Probability Models and Outcomes

Overview

In this second lesson of the nine-lesson unit Probability, Patterns, and Uncertainty, students develop formal probability language by identifying sample spaces and assigning probabilities to equally likely outcomes. They connect physical models, lists, tables, spinners, dice and diagrams with symbolic representations such as fractions.

Learning intentions

• WALT identify the sample space for a chance experiment.
• WALT recognise when outcomes are equally likely.
• WALT calculate and justify simple theoretical probabilities.
• WALT represent probability situations using lists, tables, diagrams and physical models.

Success criteria

• I can create a complete sample space without missing or repeating outcomes.
• I can explain whether outcomes are equally likely.
• I can calculate a theoretical probability as a fraction.
• I can justify my answer using a model, diagram or clear reasoning.

Curriculum links

• Mathematics and Statistics — Probability: exploring chance, outcomes, sample spaces and theoretical probability.
• Mathematics and Statistics — representing mathematical ideas using symbolic, visual and physical representations.
• Mathematics and Statistics — communicating reasoning, justifying conclusions and connecting representations.
• Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.

Lesson structure (60 minutes)

1. 0–7 min · Hook and prior knowledge. Teacher opens with the hook and learning intention slides and displays two questions: “Is rolling a 6 more likely than rolling an odd number?” and “How could we prove it?” Students make an individual prediction, then explain their thinking to a partner using words such as outcome, chance and likely.

2. 7–17 min · Model key ideas. Teacher uses a fair six-sided die and the sample space teaching slides to model the experiment “roll a die once”. Explicitly define experiment, outcome, sample space, equally likely and theoretical probability. Build the sample space {1, 2, 3, 4, 5, 6}, then calculate (P(6)=1/6) and (P(\text{odd})=3/6=1/2). Students copy or annotate the model and identify the favourable outcomes in each example.

3. 17–30 min · Guided representations. Teacher demonstrates how the same experiment can be represented as a list, outcome table, labelled die diagram and fraction calculation, referring to the representation examples. Distribute the probability models worksheet. Students complete the first guided questions, including a coin toss, a spinner with equal sections and a two-colour counter experiment. Pause to check that students distinguish the sample space from the favourable outcomes.

4. 30–44 min · Pair investigation. Teacher gives each pair a die, coin and counters, and directs students to complete the worksheet’s investigation tasks. Students construct complete sample spaces for one-step and two-step experiments, represent at least one situation with a table or simple diagram, and calculate probabilities such as (P(\text{red})), (P(\text{sum of 7})) or (P(\text{at least one head})), according to the task selected. Partners must explain how they know outcomes are equally likely.

5. 44–53 min · Reasoning and discussion. Teacher displays the reasoning prompts in the discussion and misconception slides: “A student says (P(\text{rolling a 1 or 2})=1/6). Do you agree?” and “A spinner has four equal sections labelled A, A, B and C. Are the letters equally likely?” Students compare answers, identify errors and justify corrections using a sample space and favourable-outcome count. Invite several pairs to share different representations.

6. 53–60 min · Exit check and review. Teacher displays the plenary and exit-ticket slide. Students complete the final worksheet question independently: “A bag contains three equally likely blue outcomes and two equally likely yellow outcomes. Create the sample space and find (P(\text{yellow})). Explain why.” Students then self-assess against the success criteria and hand in their response.

Resources

• the probability models and outcomes slide deck
• the probability models worksheet
• Fair six-sided dice, one per pair
• Coins, one per pair
• Two-colour counters or small coloured objects
• Mini-whiteboards and pens
• Calculators
• Visualiser or board space
• Optional speech-to-text device or teacher-recorded instructions

Assessment

• During modelling, ask students to identify the sample space and favourable outcomes before calculating.
• Circulate during pair work, checking completeness, correct use of “equally likely” and the connection between models and fractions.
• Use the independent exit response to identify students needing a follow-up lesson on sample spaces, and students ready for compound-event work.

Differentiation

• Support students with physical dice, coins and counters before moving to symbols; provide partially completed sample spaces and the worked example on the worksheet.
• Give one-step instructions, boxed key terms, large accessible diagrams and sentence starters such as “The sample space is…” and “The outcomes are equally likely because…”.
• Offer calculator use, speech-to-text, oral rehearsal with a partner and teacher-recorded explanations. Read instructions aloud and avoid dense blocks of text for dyslexic learners.
• For students requiring additional support, reduce the number of outcomes and work with a teacher-led example before returning to the pair task.
• Advanced learners design a fair spinner or chance game, then prove it is fair by listing the sample space, showing the probability of each outcome and explaining why the outcomes are equally likely.

Extension

• Design a spinner with at least four labelled regions for a target set of probabilities, such as (1/2), (1/4) and (1/4).
• Write a short proof explaining why the game is fair, then challenge another pair to test whether the model matches the intended probabilities.