Sample Spaces and Outcomes

From a Chance Process to Its Possible Results

In Full Simulation Problem Walkthrough for the AP Exam, each simulated trial was built from the possible results of a chance process. Before calculating probabilities or simulating a process, it helps to name those results carefully. A complete list makes clear what one result looks like and helps prevent outcomes from being accidentally left out or counted more than once.

For example, when two dice are rolled, is a result just the sum, or does it specify what each die shows? Those descriptions capture different information. In this tutorial, we will write sample spaces for familiar chance processes and count their outcomes. The key is to describe exactly what is recorded in one result.

Definition: A sample space is the set of all possible outcomes of a chance process. An outcome is one complete possible result of that process. We often use \(S\) to name a sample space.

A sample space should be complete: every possible result allowed by the stated process appears in it. Its outcomes should also be described precisely enough to tell them apart. A process might have a small sample space that can be listed directly, or a larger one that is easier to count by considering its stages.

Describe What Counts as One Outcome

The outcome is not always a single number or symbol. For a multi-step process, one outcome can be an ordered list that records what happened at each step. “Ordered” means that the position of each result matters: the result at the first step is recorded separately from the result at the second step.

The sample space depends on the process’s recording rule. If two dice are rolled and we record which number appears on each die, a result is an ordered pair. If instead we record only the sum, a result is a sum. Both descriptions can be correct for their respective recording rules, but they do not contain the same information.

Key idea: Before listing outcomes, finish the sentence “One outcome records ...” If the process has several stages, specify what is recorded at each stage and whether the order matters.

A sample space is a model of the results the process can produce. It is not necessarily a list of equally likely outcomes. For example, we will assume fair dice and fair coins in the worked examples below, but the definition of a sample space does not require fairness. Listing the outcomes and deciding whether they are equally likely are separate questions.

Counting Outcomes from Stages

When a chance process has several stages, the multiplication principle is a convenient way to count its possible outcomes. If the first stage can have \(a\) possible results and, for each first-stage result, the next stage can have \(b\) possible results, then the pairs of results can be counted by multiplying: \(a \times b\). For more stages, continue multiplying the number of possibilities at each stage.

This count works when each stage has the stated number of available possibilities for every earlier result. It counts complete sequences of stage results. When you write the sample space, keep the stage order clear so that each sequence represents exactly one complete outcome.

Formula: If a process has stages with \(a_1, a_2, \ldots, a_k\) possible results at each stage, respectively, the multiplication principle counts \(a_1 \times a_2 \times \cdots \times a_k\) complete outcomes, provided those stage counts apply throughout the process.

Worked Example: List the Outcomes for Two Dice

Imagine rolling two fair, six-sided dice. For this process, record the number on the first die and the number on the second die. One outcome is therefore an ordered pair \((d_1,d_2)\), where \(d_1\) is the first die’s result and \(d_2\) is the second die’s result. The order distinguishes the dice, even if the dice show the same two numbers in opposite positions.

The first die can show any number from 1 to 6. For each of those results, the second die can also show any number from 1 to 6. The multiplication principle gives \(6 \times 6=36\) ordered outcomes. Here is the full sample space, organized by the first die’s result:

First die	Outcomes as ordered pairs (first die, second die)
1	(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
2	(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
3	(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
4	(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
5	(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
6	(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Each row has six outcomes, one for each possible result on the second die. There are six rows, one for each possible result on the first die, so the table contains \(6 \times 6=36\) outcomes. For example, \((2,5)\) and \((5,2)\) are different outcomes under this recording rule: the first die and second die have different results in the two cases.

If the process records only the sum, the possible recorded results are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. That is a different sample space, with 11 possible recorded sums. The ordered-pair sample space and the sum-only sample space answer different descriptions of what one outcome records. In particular, a sum such as 7 can arise from several ordered pairs; treating the sum as the entire outcome leaves out which pair occurred.

List the Outcomes for Three Coin Flips

For three flips of a coin, record the result of each flip in order. Use \(H\) for heads and \(T\) for tails. A complete outcome is a three-letter sequence: the first letter records flip 1, the second records flip 2, and the third records flip 3.

Each flip has two possible results. Thus, the multiplication principle gives \(2 \times 2 \times 2=8\) sequences. Listing by the first two flips makes it easier to check that the final flip has both possibilities for every pair of earlier results:

S=\{HHH,\ HHT,\ HTH,\ HTT,\ THH,\ THT,\ TTH,\ TTT\}

For instance, \(HTH\) means heads on the first flip, tails on the second, and heads on the third. It is a different outcome from \(HHT\), even though each sequence contains two heads and one tail. The positions preserve the order in which the results occurred.

