---
title: "Venn Diagrams and Counting Outcomes"
description: "Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined eve"
canonical: https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H
source: https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H.md
retrieved: 2026-10-19
---

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# Venn Diagrams and Counting Outcomes

Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events

Subject: Mathematics · Area: Probability · Ages 12 to 13
Page: https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H

## Ready when they can

- Draw a two-circle Venn diagram to sort 30 students by whether they like football, like cricket, or like both
- Shade the intersection A ∩ B and the union A ∪ B on a Venn diagram and explain what each region represents
- Use a completed Venn diagram to calculate P(A), P(B), P(A ∩ B), and P(A ∪ B)

## Lesson: Counting with Venn diagrams

You've already drawn Venn diagrams to sort things into groups. Now you'll use the same diagram to count real groups of people, like students who play different sports, and to work out how likely you are to pick one of them at random.

*(drawing: These four bars are the four regions of a Venn diagram for football and cricket: only football, only cricket, both, and neither.)*

**Example.** A class of 30 students is asked about football and cricket. 18 like football, 15 like cricket, and 8 like both. The 8 who like both go in the middle, the intersection. Only football is 18 minus 8, which is 10. Only cricket is 15 minus 8, which is 7. Adding 10 + 7 + 8 gives 25 students who like at least one sport, so 30 minus 25 leaves 5 who like neither. The probability of picking a student who likes both sports is 8 out of 30, so P(A ∩ B) = 8/30.

**Tip.** When two circles overlap, count the middle part once, not twice. Add only-A, only-B, and both together to get the total for A or B.

**Recap.** The intersection is the overlap counted once, the union is everyone in A, B, or both, and a region's count divided by the total gives its probability.

## Practice

16 questions on this page, each with its working shown.

## Opens up

- [Sets & Venn Diagrams](https://lightmysky.com/learn/mathematics/sets-and-venn-diagrams-mt_1YwOCMMwD8)
- [Counting Outcomes with the Product Rule](https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p)