Venn Diagrams

Statisticians use Venn diagrams to depict relationships between events in a sample space.

What is a Venn Diagram?

In a Venn diagram, the sample space is represented by a rectangle. Events within the sample space are often represented by circles within the rectangle. Here is a simple Venn diagram:

Venn diagram showing event A within sample space S

In this diagram, the blue circle represents Event A; and the area in white around the circle represents A', the complement of Event A. That is, the white space represents all of the events in the sample space that are not represented by Event A.

Why Use Venn Diagrams?

Relationships between events can get complicated. Within a sample space, some events might be mutually exclusive. Other events might have elements in common. Venn diagrams make it possible to see the big picture.

With a Venn diagram, you can display relationships visually. Events that have elements in common appear as partially-overlapping circles. For example, in the Venn diagram below, the overlap between Events A and B represents the intersection of A and B, A B.

Venn diagram showing intersection of Events A and B

And events that are mutually exclusive appear as non-overlapping circles. In the Venn diagram below, Events X and Y have no elements in common. Therefore, they are mutually exclusive.

Venn diagram showing two mutually exclusive events

It's not necessary to draw a Venn diagram every time you try to solve a probability problem. But it's often helpful. When you can see the sets, unions, and intersections that make up a sample space, it is easier to develop a plan to compute related probabilities.

Test Your Understanding

Problem 1

Look at the Venn diagram shown below. It shows three events in a sample space. What part of the Venn diagram represents the complement of the union of Events A and B?

Venn diagram

(A) The intersection of the blue and red circles.
(B) The non-intersecting part of the blue and red circles.
(C) The green circle.
(D) The white space around the circles.
(E) The white space around the circles, plus the green circle.

Solution

The correct answer is E. The union of Events A and B is represented by the blue and red circles, including the portion that overlaps. The complement of that union is everthing else. Thus, the complement is the white space around the circles, plus the green circle.

Remember, the box that holds the circles represents the entire sample space. Each circle within the box represents an event in the sample space. The white space around the circles represents events in the sample space that are not represented by the circles. Thus, the white space and the green circle represent all the events that are not represented by the blue and red circles.