Probability & Venn Diagrams

How do I find probabilities from Venn diagrams?

Count the number of elements you want and divide by the total number of elements
For the Venn diagram shown below,
- The probability of being in A is
  - There are 5 elements in A out of 11 in total
- The probability of being in both A and B is
  - There are 2 elements in A and B (the intersection)
- The probability of being in A, but not B, is
  - 3 elements are in A but not B
Some harder questions are not out of the total number, but out of a restricted number
- The probability of being in B, given that you are already in A, is
  - You are only interested in elements in A
  - There are 5 elements in A, out of which only 2 are also in B

Two sets, A and B, represented on a Venn diagram

Examiner Tip

Be careful when filling in numbers for a Venn diagram
- Some of the given numbers may need to be split between two sections of the Venn diagram
Suppose 10 people have a cat, 8 people have a dog and 6 people have both a cat and a dog
- Out of the 10 people who have a cat
  - 6 also have a dog
  - 4 do not have a dog
- Out of the 8 people who have a dog
  - 6 also have a cat
  - 2 do not have a cat

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.

(a)

Draw a Venn diagram to show this information.

Draw the Venn diagram with its rectangular box and two (labelled) overlapping circles
3 students study both Spanish and German, so start here and work outwards
12 must study Spanish but not German (to get 15 in total for Spanish)
7 study neither, so this goes outside of the circles
To get 30 in total, 8 must study German but not Spanish

Venn diagram showing the numbers of students studying Spanish and German

(b)

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

It helps to highlight Spanish but not German

Venn diagram showing the numbers of students studying Spanish and German with the the number studying Spanish only, highlighted

Divide the number of students studying Spanish but not German by the total number of students

Students studying Spanish but not German = 12
Total number of students = 30

P(Spanish but not German) $= \frac{12}{30} (= \frac{2}{5})$

Probability & Venn Diagrams (CIE IGCSE Maths: Core)

Revision Note