Year 10 Interactive Maths - Second Edition

Representation of a Sample Space

We use tree diagrams, Venn diagrams and graphs to represent sample spaces.

Tree Diagrams

A tree diagram enables us to find the elements of the sample space.

E.g.  If an experiment's trial involves tossing a fair coin, then the possible outcomes are head and tail.  This information can be depicted in the following tree diagram.

Clearly, S = {H, T}

Recall that each element of the sample space is called a sample point.  So, the sample points are H and T.

Consider the experiment of tossing a coin twice where we are interested in the number of heads. The tree diagram is shown below.

Note that the tree diagram representation of this experiment involves two parts, 'the first toss of the coin' and 'the second toss of the coin'.

Venn Diagrams

The sample space and an event may be represented on a Venn diagram.

For the experiment of tossing a fair coin, the possible outcomes are head and tail.  So, the following Venn diagram represents the experiment's sample space.

If A is the event 'a head falls', then we can use the following Venn diagram to represent it.

Graphical Representation

If an experiment's trial involves rolling a die twice, then there are 36 elements in the sample space as shown below.

As the tree diagram will be very large, the following simple graphical representation can be used instead.

Key Terms

tree diagram, Venn diagram, graphical representation