Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 4 - Conditional probability and simulation

Conditional probability and simulation

Use two-way tables, tree diagrams, conditional probabilities and simulations to solve multi-stage chance problems.

Updated 2026-07-26 - 11 min read

Conditional probability and simulation is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

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.

Calculate conditional probability

P(A|B) uses only outcomes within condition B as the denominator. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate conditional probability 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: Condition B restricts the sample space before the proportion for A is calculated.

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

Of 40 students, 24 play sport and 15 of those play soccer. Find P(soccer | sport).

Step 1 - identify the governing idea: P(A|B) uses only outcomes within condition B as the denominator.

Step 2 - apply it to this evidence: The condition restricts the denominator to the 24 sport players.

Result: 15/24

The relationship is visible in the working: The condition restricts the denominator to the 24 sport players. 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:

  • 15/40 — It does not agree with the required relationship: The condition restricts the denominator to the 24 sport players.
  • 24/40 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 9/24 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A table has 30 red items, 12 of them large. Find P(large | red).

Step 1 - identify the governing idea: P(A|B) uses only outcomes within condition B as the denominator.

Step 2 - apply it to this evidence: 12/30 = 0.4.

Result: 0.4

The relationship is visible in the working: 12/30 = 0.4. 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:

  • 12/total items — It does not agree with the required relationship: 12/30 = 0.4.
  • 0.6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2.5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

If A and B are independent, what is P(A|B)?

Step 1 - identify the governing idea: P(A|B) uses only outcomes within condition B as the denominator.

Step 2 - apply it to this evidence: Knowing B does not change A's probability under independence.

Result: P(A)

The relationship is visible in the working: Knowing B does not change A's probability under independence. 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:

  • P(B) — It does not agree with the required relationship: Knowing B does not change A's probability under independence.
  • P(A)+P(B) — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Use probability trees

Multiply probabilities along a branch and add mutually exclusive branches that form the event. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by use probability trees 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: A branch describes successive conditions, so joint probability multiplies; alternative branches are then added.

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

Two independent fair coins are tossed. Find P(HH).

Step 1 - identify the governing idea: Multiply probabilities along a branch and add mutually exclusive branches that form the event.

Step 2 - apply it to this evidence: Multiply 1/2 × 1/2.

Result: 1/4

The relationship is visible in the working: Multiply 1/2 × 1/2. 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:

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

Example 2.2

A bag has 3 red and 2 blue. Two draws without replacement. Find P(red then red).

Step 1 - identify the governing idea: Multiply probabilities along a branch and add mutually exclusive branches that form the event.

Step 2 - apply it to this evidence: 3/5 × 2/4 = 6/20 = 3/10.

Result: 3/10

The relationship is visible in the working: 3/5 × 2/4 = 6/20 = 3/10. 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:

  • 9/25 — It does not agree with the required relationship: 3/5 × 2/4 = 6/20 = 3/10.
  • 6/25 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 1/2 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

For two fair dice, find P(sum 7).

Step 1 - identify the governing idea: Multiply probabilities along a branch and add mutually exclusive branches that form the event.

Step 2 - apply it to this evidence: Six of 36 ordered outcomes sum to 7.

Result: 1/6

The relationship is visible in the working: Six of 36 ordered outcomes sum to 7. 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:

  • 7/36 — It does not agree with the required relationship: Six of 36 ordered outcomes sum to 7.
  • 1/7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 6/12 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Design and interpret conditional simulations

Make random outcomes match each stage's conditional probabilities, run many trials and compare relative frequencies with theoretical expectations. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by design and interpret conditional simulations 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: Simulation produces an estimate with random variation; more trials usually improve stability but do not prove exactness.

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

How can digits 0–9 simulate a 20% event?

Step 1 - identify the governing idea: Make random outcomes match each stage's conditional probabilities, run many trials and compare relative frequencies with theoretical expectations.

Step 2 - apply it to this evidence: Two of ten equally likely digits represent probability 0.2.

Result: Assign two digits to success

The relationship is visible in the working: Two of ten equally likely digits represent probability 0.2. 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:

  • Assign one digit — It does not agree with the required relationship: Two of ten equally likely digits represent probability 0.2.
  • Assign five digits — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Use only even trials — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A simulation has 468 successes in 1200 trials. Estimate probability.

Step 1 - identify the governing idea: Make random outcomes match each stage's conditional probabilities, run many trials and compare relative frequencies with theoretical expectations.

Step 2 - apply it to this evidence: 468/1200 = 0.39.

Result: 0.39

The relationship is visible in the working: 468/1200 = 0.39. 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:

  • 0.468 — It does not agree with the required relationship: 468/1200 = 0.39.
  • 2.564 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 39 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Why compare simulation with theory?

Step 1 - identify the governing idea: Make random outcomes match each stage's conditional probabilities, run many trials and compare relative frequencies with theoretical expectations.

Step 2 - apply it to this evidence: Large systematic differences can reveal coding or assumption problems.

Result: To evaluate model implementation and expected random variation

The relationship is visible in the working: Large systematic differences can reveal coding or assumption problems. 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:

  • To force exact agreement — It does not agree with the required relationship: Large systematic differences can reveal coding or assumption problems.
  • To remove randomness — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • To change observed data — 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. A table has 30 red items, 12 of them large. Find P(large | red).
  2. A bag has 3 red and 2 blue. Two draws without replacement. Find P(red then red).
  3. A simulation has 468 successes in 1200 trials. Estimate probability.

Answers

  1. 0.4 — 12/30 = 0.4.
  2. 3/10 — 3/5 × 2/4 = 6/20 = 3/10.
  3. 0.39 — 468/1200 = 0.39.

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