GCSE Maths: Probability — Deep Revision Workbook

Inside GCSE Maths: Probability — Deep Revision Workbook

place events on the 0–1 probability scale, find theoretical probabilities of single events, use the ‘total = 1’ rule, calculate experimental probability, relative frequency and expected frequency, read sample space and two-way tables, apply the AND (×) and OR (+) rules, and work through tree and Venn diagrams with set notation. About 45 minutes.

Learning objectives

• placing events on the 0–1 probability scale and finding the theoretical probability of single events, using the fact that all probabilities sum to 1;
• calculating experimental probability and relative frequency, and using expected frequency = probability × number of trials;
• reading sample space diagrams and two-way tables, and applying the AND (×) rule for independent events and the OR (+) rule for mutually exclusive events;
• working with higher-tier tree diagrams (with and without replacement) and Venn diagrams and set notation, including union (∪) and intersection (∩).

Activities in this book

1. Section 1 — The Probability Scale & Single Events

   every probability sits between 0 (impossible) and 1 (certain). For a fair situation the theoretical probability is the number of ways an event can happen divided by the total number of equally likely outcomes. The probabilities of all the outcomes always add up to 1.

2. Section 2 — Experimental Probability & Expected Frequency

   experimental (relative) frequency = number of times an event happened ÷ total number of trials. Expected frequency = probability × number of trials. With many independent trials under the same conditions, relative frequency tends to settle near the true probability; it need not get closer after every trial.

3. Section 3 — Sample Space, Two-Way Tables & the AND/OR Rules

   a sample space lists every equally likely outcome. For independent events use the AND rule and multiply the probabilities; for mutually exclusive events use the OR rule and add them.

4. Section 4 — Tree Diagrams & Venn Diagrams (Higher)

   follow the branches of a tree diagram and multiply along them. ‘With replacement’ means the probabilities stay the same; ‘without replacement’ means the second denominator drops by one. For a Venn diagram, the overlap is the intersection (both) and the whole shaded region is the union (at least one).

5. Section 5 — Extended Exam-Style Tree Diagram

   a multi-branch tree diagram sets out every stage of a problem. Multiply along a path to find the probability of that combined outcome, and add the probabilities of the separate paths that satisfy the event. The branches leaving any point must add up to 1.

Suggestions for teaching

• Language first. Fix the vocabulary early — impossible, unlikely, even chance, likely, certain — and link each to a value between 0 and 1.
• Fractions in lowest terms. Model writing a probability as a fraction of equally likely outcomes and simplifying, e.g. 4/36 = 1/9.
• Sample space then table. Build the 6×6 grid of two-dice totals so learners see why there are 36 outcomes before reading a two-way table.
• Branch by branch. For tree diagrams, insist each set of branches sums to 1, and that we multiply along a path and add between paths.
• Venn regions. Fill the intersection first, then the ‘only’ regions, then ‘neither’ — and connect union and intersection to the words ‘or’ and ‘and’.