Two-Step Compound Events and Sample Spaces — AC9M9P01

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

Recall simple probability, fractions, complements and equally likely outcomes.

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.

• 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. List outcomes for tossing a coin then spinning a 1–3 spinner.
2. Draw a tree for two fair coin tosses.
3. Find P(RB) without replacement from 3R and 2B.
4. Repeat with replacement and compare.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. List outcomes for two draws from A,B,C with replacement.
2. Create a two-way array for two dice.
3. Find P(two red) without replacement from 4R and 3B.
4. Find P(one of each colour) without replacement.
5. Explain why replacement changes second-branch probabilities.
6. State when outcome counting can replace branch multiplication and when it cannot.

A student lists RR, RB, BR and BB and assigns probability 1/4 to each when drawing from a bag with unequal colours. Explain why the labels do not prove equal likelihood and calculate using branch probabilities instead.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

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.

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.

• I can list complete ordered outcomes.
• I can update probabilities without replacement.
• I can multiply along branches and add target paths.
• I can identify when outcomes are not equally likely.
• I can choose a clear sample-space representation.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

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.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9P02 — Relative Frequency and AND/OR Events
• AC9M9P03 — Repeated Chance Experiments and Probability Simulation

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

Curriculum equivalents for List all outcomes for compound events both with and without...

Mapped skill: list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes

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.