent sample spaces using trees, tables or arrays
  • assign probabilities to outcomes and combined events
Key conceptTeach from the board

A compound event has more than one stage

Represent stages systematically so every ordered outcome is considered.

Replacement controls whether probabilities reset

With replacement, the composition returns to its start. Without replacement, the first result changes later-stage probabilities.

Representation prevents mistakes

Tree diagrams show sequential probability; tables and arrays show combinations compactly.

Outcome probabilities may not be equal

In without-replacement or uneven experiments, calculate branch probabilities rather than counting labelled outcomes blindly.

Worked examplesWe do

Worked examples

AC9M9P01 - Two-Step Compound Events and Sample Spaces
Example 1

Two coins HH, HT, TH, TT, each with probability 1/4 for fair independent coins.

Example 2

Two dice A 6×6 array lists 36 ordered outcomes.

Example 3

Without replacement From 3 red and 2 blue, P(R then B)=3/5×2/4=3/10.

Example 4

With replacement P(R then B)=3/5×2/5=6/25.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9P01: list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes

Victoria: VC2M9P01 — Level 9 Probability. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Probability; Path — Further probability. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9P01VC2M9P01 — Level 9 ProbabilityStage 5 Core — Probability; Path — Further probability
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
What changes without replacement?
The composition and therefore later-stage probabilities.
What operation combines stages on one path?
Multiplication.
When do I add paths?
When different mutually exclusive paths all satisfy the target event.
Practice and reviewReady for practice
  • RB and BR are always the same outcome: In ordered two-step experiments they are different sequences.
  • Without replacement keeps probabilities unchanged: The sample composition changes after the first draw.
  • All listed outcomes are automatically equally likely: Only if the underlying stages make them equally likely.
  • Tree branch probabilities are added along a path: Multiply along a path; add mutually exclusive target paths.
  1. [5 marks] Construct a complete tree diagram for a two-step experiment and label branch probabilities.
  2. [6 marks] Compare a probability with and without replacement and explain the difference.
  3. [7 marks] Solve a compound-event problem using two representations and show they give the same probability.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

Curriculum alignmentStart here

Two-step chance experiments require a complete sample space. Replacement changes second-step probabilities; tree diagrams, tables and arrays help list outcomes without omissions or duplicates.

By the end of this lesson, you should be able to:

  • list all outcomes for two-step chance experiments
  • distinguish experiments with and without replacement
  • represent sample spaces using trees, tables or arrays
  • assign probabilities to outcomes and combined events
Related resourcesKeep teaching
Teach & ExplainTeaching slides and samples

Teach this topic step by step

Explore optional teaching slide packs for classroom lessons and explanations at home.

Browse Teach & Explain · Browse Print & Go

Teachers: follow SkillrHub on TPT, then email us to request a free sample before buying. Include the year, subject and topic or curriculum code.

Request a free sample

After trying the sample, honest feedback is welcome. A TPT review is optional, where available, and does not need to be positive.