Chance in Action

Overview

Students investigate how experimental probability approaches theoretical probability as the number of trials increases. Working in pairs, they simulate repeated chance experiments, visualise results, compare predictions with outcomes, and then independently explain the reliability of their conclusions.

Learning intentions

• WALT design and carry out a probability simulation.
• WALT compare experimental and theoretical probabilities.
• WALT use tables, fractions and graphs to visualise probability results.
• WALT make and justify predictions using evidence from data.

Success criteria

• I can identify the possible outcomes and calculate a theoretical probability.
• I can record simulation results accurately and represent them in a suitable graph.
• I can compare experimental and theoretical probabilities using mathematical language.
• I can explain why larger numbers of trials generally produce more reliable estimates.

Curriculum links

• Mathematics and Statistics: probability investigations involving chance, outcomes, theoretical probability and experimental probability.
• Mathematics and Statistics: representing and interpreting data from simulations to identify patterns and variation.
• Mathematics and Statistics: communicating mathematical reasoning and evaluating the reliability of predictions.
• Key competencies: thinking; managing self; participating and contributing; relating to others.

Lesson structure (60 minutes)

1. 0–7 min · Hook and prediction. Display a picture of a balanced coin and ask, “If a coin is tossed 20 times, must it land heads exactly 10 times?” Use the hook and prediction slide to collect quick predictions, then invite students to justify whether they expect 10 heads, fewer, or more. Students record an individual prediction and briefly compare it with a partner.

2. 7–15 min · Explicit teaching. Use the probability teaching slides to revise outcome, sample space, theoretical probability and experimental probability. Model a coin-toss simulation: theoretical probability of heads is (1/2), while an experiment may produce (8/20) or (13/20). Explain that variation is expected, and that repeated trials can make an estimate more stable. Students calculate the theoretical probability and identify what information should be recorded.

3. 15–32 min · Paired simulation. Distribute the probability simulation investigation to pairs. Each pair uses a digital random-number generator or two-colour counters to simulate 20, 50 and then 100 coin tosses, recording the cumulative number of heads, the experimental probability, and the difference from (1/2). One student operates the simulation while the other records; they swap roles after each trial set. Pause midway to ask pairs to predict whether the difference will increase, decrease or stay similar as trials continue.

4. 32–42 min · Visualise and compare. Students complete the table and graph on the probability simulation investigation, plotting number of trials against experimental probability. Display the graphing and discussion slides to model suitable scales and labels. Pairs compare their graphs with another pair, identifying similarities, differences and any unusual results. Emphasise that two fair simulations will not usually produce identical graphs.

5. 42–52 min · Individual reasoning task. Students complete the independent questions on the probability simulation investigation without discussing answers. They explain which of three claims is most convincing: “20 trials are enough to know the exact probability”, “100 trials usually give a more reliable estimate”, or “every experiment should match the theoretical probability”. Require evidence from their graph and at least one comparison using fractions or decimals.

6. 52–60 min · Plenary and exit check. Use the plenary and exit-question slide to revisit the opening prediction. Students complete the final exit response on the worksheet: “A simulation is useful because …” and “My evidence shows …”. Invite two students to share contrasting results, then summarise that experimental probability varies, while increasing the number of trials generally improves the estimate of theoretical probability.

Resources

• Teacher computer and projector
• the probability simulation and discussion deck
• the probability simulation investigation
• Digital random-number generator, spreadsheet or simulation tool
• Two-colour counters or coins for pairs who cannot access a device
• Pencils, rulers and coloured pens
• Board or shared display for collecting predictions

Assessment

• Listen to partner discussions for correct use of outcome, theoretical probability, experimental probability, variation and reliability.
• Check pair tables and graphs for accurate recording, appropriate scales, correct fractions or decimals, and a clear comparison with (1/2).
• Use the independent explanation and exit response to assess whether students can connect the number of trials with the reliability of a prediction.

Differentiation

• Support students with a partially completed model table, a displayed formula such as experimental probability = favourable outcomes ÷ total trials, and sentence starters: “The theoretical probability is …”, “Our result differs because …”, and “As the number of trials increases …”.
• Pair students strategically and provide counters or a teacher-led simulation for students who find digital tools difficult. Allow decimal, fraction or percentage representations, provided the meaning is clear.
• Extend confident students by asking them to run two separate 100-trial simulations, compare their variability, and explain why neither result proves that the coin is fair.
• Support EAL learners with visual examples, repeated oral modelling and a small word bank. Provide graph axes and labels in advance for students needing fine-motor, processing or organisational support.