What is a Venn diagram?

A Venn diagram, named after John Venn in the 19th century, provide a convenient way to represent a sample space. Click here to remind yourself of what a sample space is.

A Venn diagram is a rectangle representing the whole space and circles inside representing various subspaces.

Venn diagrams

For example, A could represent the event that a person had blue eyes. Event B could be that person have brown hair.

Events on a Venn diagram



This Venn diagram shows the complement of A. The part shaded with diagonal lines is everything that is not in A and we call this the complement. Note the spelling – it doesn’t say ‘compliment’ as charming as event A might be.

The complement of A is written as:

A^C or A



In this Venn diagram we see the union of events A and B. This means that the shaded part represents all outcomes where either event A or event B has occurred.

The union of A and B is written as:

A\cup B



Alternatively to the union, there is also the intersection of two events. The intersection represents the set of all outcomes where BOTH events A and B have occurred.

The intersection of A and B is written as:

A\cap B



It follows from the three previous definitions that it is possible to combine complement, union or intersection to get particular areas of a Venn diagram.

The above combination is written as:


Example 1

A card is chosen at random from a standard deck of 52 playing cards. Let K be the event that the card is a King. Let H be the event that the card is a heart. Find:

  1. P(K\cap H)
  2. P(H
  3. P(K\cup H)
  4. P(K

Example 2

A school has 204 students. 85 of them have chosen to study Maths, 56 of them have chosen to study French and 68 have chosen to study History. It turns that 35 study both French and Maths, 23 study both Maths and History and 27 study both History and French. 7 study all three subjects.

  1. Draw a Venn diagram to represent this information.
  2. Find the probability that a student does not study Maths.
  3. Find the probability that a student does none of these three subjects.