Skill: List outcomes in a sample space

A sample space is a list of every possible outcome of a random experiment. Each outcome should be:

• included exactly once,
• written clearly, and
• different from every other outcome.

For a two-step experiment, keep the outcomes in the order the steps happen.

Example: A coin is flipped, and then a spinner with the numbers $(1,2,3)$ is spun. List the sample space.

Step 1: List the results of the first step.

The coin can land:

$$H \text{ or } T$$

Step 2: For each coin result, pair it with every spinner result.

If the coin is $H$, the outcomes are:

$$(H,1),\ (H,2),\ (H,3)$$

If the coin is $T$, the outcomes are:

$$(T,1),\ (T,2),\ (T,3)$$

Step 3: Write the complete sample space.

$$S=\{(H,1),(H,2),(H,3),(T,1),(T,2),(T,3)\}$$

There are $6$ possible outcomes. The pair $((H,2))$, for example, means the coin landed heads and the spinner landed on $2$.

The order matters because the first part records the coin and the second part records the spinner. The sample space lists individual outcomes; an event would be a group of outcomes, such as “the spinner lands on $2$”:

$$\{(H,2),(T,2)\}$$