Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 0 - Single-stage probability and simulations

Single-stage probability and simulations

List sample spaces, assign probabilities, predict relative frequencies and explain simulation variation.

Updated 2026-07-24 - 7 min read

List sample spaces, assign probabilities, predict relative frequencies and explain simulation variation. This draft teaching resource is aligned to the cited Year 7 content descriptions and still requires named human curriculum approval.

Keep the investigative question connected to the data, display and conclusion so the final claim cannot exceed the evidence.

1. Sample spaces and probabilities

A sample space contains every possible outcome once. For equally likely outcomes, probability is favourable outcomes divided by total outcomes.

Decision check: Apply this lesson's rule to the worked example: Check that probabilities are between 0 and 1 and match the complete sample space. Identify the exact value, label or property that satisfies the rule.

2. Predict and observe relative frequency

Predicted frequency is probability times trial count. Observed relative frequency is actual successes divided by trials. Label both values before comparing them.

Decision check: Apply this lesson's rule to the worked example: Keep prediction and observation as different quantities. Identify the exact value, label or property that satisfies the rule.

3. Run and interpret simulations

Random processes do not usually match theoretical probability exactly in a small run. Larger trial counts tend to reduce short-run fluctuation.

Decision check: Apply this lesson's rule to the worked example: A difference from prediction is not automatically proof that the model is unfair. Identify the exact value, label or property that satisfies the rule.

Worked examples with complete reasoning

Worked example 1

Question: Find the probability of rolling a 4 on a fair six-sided die. Give a decimal to three decimal places if needed.

  1. Probability = favourable outcomes ÷ total equally likely outcomes.
  2. 1 ÷ 6 = 0.167.

Answer: 0.167

Check: Count the sample space and favourable outcomes.

Worked example 2

Question: A simulation predicted 50 occurrences but observed 46 in 100 trials. Which explanation is best?

  1. Theoretical probability describes a long-run pattern.
  2. Finite experiments vary by chance, even when the model is fair.

Answer: Random variation can make observed results differ from predictions.

Check: Compare over more trials before treating a difference as evidence of a flawed model.

Worked example 3

Question: An event has probability 0.25. Predict its frequency in 80 trials.

  1. Predicted frequency = probability × trials.
  2. 0.25 × 80 = 20.

Answer: 20

Check: A prediction is an expected count, not a guarantee.

Method map for this topic

  1. Sample spaces and probabilities: Check that probabilities are between 0 and 1 and match the complete sample space.
  2. Predict and observe relative frequency: Keep prediction and observation as different quantities.
  3. Run and interpret simulations: A difference from prediction is not automatically proof that the model is unfair.

Common mistakes and how to correct them

  • single event probability: Not yet. Count the sample space and favourable outcomes. Recheck the single event probability step, then try again.
  • sample space: Not yet. A sample space is a set of outcomes, not a list of probabilities. Recheck the sample space step, then try again.
  • relative frequency: Not yet. A prediction is an expected count, not a guarantee. Recheck the relative frequency step, then try again.
  • observed frequency: Not yet. Use the number of trials as the denominator. Recheck the observed frequency step, then try again.
  • random variation: Not yet. Compare over more trials before treating a difference as evidence of a flawed model. Recheck the random variation step, then try again.
  • simulation size: Not yet. Compare trial counts as well as observed proportions. Recheck the simulation size step, then try again.

Guided interactive lesson studio

The Learn lesson for this topic includes a structured action rather than asking you only to read an explanation or select an answer.

Fair-coin simulation checkpoints

Reveal the fixed fair-coin trial sequence one result at a time. At 10, 20 and 40 trials, record the cumulative number and relative frequency of heads before comparing with the theoretical probability.

You will complete: Cumulative heads counts; Cumulative relative frequencies; Explain the changing gap from 0.5.

Accessible route: Each outcome is announced as the letter H or T and added to a text log. Learners can use a Reveal next button and labelled number fields.

Evidence boundary: the automatically checkable fields give immediate formative feedback. Explanation and reflection fields remain formative and require rubric or teacher review; this action cannot create mastery evidence by itself.

Practice

  1. Find the probability of rolling a 4 on a fair six-sided die. Give a decimal to three decimal places if needed.
  2. Find the probability of selecting one of 3 red counters from 8 equally likely counters. Give a decimal to three decimal places if needed.
  3. A simulation predicted 50 occurrences but observed 46 in 100 trials. Which explanation is best?
  4. A simulation predicted 25 occurrences but observed 31 in 50 trials. Which explanation is best?
  5. An event has probability 0.25. Predict its frequency in 80 trials.
  6. An event has probability 0.6. Predict its frequency in 50 trials.

<details> <summary>Answers and reasoning</summary>

  1. 0.167 — Probability = favourable outcomes ÷ total equally likely outcomes. 1 ÷ 6 = 0.167.
  2. 0.375 — Probability = favourable outcomes ÷ total equally likely outcomes. 3 ÷ 8 = 0.375.
  3. Random variation can make observed results differ from predictions. — Theoretical probability describes a long-run pattern. Finite experiments vary by chance, even when the model is fair.
  4. Random variation can make observed results differ from predictions. — Theoretical probability describes a long-run pattern. Finite experiments vary by chance, even when the model is fair.
  5. 20 — Predicted frequency = probability × trials. 0.25 × 80 = 20.
  6. 30 — Predicted frequency = probability × trials. 0.6 × 50 = 30.

</details>

Checklist and vocabulary

| Skill | Observable evidence | | --- | --- | | assign probabilities simple events place | List a single-stage sample space, assign probabilities to outcomes and place them on a probability scale. | | compare theoretical probability results small | Compare theoretical probability with observed simulation results and explain trial-count effects. | | assign probability predict frequency | Assign probabilities to equally likely outcomes and predict relative frequencies. |

Retrieval prompts

  • State the defining relationship for each method above without looking.
  • Give one example where two visually similar representations mean different things.
  • Diagnose one misconception from the list and explain the first corrective step.
  • Name a check that is independent of simply repeating the same calculation.

Teach-back and self-check

Close the worked examples and explain single-stage probability and simulations as if you were helping another Year 7 student. Begin by naming the clue that tells you which relationship, representation or operation is needed. Then show one complete example with labels and units where they matter. At each line, explain what stayed equivalent and why the next step is allowed.

Use this four-part check before calling the method secure:

  1. Identify: What information is given, and what must be found or justified?
  2. Represent: Which number line, diagram, table, expression, graph or organised list makes the relationship visible?
  3. Reason: Which rule connects the representation to the required answer?
  4. Verify: Can you check by estimation, substitution, an inverse operation, a second representation or the original context?

If the explanation depends on “I just knew”, return to the method map and name the missing decision. A durable method is one you can explain, check and use in a changed context after a delay.

Transfer task

Design a small investigation or simulation using single-stage probability and simulations. Record the data or outcomes in an accessible text format, analyse them, and write a conclusion that does not claim more than the evidence supports.

Sources