What are Venn diagrams?
A Venn diagram is a rectangle representing the whole space and circles inside representing various subspaces.
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. See Example 1. 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. See Example 2. The union of A and B is written as:
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. See Example 3. The intersection of A and B is written as:
Venn Diagram Examples
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:
- $P(K\cap H)$
- $P(K\cup H)$
- $P(K’\cap H’)$
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.
- Draw a Venn diagram to represent this information.
- Find the probability that a student does not study Maths.
- Find the probability that a student does none of these three subjects.