Represent sample spaces using tables

A table represents the finite sample space of a two-stage random experiment by pairing each outcome from one stage with each outcome from the other, with rows and columns making outcomes systematic, exhaustive, and nonrepeating. It supports identifying events as sets of outcomes and, when outcomes are equally likely, finding theoretical probabilities by counting favorable entries over total entries; continuous, infinite, and more abstract multidimensional sample spaces are outside this scope.

Detailed Explanation: Represent sample spaces using tables

A sample space is the list of every possible outcome of a random experiment. When an experiment has two stages, use a table to pair each outcome from the first stage with each outcome from the second stage.

Example

A fair coin is flipped, and then a fair six-sided die is rolled. Represent the sample space in a table. Then find the probability of getting heads and an even number.

Step 1: List the outcomes for each stage.

Coin outcomes: $H$ or $T$
Die outcomes: $(1,2,3,4,5,6)$

Step 2: Use one stage for the rows and the other stage for the columns.

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

	$1$	$2$	$3$	$4$	$5$	$6$
$H$	$((H,1))$	$((H,2))$	$((H,3))$	$((H,4))$	$((H,5))$	$((H,6))$
$T$	$((T,1))$	$((T,2))$	$((T,3))$	$((T,4))$	$((T,5))$	$((T,6))$

Each cell is one outcome. For example, $((H,4))$ means the coin landed heads and the die showed $4$ .

Step 3: Check that the table is complete and has no repeats.

There are $2$ coin outcomes and $6$ die outcomes, so the table has

2 \times 6=12

possible outcomes. Every coin outcome is paired with every die outcome exactly once.

Step 4: Identify the event.

The event “heads and an even number” includes the entries with $H$ and an even die number:

(H,2),\ (H,4),\ (H,6)

There are $3$ favorable entries out of $(12)$ total entries.

P(\text{heads and even})=\frac{3}{12}=\frac14

A table makes it easy to see all possible outcomes and count the outcomes that belong to a particular event.

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Probability Sample Space - Definition to Sample Space List

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