Probabilities from Venn Diagrams (Edexcel GCSE Maths)

Revision Note

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Written by: Mark Curtis

Reviewed by: Dan Finlay

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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 $\frac{5}{11}$
  - There are 5 elements in A out of 11 in total
- The probability of being in both A and B is $\frac{2}{11}$
  - There are 2 elements in A and B (the intersection)
- The probability of being in A, but not B, is $\frac{3}{11}$
  - 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 $\frac{2}{5}$
  - 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 Tips and Tricks

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