Probability: P2-3 Venn Diagrams - using Venn diagrams to determine numbers and probabilities

Oct 26, 2026

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Venn diagrams for probability show how items or people distribute across two or three groups, including overlaps where someone belongs to multiple categories. The material explains dot-based diagrams, why overlaps prevent double-counting, and how number-filled regions simplify larger datasets. It then guides learners through practice questions that require calculating exact counts for intersections, differences, unions, and complements, using both two-circle and three-circle Venn diagrams with real examples.

LanguageEnglish

CategoryExam

FormatPDF

Pages11

Reading Time28 min

AccessFree to download

Probability: P2-3 Venn Diagrams

Using Venn diagrams with dots

Venn diagrams with numbers

Drawing Venn diagrams

Venn Diagrams with 3 Circles

Practice questions

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[M1Maths.com](M1Maths.com) homepage Other probability modules
Probability: P2-3 Venn Diagrams
• using Venn diagrams to determine numbers and probabilities
Summary Learn Solve Revise Answers | | | --- | --- | ~~ ~~ Summary ~~ ~~ A Venn diagram shows which of two or three groups certain items or people belong to. Venn diagrams can be used to work out probabilities. ~~ ~~ Learn ~~ ~~Venn Diagrams with Dots A Venn diagram is a visual way of showing which of two or three groups certain items or people belong to. For example, a school might have 12 teachers. Suppose 4 of them teach English, 3 teach maths and 5 teach neither. We can show this visually on a Venn diagram like this: English Maths The way the diagram works is that those who teach English are represented by dots in the English circle; those who teach Maths are represented by dots in the maths circle and those who teach neither are represented by dots in neither circle. But suppose one of the Maths teachers was given an English class as well. That teacher then needs to be in both circles. Giving her two dots might be misleading because we would have 13 dots and so might think that there are 13 teachers. Instead we make the circles overlap, so that one dot can be in both circles, like this: English Maths This Venn diagram now has 5 dots in the English circle, showing that 5 of the teachers teach English. It has 3 dots in the Maths circle, showing that 3 of the teachers teach Maths. And it has 5 teachers not in either circle, showing that 5 teachers don’t teach either. And there are still just 12 dots showing that there are just 12 teachers. Practice Q1 This Venn diagram shows the students in Class 8C who have a cat or a dog at home. (a) How many have a cat and a dog? (b) How many have a cat? (c) How many don’t have a cat? (d) How many have a cat, but not a dog? (e) How many have a dog? (f) How many don’t have a dog? (g) How many have a dog but not a cat? (h) How many have neither? (i) How many students in the class? Make sure you can get all parts of Q1 right before you go on. Venn Diagrams with Numbers Now, using dots is ok when there are just a few people. But suppose the Venn diagram for English and Maths teachers above was for a big school where 12 teachers teach just English, 14 teach just maths, 5 teachers teach both English and Maths and 49 teachers teach neither. Using dots would be messy and anyone who wants to read the diagram will have to do a lot of counting. So instead, we tend to just write the numbers in each region of the diagram instead of using dots. In fact, we tend to use numbers even when the numbers are small. So the Venn diagram for the big school would look like this: English Maths 12 5 14 49 Practice Q2 This Venn diagram shows the students that play guitar and volleyball at Cabbage Tree High School. Guitar Volleyball 1 8 3 106 (a) How many play just guitar? (b) How many play guitar? (c) How many don’t play guitar? (d) How many play volleyball? (e) How many play volleyball but not guitar? (f) How many don’t play volleyball? (g) How many play guitar and volleyball? (h) How many play neither? (i) How many students in the school? Drawing Venn Diagrams We can draw Venn diagrams from information about how many people are in each group. For example, if we are told that there were 22 people in a club, that 5 of them were over 50, 13 of them were female and 6 of them were neither, we can draw the diagram like this: Note that we have to work out that there are 2 in the overlap by seeing that 5 + 13 + 6 = 24 and thus that 2 people must have been counted twice. Once we know this, we can work out that there must be 3 in the rest of the Over 50 circle and 11 in the rest of the female section. Practice Q3 Draw a Venn diagram to show the following information. There are 17 people in the swimming club. 5 race freestyle, 4 race breaststroke, 10 r

FAQs

What does a Venn diagram represent when calculating probabilities?

It represents which items or people belong to one or more of two or three groups, with overlaps showing shared membership that must be counted correctly.

Why do overlapping circles matter in a Venn diagram?

Overlaps prevent double-counting; a person in both groups is shown in the intersection so totals match the given overall number.

How do you use a three-circle Venn diagram to find totals for a category?

Add all numbers inside the circle for that category, then use intersections and remaining regions to answer questions about combinations and exclusions.

Oct 26, 2026

Upload by Hazel