Repeated Chance Experiments and Probability Simulation — AC9M9P03

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

Recall relative frequency, compound events, random-number generation and basic data recording.

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.

• 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. Design a digit mapping for probability 0.4.
2. Describe a simulation of a two-stage event for 100 trials.
3. Compare expected stability for 20, 200 and 2000 trials.
4. Validate against an exact probability where possible.

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. Design a simulation for a 15% event using random integers 1–100.
2. Describe how to simulate drawing with replacement.
3. Explain what must change for without replacement.
4. Compare estimates from two different trial counts.
5. Identify a biased mapping in a supplied simulation.
6. State one validation check for a digital simulation.

A simulation with one million trials disagrees strongly with a simple exact probability. Explain why increasing the trial count further is not the first response and list model or coding checks that should be investigated first.

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

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.

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.

• I can map random outcomes faithfully.
• I can define one complete trial.
• I can use relative frequency as an estimate.
• I can explain why larger samples are more stable.
• I can audit a simulation before trusting its result.

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

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.

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

• AC9M9P01 — Two-Step Compound Events and Sample Spaces
• AC9M9P02 — Relative Frequency and AND/OR Events

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

Curriculum equivalents for Design and conduct repeated chance experiments and simulations, using digital...

Mapped skill: design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events, and describe results

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.