Skill: Represent sample spaces using organized lists

A sample space lists every possible outcome of a chance experiment exactly once. An organized list helps you avoid missing outcomes or listing any outcome more than once.

Example

A fair six-sided die is rolled, and then a coin is flipped. Write the sample space.

Step 1: Identify the stages

There are two stages:

1. Roll the die: possible results are $1,2,3,4,5,6$.
2. Flip the coin: possible results are $H$ or $T$.

Because the die is rolled first and the coin is flipped second, write each outcome as an ordered pair:

$$(\text{die result},\text{coin result})$$

For example, $(3,H)$ means the die shows $3$ and the coin shows heads.

Step 2: List the coin results for each die result

Start with die result $1$:

$$(1,H),\ (1,T)$$

Then die result $2$:

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

Continue in the same organized pattern:

$$\begin{aligned} S=\{&(1,H),(1,T),\\ &(2,H),(2,T),\\ &(3,H),(3,T),\\ &(4,H),(4,T),\\ &(5,H),(5,T),\\ &(6,H),(6,T)\} \end{aligned}$$

This set $S$ is the sample space.

Step 3: Check the list

There are $6$ possible die results and $2$ possible coin results, so the list should contain

$$6 \times 2=12$$

outcomes. The organized list has $12$ outcomes, with no repetitions or omissions.

The order matters: $(3,H)$ describes die first and coin second, while $(H,3)$ would describe a different order and is not how this experiment was defined.