Year 10: Sample spaces and outcomes

A strong Year 10 probability answer starts with a clear sample space. In Australian classrooms, this is often the first thing you are expected to write before calculating a probability. If you list the possible outcomes carefully, the rest of the question becomes much easier.

The sample space is the set of all possible outcomes. An event is a selection of outcomes from that sample space.

What is a sample space?

A sample space records everything that could happen in a random experiment. For a coin toss, it is small. For two dice, it is much larger. The main skill is being systematic so that no outcomes are missed or double-counted.

Example 1: Write the sample space for rolling a fair six-sided die.

The possible outcomes are $1, 2, 3, 4, 5, 6$ , so $S = {1, 2, 3, 4, 5, 6}$ .

Events and favourable outcomes

An event is the part of the sample space you care about. For example, if the question asks for the probability of an even number, you only keep the even outcomes.

Example 2: For one die roll, let event

E

be “roll an even number”.

Then $E = {2, 4, 6}$ .

Systematic listing

When there is more than one step in an experiment, systematic listing matters. Tables, tree diagrams, and ordered pairs are all helpful Year 10 tools.

Example 3: Write the sample space for tossing a coin and then rolling a die.

The coin outcomes are $H$ and $T$ .
The die outcomes are $1$ to $6$ .
Combine them: $(H, 1), (H, 2), . . ., (H, 6), (T, 1), . . ., (T, 6)$ .
There are $12$ outcomes in total.

Why this matters for probability

Once the sample space is complete, you can count favourable outcomes and total outcomes. That is the foundation of simple and compound event probability.

Common mistakes

Leaving out outcomes when there is more than one stage.
Listing outcomes twice in a messy order.
Confusing an event with the full sample space.
Not stating whether order matters.

Study routine

Before calculating any probability, pause and write the sample space or at least describe it clearly. If there are multiple stages, use a table or tree diagram. This habit prevents most counting errors.