Venn diagrams

In a nutshell

Venn diagrams are an intuitive way to represent the events in a sample space. These events may or may not overlap.

Drawing Venn diagrams

​Venn diagrams are represented with circles (representing the events) and a rectangle to represent the sample space.

​	$$A,B,C$$​​ Events of the sample space. ​$$E$$​​ The sample space.

Note: Venn diagrams can either contain probabilities or frequencies.

Venn diagram notation

There is notation to represent certain parts of a Venn diagram.

name	notation	illustration
Intersection	$$A \cap B$$​​
Union	​$$A \cup B$$​​
Complement	​$$\overline{A}$$​ or $$A'$$​

Venn diagram problems

You may be asked to fill out Venn diagrams and find probabiities.

Example 1

​$$100$$​ students were asked whether they prefer reading books ($$B$$) or playing games ($$G$$). The results were as follows:

• $$52$$​ students said they prefer reading books.
• ​$$43$$​ students said they prefer playing games.
• ​$$20$$ students said they have no preference (i.e. they like both).

i) Draw a Venn diagram with this information.

ii) If one of these students are selected at random, what is the probability that the student likes games or likes books?

Part i):

Start at the intersection, and work out the other numbers via subtraction:

$$52-20=32$$​ students like books and not games.

​$$43-20=23$$​ students like games and not books.

​$$100-23-32-20=25$$​ don't like either.

Part ii):

Identify the correct region:

"Games or books" corresponds to the union, or $$B\cup G$$.

Find the probability corresponding to the region:

​$$P(B \cup G) = \dfrac{23+32+20}{100}=\dfrac{75}{100}$$​​

​$$P(B\cup G) =\underline{\dfrac34}$$​​