Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 4 - Compound events, relative frequency and simulation

Compound events, relative frequency and simulation

List compound outcomes, assign probabilities and compare theoretical models with repeated simulations.

Updated 2026-07-26 - 11 min read

Compound events, relative frequency and simulation is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Represent compound outcomes

With replacement, later probabilities stay the same; without replacement, available outcomes and denominators change.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by represent compound outcomes and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Replacement restores the original composition; without it, later branches depend on earlier outcomes.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Two fair coin tosses: number of ordered outcomes?

Step 1 - identify the governing idea: A compound sample space records each ordered outcome once with probabilities that reflect the experiment.

Step 2 - apply it to this evidence: HH, HT, TH and TT.

Result: 4

The relationship is visible in the working: HH, HT, TH and TT. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 3 — It does not agree with the required relationship: HH, HT, TH and TT.
  • 2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Draw 2 from 3 red and 2 blue without replacement. P(R then R)?

Step 1 - identify the governing idea: A compound sample space records each ordered outcome once with probabilities that reflect the experiment.

Step 2 - apply it to this evidence: One red and one total item are removed after first red.

Result: 3/4×2/3=1/2

The relationship is visible in the working: One red and one total item are removed after first red. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 3/4×3/4=9/16 — It does not agree with the required relationship: One red and one total item are removed after first red.
  • 2/4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 3/8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A die then coin: number of equally likely ordered outcomes?

Step 1 - identify the governing idea: A compound sample space records each ordered outcome once with probabilities that reflect the experiment.

Step 2 - apply it to this evidence: 6 die outcomes×2 coin outcomes.

Result: 12

The relationship is visible in the working: 6 die outcomes×2 coin outcomes. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 8 — It does not agree with the required relationship: 6 die outcomes×2 coin outcomes.
  • 6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 36 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Calculate AND and OR probabilities

For inclusive OR, P(A∪B)=P(A)+P(B)−P(A∩B). Independence is needed before multiplying separate event probabilities directly.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate and and or probabilities and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Subtract overlap for inclusive OR, and identify whether events are independent before multiplying.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Fair die: P(even or greater than 4), inclusive OR?

Step 1 - identify the governing idea: AND means intersection; inclusive OR includes outcomes in either event including overlap; exclusive OR excludes overlap.

Step 2 - apply it to this evidence: Even={2,4,6}; >4={5,6}; union={2,4,5,6}.

Result: 2/3

The relationship is visible in the working: Even={2,4,6}; >4={5,6}; union={2,4,5,6}. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 5/6 — It does not agree with the required relationship: Even={2,4,6}; >4={5,6}; union={2,4,5,6}.
  • 1/2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Fair die: P(even and greater than 4)?

Step 1 - identify the governing idea: AND means intersection; inclusive OR includes outcomes in either event including overlap; exclusive OR excludes overlap.

Step 2 - apply it to this evidence: Only outcome 6 satisfies both.

Result: 1/6

The relationship is visible in the working: Only outcome 6 satisfies both. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2/3 — It does not agree with the required relationship: Only outcome 6 satisfies both.
  • 1/2 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 5/6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Two independent events have P(A)=0.4, P(B)=0.5. P(A and B)?

Step 1 - identify the governing idea: AND means intersection; inclusive OR includes outcomes in either event including overlap; exclusive OR excludes overlap.

Step 2 - apply it to this evidence: Independence permits 0.4×0.5.

Result: 0.2

The relationship is visible in the working: Independence permits 0.4×0.5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 0.9 — It does not agree with the required relationship: Independence permits 0.4×0.5.
  • 0.45 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.1 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Simulate and interpret relative frequency

Simulation validity depends on matching the intended sample space, replacement and dependence conditions.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by simulate and interpret relative frequency and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Random variation is expected; evaluate convergence and model design over many trials.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Event occurs 126 times in 300 trials. Relative frequency?

Step 1 - identify the governing idea: Relative frequency=event count/trials. Larger well-designed runs tend to give more stable estimates of theoretical probability.

Step 2 - apply it to this evidence: 126/300=0.42.

Result: 0.42

The relationship is visible in the working: 126/300=0.42. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 0.238 — It does not agree with the required relationship: 126/300=0.42.
  • 1.26 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 174 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Theoretical probability 0.4; simulation gives 0.55 after 20 trials. Best conclusion?

Step 1 - identify the governing idea: Relative frequency=event count/trials. Larger well-designed runs tend to give more stable estimates of theoretical probability.

Step 2 - apply it to this evidence: Twenty trials can vary substantially.

Result: The difference may be short-run variation; run more valid trials

The relationship is visible in the working: Twenty trials can vary substantially. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • The theory is disproved — It does not agree with the required relationship: Twenty trials can vary substantially.
  • The model is certainly biased — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.55 must become 0.4 next trial — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

To simulate drawing without replacement, best digital design?

Step 1 - identify the governing idea: Relative frequency=event count/trials. Larger well-designed runs tend to give more stable estimates of theoretical probability.

Step 2 - apply it to this evidence: Dependence must match the real experiment.

Result: Remove the selected item before the next draw in each trial

The relationship is visible in the working: Dependence must match the real experiment. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Reset before every draw — It does not agree with the required relationship: Dependence must match the real experiment.
  • Use one coin regardless of composition — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Ignore item counts — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Draw 2 from 3 red and 2 blue without replacement. P(R then R)?
  2. Fair die: P(even and greater than 4)?
  3. Theoretical probability 0.4; simulation gives 0.55 after 20 trials. Best conclusion?

Answers

  1. 3/4×2/3=1/2 — One red and one total item are removed after first red.
  2. 1/6 — Only outcome 6 satisfies both.
  3. The difference may be short-run variation; run more valid trials — Twenty trials can vary substantially.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources