Year 9 Mathematics · AC9M9P03

Repeated Chance Experiments and Probability Simulation

Simulation estimates probabilities that are difficult or impossible to determine exactly. A valid simulation must model the original chance process faithfully and use…

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Learning goalsSay it simply

Simulation estimates probabilities that are difficult or impossible to determine exactly. A valid simulation must model the original chance process faithfully and use enough trials to produce stable relative frequencies.

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

  • design a simulation that matches a chance process
  • use digital tools to run repeated trials
  • estimate probability from long-run relative frequency
  • evaluate model validity, trial count and variability
Key conceptTeach from the board

A simulation needs a faithful mapping

Define how random digits, spinners or software outcomes correspond to the real event and preserve the correct probabilities.

Repeat many independent trials

Each trial should reset appropriately. Record the target event consistently.

Relative frequency stabilises with more trials

Small samples can fluctuate widely. Large samples usually provide more stable estimates, although simulation error never becomes literally zero.

Validate the model

When an exact small case is available, compare simulation output with theoretical probability. Check coding, mapping and reset rules.

Worked examplesWe do

Worked examples

AC9M9P03 - Repeated Chance Experiments and Probability Simulation
Example 1

Digit mapping For probability 0.3, map digits 0–2 to event and 3–9 to no event.

Example 2

Compound event Simulate both stages per trial and record whether the target combination occurs.

Example 3

Convergence 20 trials might estimate 0.45; 20,000 trials will usually be more stable around the long-run probability.

Example 4

Design a digit mapping for probability 0.4.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9P03: design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events, and describe results

Victoria: VC2M9P03 — 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 examplesAC9M9P03VC2M9P03 — 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 makes a simulation valid?
Its random mechanism faithfully represents the probabilities and dependence of the real process.
Why use many trials?
Long-run relative frequency is usually more stable.
How can I validate?
Compare with an exact case when possible and audit mapping, reset and event-recording rules.
Practice and reviewReady for practice
  • More trials guarantee the exact answer: They improve stability but still produce random variation.
  • Any random generator automatically models the context: The mapping must reproduce probabilities and dependence.
  • A changed state can be reused when each trial should reset: Define trial boundaries and replacement or reset rules.
  • A simulation result proves a theoretical value: It is empirical evidence or an estimate.
  1. [6 marks] Design a digital simulation for a stated probability problem, specifying random mapping, trial structure and recorded event.
  2. [7 marks] Analyse output from different trial counts and discuss variability and convergence.
  3. [8 marks] Critique a flawed simulation, identify at least two modelling or coding problems, redesign it and state a validation strategy.

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

Simulation estimates probabilities that are difficult or impossible to determine exactly. A valid simulation must model the original chance process faithfully and use enough trials to produce stable relative frequencies.

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

  • design a simulation that matches a chance process
  • use digital tools to run repeated trials
  • estimate probability from long-run relative frequency
  • evaluate model validity, trial count and variability
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