Skill: Represent sample spaces using tables

A sample space is the list of every possible outcome of a chance experiment. A table is useful when the experiment has two stages, because:

• the rows can represent the outcomes of the first stage;
• the columns can represent the outcomes of the second stage;
• each cell represents one combined outcome.

Example

A fair coin is flipped, and a fair spinner numbered $(1,2,3)$ is spun. Find the probability of getting heads and an odd number.

Step 1: List the outcomes for each stage

The coin can land:

$$H \text{ or } T$$

The spinner can land:

$$1,\ 2,\ \text{or } 3$$

Step 2: Make a table

Put the coin outcomes in the rows and the spinner outcomes in the columns.

Coin \ Spinner	$1$	$2$	$3$
$H$	$(H1)$	$(H2)$	$(H3)$
$T$	$(T1)$	$(T2)$	$(T3)$

Each cell is one outcome, such as $(H2)$, meaning “heads and 2.”

The table is:

• exhaustive because it includes every possible outcome;
• non-overlapping because each result belongs to exactly one cell.

Thus, the sample space has $6$ outcomes:

$$\{H1,H2,H3,T1,T2,T3\}$$

Step 3: Identify the event

The event is “heads and an odd number.” The odd spinner numbers are $1$ and $3$, so the successful outcomes are:

$$H1 \text{ and } H3$$

These are two outcomes, not one outcome.

Step 4: Find the probability

All $6$ outcomes are equally likely. Therefore,

$$P(\text{heads and odd})=\frac{\text{number of successful outcomes}}{\text{total number of outcomes}} =\frac{2}{6} =\frac{1}{3}$$

So, the probability of getting heads and an odd number is

$$\boxed{\frac{1}{3}}$$