Probability & Venn Diagrams

What is a Venn diagram?

Venn diagrams allow us to show two characteristics of a situation where there is overlap between the characteristics
For example, students in a sixth form can study biology or chemistry but there may be students who study both

What might I be asked to do with a Venn diagram?

You can be asked to
- draw a Venn diagram and/or
- interpret a Venn diagram
Strictly speaking the rectangle (box) is always essential on a Venn diagram
- it represents everything that can happen in the situation
- you may see the letter $E$ or $ξ$ written inside or just outside the box
  - this means “the set of all possible outcomes” - i.e. “everything”!
  - sometimes the letters U or S are used instead
The words AND and OR become very important in both drawing and interpreting Venn diagrams
You will need to be familiar with the symbols ∩ and ∪
- ∩ is intersection
- ∪ is union
- these mean AND and OR (respectively)

How do I draw a Venn diagram?

Start with a “box” and overlapping “bubbles”
- there will be two bubbles
Work through each sentence/piece of information given in a question to begin completing sections of your Venn diagram
- pieces of information may have to be combined before you can enter a value into the diagram
- not all values will be given directly
  - some may need working out
  - you will be expected to do this to complete your Venn diagram
Remember to consider AND and OR

How do I interpret a Venn diagram?

Use the information in the question to identify the parts of the Venn diagram needed to answer it
Shading the relevant parts of a Venn diagram can be helpful
Be careful with probability notation such as
- the symbols ∩ and ∪
  - A∩B is just the middle of the two bubbles
  - A∪B is anything that is in at least one of the bubbles

Exam Tip

You may have to use your Venn diagram more than once in a question
- so shading the original diagram can become confusing if you're trying to use it more than once
- draw a 'mini'-Venn diagram (a small quick sketch just showing the box and bubbles but no values) and shade that

Worked example

In a class of 30 students, 15 students study Spanish, and 3 of the Spanish students also study German.
7 students study neither Spanish nor German.

Draw a Venn diagram to show this information.

We start with the 3 in the intersection ("overlap"); we can then deduce the "Spanish only" section is 12.
7 needs to be outside both bubbles but within the box.
With a total of 30 we can work out how many students study "German only" and complete the diagram.

Use your Venn diagram to find the probability that a student, selected at random from the class, studies Spanish but not German.

Highlight the part "Spanish only".

Venn-Q1b-Working, downloadable IGCSE & GCSE Maths revision notes

Pick out the numbers you need carefully.

Students studying "Spanish only" = 12
Total number of students = 30

P(Spanish only) $= \frac{12}{30} (= \frac{2}{5})$

Probability & Venn Diagrams (CIE IGCSE Maths: Core)

Revision Note