Combined Events Probabilities

Overview

This lesson introduces combined events (using “and” / “or”) and teaches students how to calculate theoretical probabilities using sample spaces and counting outcomes. It builds directly from understanding single-event probability and the idea that all probabilities come from equally likely outcomes.

Learning intentions

• Students will identify what a “combined event” means when it is written using and or or.
• Students will generate outcomes for combined events by using a systematic list or table/grid.
• Students will calculate theoretical probabilities for combined events using the rule: probability = favourable outcomes / total equally likely outcomes.
• Students will check that probabilities are sensible (e.g., between 0 and 1) and consistent with the sample space.

Success criteria

• I can distinguish between “A and B” and “A or B” combined events.
• I can list or count the outcomes for combined events systematically.
• I can calculate a correct theoretical probability for a combined event.
• I can explain how my total outcomes come from the sample space.

Curriculum links

• Probability — generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities.
• Probability — enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams.
• Probability — understand that the probabilities of all possible outcomes sum to 1.
• Probability — use appropriate language to describe outcomes.

Lesson structure (60 minutes)

1. 0–8 min · Do Now (individual recap). Teacher displays 3 quick questions on the board (e.g., “Toss a fair coin: P(heads)?” and a 2-die grid question) and students work silently. Students answer on mini-whiteboards, then teacher cold-calls 2 pupils to explain how they counted outcomes.

2. 8–18 min · I Do (combined events with and/or). Teacher models a combined event scenario: “Roll a fair die. Let A = ‘even’ and B = ‘greater than 3’.” Students watch the sample space listing {1,2,3,4,5,6}, then teacher explicitly shows:

• A and B: outcomes that satisfy both conditions
• A or B: outcomes that satisfy at least one condition (without double-counting) Teacher uses a table/grid to show outcomes for counting (and optionally a small Venn-style overlap explanation).

3. 18–27 min · We Do (class example, two events). Teacher writes one example: “A = ‘multiple of 3’ and B = ‘odd’ when rolling one die. Find P(A and B) and P(A or B).” Students do a short turn and talk to agree the outcomes, then teacher leads the full count with the class (favourable outcomes ÷ total outcomes).

4. 27–45 min · You Do (small group challenge; two questions). Teacher puts pupils into groups of 3–4 and gives a worksheet with two tasks:

• Task 1: Two-event combined probability using a 2-event die setup (equally likely, mutually exclusive outcomes).
• Task 2: A combined event written in words that must be translated into set logic (“and”/“or”), then counted using a table/grid. Students work using systematic listing (or a grid), record the favourable counts, and calculate the probabilities.

5. 45–55 min · Check & Correct (targeted feedback). Teacher circulates and selects 2–3 group methods (one correct and one common mistake, such as double-counting for “or”). Students compare reasoning in a brief turn and talk: “What went wrong and how do we fix it?”

6. 55–60 min · Exit ticket (independent). Teacher gives one final combined event problem on paper: “Roll a die. A = ‘even’, B = ‘prime’. Find P(A and B).” Students submit; teacher uses answers to identify whether pupils can compute “and” correctly.

Resources

• Mini-whiteboards and pens
• Worksheet with two “You Do” combined-event challenges
• Sample space cards (e.g., dice outcomes 1–6) for quick manipulation
• Table/grid templates for listing combined outcomes
• Display slide with worked example of “and/or” using a die
• Dyslexia-friendly alternative print option (larger font, simplified sentence layouts)
• Group roles prompt sheet (e.g., counter, checker, explainer)

Assessment

• Formative: observe Do Now boards for correct counting of outcomes
• Formative: during We Do, listen for correct interpretation of “and” vs “or” and evidence of systematic listing
• Formative: during You Do, check groups’ favourable counts and whether “or” outcomes are double-counted
• Exit ticket: single “and” combined event to confirm mastery

Differentiation

• Support: sentence starters such as “A and B means outcomes that are…” and “For A or B, we include outcomes that…”; provide partially completed tables for the first task.
• Support: offer dyslexia-friendly reading options (larger print, reduced text, key terms highlighted: and/or; equally likely).
• Stretch/extension for advanced learners: add a third event condition (e.g., “A = even, B = prime, find P(A or B) and state whether your answer equals 1/2, 2/3, etc.”) and ask them to justify using counts.
• Additional extension challenge: ask learners to represent the same combined events using a Venn diagram or set notation-style reasoning (in words, not formal notation), then compare to the table count.
• EAL support: allow pupils to demonstrate reasoning with drawings/diagrams; pre-teach “favourable outcomes” as “the outcomes that match the event”.

Success criteria for this lesson (at-a-glance)

• I can translate “A and B” / “A or B” into a set of outcomes.
• I can count favourable outcomes from a sample space/table correctly.
• I can calculate probability as favourable ÷ total and get an answer that makes sense.