If a question instead recorded only the number of heads in the three flips, then the recorded values could be 0, 1, 2, or 3. That would be a different description of a result from listing the full flip sequence. For example, \(HTH\) and \(HHT\) would both be summarized as “two heads.” When defining a sample space, do not replace detailed results with a summary unless the process is specifically defined to record that summary.

Worked Example: Count and List a Two-Stage Selection

A student designing a fictional event badge will choose one of three layouts—circle, square, or triangle—and then choose one of four colors—blue, green, orange, or purple. Suppose the chance process records both choices, with the layout first and the color second. One outcome is an ordered pair \((\text{layout},\text{color})\).

There are three possibilities for the layout and, for each layout, four possibilities for the color. The multiplication principle gives \(3 \times 4=12\) complete outcomes. The sample space is:

\begin{aligned} S=\{&(\text{circle},\text{blue}),(\text{circle},\text{green}),(\text{circle},\text{orange}),(\text{circle},\text{purple}),\\ &(\text{square},\text{blue}),(\text{square},\text{green}),(\text{square},\text{orange}),(\text{square},\text{purple}),\\ &(\text{triangle},\text{blue}),(\text{triangle},\text{green}),(\text{triangle},\text{orange}),(\text{triangle},\text{purple})\} \end{aligned}

There are four listed pairs for each of the three layouts, for a total of \(4+4+4=12\). This agrees with \(3 \times 4=12\). The check is useful: the direct list and the stage-by-stage count match. An outcome such as \((\text{square},\text{orange})\) is one complete badge choice, not two separate outcomes.

Check a Sample Space for Completeness

A list can look plausible and still be incomplete or ambiguous. Use the following checks whenever you write one:

Match the stated process. Include every result the process allows, and do not include results it cannot produce.
Record all the required stages. For two dice, include the result on each die if the process records both dice.
Keep order when the stages are distinguished. The sequence \(HTH\) records a different order of coin results from \(HHT\).
Check the count independently. Use the multiplication principle when the process has a fixed number of possibilities at each stage, then compare that count with the number of listed outcomes.
Do not confuse a detailed result with a summary. A dice pair and its sum, or a coin sequence and its number of heads, are different ways to record results.

It can help to organize a list as a table, as in the two-dice example, or to group results by their first stage, as in the three-coin example. An organized list makes omitted outcomes easier to spot. If the list is too long to write out, clearly naming the form of one outcome and using the multiplication principle still gives a precise description and count.

Common Mistakes and AP Exam Tips

Listing only a summary when the process records each step. For two dice, listing possible sums does not list the outcomes if the result is defined as the number on each die. State what one outcome records, then use the matching sample space.
Ignoring order. If the dice are distinguished as first and second, \((2,5)\) and \((5,2)\) are separate outcomes. For coin flips, \(HTH\) and \(HHT\) are separate sequences.
Counting the stages but not complete outcomes. Three coin flips do not have \(2+2+2=6\) sequences. Each first-flip result can be followed by either second-flip result, and each two-flip sequence can be followed by either third-flip result, so there are \(2 \times 2 \times 2=8\).
Giving an incomplete list without saying so. If asked to list the sample space, include every outcome. If a space is large, clearly define the form of its outcomes and show the count rather than presenting a few examples as though they were the full list.
Assuming the outcomes must be equally likely. A sample space lists what can happen. Whether its outcomes have equal chances depends on the chance process, not on the definition of a sample space.
Failing to verify the count. In a table, count the entries across and down. In a sequence process, multiply the number of choices at each stage. If those counts disagree with the written list, look for a missing or repeated outcome.

A clear AP response identifies the recording rule, gives a precise form for one outcome, and lists or counts all possibilities. For a staged process, show the multiplication that supports the total. If a different recording rule would produce a different sample space, say so rather than silently switching descriptions.

Key takeaway: A sample space lists every possible complete outcome of a chance process. Define what one outcome records, preserve the order of distinct stages, and use the multiplication principle to count outcomes when the process has multiple stages.

Check Your Understanding

For each question, focus on what one outcome records before listing or counting possibilities.

Two dice are rolled. If an outcome records the result on each die in order, how many outcomes are in the sample space? Explain the count.
Write the sample space for two coin flips, recording the first flip before the second.
Three coins are flipped. How many possible sequences are there? Show the multiplication used to count them.
A process chooses one of four sticker shapes and then one of two patterns. If both choices are recorded in order, how many outcomes are possible?
Explain why \((3,6)\) and \((6,3)\) are different outcomes when the first and second dice are recorded separately, but have the same sum.

What You'll Learn