Theoretical Probability

Overview

This is lesson 3 of 9 in Probability, Patterns, and Uncertainty. Students build on probability language and sample spaces by calculating theoretical probability as favourable outcomes divided by total equally likely outcomes. They apply the model to coins, dice, cards and spinners, then communicate answers as fractions, decimals and percentages.

Learning intentions

• WALT calculate theoretical probability using favourable outcomes over total outcomes.
• WALT express probabilities as fractions, decimals and percentages.
• WALT interpret probability in context and check that answers lie between 0 and 1.
• WALT select an appropriate model for a probability problem.

Success criteria

• I can identify the total number of possible outcomes and the favourable outcomes.
• I can calculate a probability accurately and convert it between forms.
• I can explain what my answer means in the given context.
• I can check that my probability is between 0 and 1.

Curriculum links

• Mathematics and Statistics — Mathsteasers: higher-order thinking questions that challenge advanced learners and deepen understanding.
• Mathematics and Statistics — Mathsteasers: aligned challenge tasks that extend textbook learning.
• Statistical thinking: using models to describe chance and uncertainty, communicate conclusions and evaluate whether answers are reasonable.
• Key competencies: thinking; using language, symbols and texts; managing self; participating and contributing.

Lesson structure (60 minutes)

1. 0–6 min · Hook and retrieval. Open with the probability hook and retrieval slides and ask: “Is rolling a 6 more likely than rolling an even number?” Students make a prediction, justify it using probability language, and recall that probabilities range from impossible to certain.

2. 6–17 min · Explicit teaching. Use the worked-example slides to model [ P(\text{event})=\frac{\text{number of favourable outcomes{\text{total number of equally likely outcomes. ] Work through a coin, a six-sided die and a four-colour spinner. Colour-code the numerator and denominator, show each answer as a fraction, decimal and percentage, and model checking that (0\leq P\leq1). Students annotate their copy or use the enlarged worked example rather than copying extensive notes.

3. 17–23 min · Guided practice and checks. Display the questions from the guided-practice slides: “What is the probability of an odd number on a die?” and “What is the probability of drawing a heart from a standard pack?” Students show answers on mini-whiteboards. Pause after each question to check whether they identified the complete sample space, counted favourable outcomes correctly and simplified where appropriate.

4. 23–43 min · Paired problem solving. Distribute the theoretical probability practice worksheet to mixed-ability pairs. Students solve progressively challenging problems involving coins, dice, cards and spinners. For each question they identify the model, list or represent the outcomes, calculate the probability, convert it into another form, and write one sentence interpreting it. Circulate, ask “What is the total sample space?” and “How do you know the outcomes are equally likely?”, and use targeted questioning with students who need support.

5. 43–52 min · Extension and discussion. Invite confident students to attempt the reverse problems on the challenge and reverse-problem section, such as finding the missing number of favourable outcomes when (P(\text{event})=\frac{3}{8}) from 32 equally likely outcomes. Select two contrasting solutions for students to compare. Pairs explain how the model changes when the total number of outcomes changes and identify any answer that is not a valid probability.

6. 52–60 min · Plenary and exit check. Use the plenary and exit-ticket slides to revisit the formula and ask students to complete this exit response: “A bag contains 5 red, 3 blue and 2 green counters. Find and interpret (P(\text{blue})), giving your answer as a fraction, decimal and percentage.” Students add one sentence explaining how they checked the answer. Collect responses to inform the next lesson.

Resources

• the complete probability slide deck
• the theoretical probability practice worksheet
• Mini-whiteboards, pens and erasers
• Coins, six-sided dice, playing cards and optional equal-section spinners
• Fraction walls and probability strips
• Enlarged worked examples or visualiser
• Calculators for checking decimal and percentage conversions
• Coloured pens or highlighters

Assessment

• Check predictions and mini-whiteboard responses for understanding of sample spaces, favourable outcomes and probability notation.
• Confer with pairs during worksheet work, recording students who need support with counting outcomes, simplifying fractions or converting representations.
• Use the exit response to assess model selection, calculation, interpretation and the (0)–(1) reasonableness check.

Differentiation

• Provide fraction walls, probability strips and a structured problem-solving template with prompts: “Total outcomes… Favourable outcomes… Calculation… Equivalent forms… Interpretation… Check…”.
• Use consistent notation, enlarged worked examples, colour-coded numerator and denominator, clear sans-serif text, increased spacing and short instructions for dyslexia-friendly access. Allow students to annotate, discuss or photograph examples instead of copying.
• Pair students strategically and provide concrete coins, dice and cards before moving to symbolic calculations. Read questions aloud, pre-teach key terms and check understanding frequently.
• Reduce the number of questions without reducing the mathematical thinking for students requiring additional support. Extension students solve reverse problems, compare different models and create a probability question with a specified answer.

Extension

• Find two different probability situations with (P(\text{event})=\frac{1}{4}), then explain how their sample spaces differ.
• Design a spinner or card situation with a probability of (0.35); state the favourable and total outcomes and justify the design.
• Explain why a probability greater than 1 or less than 0 cannot describe a chance event.