Relative frequency
successes/trials.
Learning goals: what you will learn
- explain the central idea: Design repeated chance experiments and digital simulations for conditional probability, then interpret long-run results and model limitations.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
Revise relative frequency, multi-step probability, sampling with and without replacement, conditional-probability language and simple digital random-number tools.
Concept teaching
Simulation approximates probability using repeated trials. The model must preserve sample space, conditional structure and whether sampling is with/without replacement. Larger trial counts usually reduce random fluctuation in relative frequency.
Design
Map random outcomes to real outcomes with correct probabilities.
Dependence
Update probabilities after outcomes when the real process changes.
Reliability
Repeat many trials, compare runs and state model assumptions/limitations.
Worked examples
1. Relative frequency
73 successes in 100 trials gives 0.73. In 10,000 trials, a stable estimate is usually closer to underlying probability than a small run, though no run is guaranteed exact.
2. With replacement
Bag 4 red,6 blue; replace after each draw. Each draw keeps P(red)=0.4, so trials are independent.
3. Without replacement
After drawing a red from 4R,6B, second-draw P(red)=3/9; simulation must update the bag state.
4. Small versus large sample
Drawing 1 item without replacement from 10 changes composition noticeably; drawing a tiny sample from a very large population changes probabilities much less.
5. Monty Hall simulation
Simulate initial choice, host opening a losing door using the real host rule, then record stay/switch outcomes. The host rule is essential; random door removal would model a different problem.
6. Birthday simulation
Generate birthdays for n people and record whether any match. Repeating thousands of groups estimates the match probability.
7. Queueing
Simulate random arrivals and service times to estimate waiting-time distribution; output depends on assumptions about those input processes.
8. Insurance risk
Simulate many policyholders using estimated claim probabilities and claim amounts to explore total-cost variability, not just one expected value.
9. Infection model
Assign infection risk conditional on contacts/status, run many virtual populations and summarise range of outcomes. It is a model, not a medical forecast without validated inputs.
10. Convergence check
Track estimate after 100,1000,10000 trials. If values stabilise around a range, confidence in simulation estimate improves; systematic modelling errors are not fixed by more trials.
Common misconceptions and corrections
- Using equal random outcomes for unequal real probabilities.
- Forgetting to update probabilities without replacement.
- Believing more trials fix a wrong model.
- Reporting one simulation run as exact probability.
- Changing a counter-intuitive problem's rules during simulation.
Guided practice
Simulate two draws without replacement from 3 red and 2 blue counters. Describe how the second-draw probabilities must be updated, what trials are retained for estimating P(second red | first blue), and why 10,000 trials should be more stable than 20.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- Find relative frequency 420/600.
- Does replacement make repeated bag draws independent if composition returns exactly?
- Bag 5R,5B: after red without replacement, P(next red)?
- Why does large population make replacement effect smaller?
- What rule must Monty Hall simulation preserve?
- How simulate birthdays?
- Why run queue simulation many times?
- What inputs matter in insurance simulation?
- Why are infection-simulation outputs conditional on assumptions?
- What cannot be fixed by increasing trial count?
Check answers and explanations
- 0.7.
- Yes, if each draw resets same probabilities.
- 4/9.
- Removing a few items changes proportions by a smaller amount.
- The host deliberately opens an eligible losing door and offers switch.
- Generate random day values for n people and check duplicates repeatedly.
- To estimate variability/distribution, not one random outcome.
- Claim probabilities, claim sizes and dependence/assumptions.
- Changing inputs/rules changes predicted outcomes.
- Systematic model/design error.
Reasoning and problem-solving task
Design a simulation for a bag with 4 red, 3 blue and 2 green counters, drawing two without replacement. Estimate P(second is red | first is blue). Specify the digital representation, update rule, trial filtering and how you would judge whether the estimate is stable.
Important questions and answers
- Find relative frequency 420/600.
- 0.7.
- Does replacement make repeated bag draws independent if composition returns exactly?
- Yes, if each draw resets same probabilities.
- Bag 5R,5B: after red without replacement, P(next red)?
- 4/9.
Assessment-style questions
Design a digital simulation to estimate a conditional probability for drawing two cards without replacement. Specify the sample-space model, condition filter, statistic recorded, trial count and one validation check against an exact small case. [7 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- model the real rules first
- filter by the condition before dividing
- update probabilities when no replacement occurs
- more trials reduce random noise, not model bias
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Have students predict before simulation, then compare small and large runs. Include one deliberately flawed model and ask students to diagnose why more trials do not repair it.
For parents and carers
Use simple cards or coloured objects first. Ask what changes after an item is removed and not replaced, then connect that idea to the digital simulation.
Curriculum alignment
Australian Curriculum: AC9M10P02 — Year 10 Probability. Design repeated chance experiments and digital simulations for conditional probability, then interpret long-run results and model limitations.
Victoria: VC2M10P01 and VC2M10P02 — Level 10 Probability; the first explicitly includes conditional-probability simulation
NSW: Stage 5 Core — Probability; Path — Further probability
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10P02 | VC2M10P01 and VC2M10P02 — Level 10 Probability; the first explicitly includes conditional-probability simulation | Stage 5 Core — Probability; Path — Further probability |
| Guided and independent practice | Builds fluency and application for AC9M10P02 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10P02: design and conduct repeated chance experiments and simulations using digital tools to model conditional probability and interpret results.
- E1: compare different sample sizes with/without replacement and when difference becomes negligible. Examples 2–4.
- E2: recognise dependency affects probability calculation. Example 3.
- E3: use simulations for counter-intuitive chance situations such as three-door/birthday problems. Examples 5–6.
- E4: identify real-life simulation decision uses such as supply-demand, insurance and queueing. Examples 7–8.
- E5: use simulation to predict likely infection numbers for flu/virus contexts. Example 9, with explicit model limitations.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
US closest CCSS HSS-CP and HSS-IC simulation standards; UK GCSE conditional probability/relative frequency; NSW Stage 5 and Victorian Level 10 probability; comparable Canadian/NZ simulations.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Monty Hall Problem - Numberphile
Numberphile — Explaining a counter-intuitive conditional-probability problem that can be tested by simulation.
As you watch: What information does the host have when opening a losing door?
Load video player Loads YouTube in this lesson. See the video notice below.
Try it: Simulate 60 games where the host always opens a losing unchosen door and always offers a switch. Compare staying and switching, then discuss sample variation.
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Curriculum equivalents: Victoria, NSW and international
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 model conditional probability and interpret 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.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10P02 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10P01 + VC2M10P02 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-PRO-P-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
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.
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Topic reference: AC9M10P02 — Conditional Probability Experiments and Simulations — AC9M10P02
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