Describing a Chance Experiment
Before calculating probabilities, we need to say exactly what a chance procedure could produce. A coin toss, a die roll, or a sequence of choices can each be described by specifying the procedure, the possible outcomes, and the full collection of those outcomes. This description is the starting point for probability.
The wording must be precise. “Toss a coin” may leave questions: Is it tossed once or twice? Are we recording the face on each toss, or just the number of heads? Those choices affect what counts as an outcome and therefore affect the sample space.
The word “experiment” does not require a laboratory. It can refer to a chance process such as rolling a die or carrying out a sequence of choices. The specification should tell us what is done and what information is recorded. An outcome is one complete result—not a partial description—and the sample space must include every result that the procedure could produce.
A result that is actually observed is one realized outcome. Before the procedure is carried out, the sample space lists the possibilities, not which one will occur. We use braces to display a sample space when it is small enough to list.
Simple Sample Spaces
For a single toss of a standard coin, suppose the recorded result is the face showing when the coin lands. There are two possible outcomes, conventionally written \(H\) for heads and \(T\) for tails. The sample space is \(S=\{H,T\}\). This description treats the coin as having only those two recorded faces; it does not attempt to record details such as how the coin spins or where it lands.
For a single roll of a standard six-sided die, if the recorded result is the number of spots showing, the sample space is \(S=\{1,2,3,4,5,6\}\). A roll yielding 4 is one outcome in this sample space. The sample space is not a list of how often each result occurs; it only identifies which results are possible under the stated recording rule.
Worked Example: Recording One Coin Toss
A hypothetical classroom demonstration tosses a standard coin once and records whether it lands heads or tails. State the experiment, an outcome, and the sample space.
Experiment. Toss the coin once and record the face showing when it lands.
Possible outcomes. The recorded face is either heads, written \(H\), or tails, written \(T\). Thus the sample space is \(S=\{H,T\}\).
Checking completeness. Under the stated recording rule, each possible recorded result is one of these two faces, so neither possibility is omitted. No other face is possible for this coin under the rule, so no additional outcome belongs in the sample space.
Worked Example: A Single Die Roll
In a hypothetical activity, a standard six-sided die is rolled once. The result is recorded as the number of spots on the upper face. Write the sample space and identify the outcome if the upper face shows five spots.
The experiment is one roll, with the upper-face number recorded. A standard six-sided die has faces numbered 1 through 6, so
The list is complete because every face has one of the six stated numbers, and each of those faces can be the upper face. If five spots show, the observed outcome is \(5\), not the whole sample space. The sample space describes the possible results before the roll.
Outcomes From More Than One Stage
An experiment can have several stages. To describe the result fully, record what happened at each stage and keep the stages in their original order. For two stages, an outcome can be written as an ordered pair \((a,b)\), where \(a\) is the first-stage result and \(b\) is the second-stage result. The order matters because \((a,b)\) and \((b,a)\) describe different stage-by-stage results when the entries differ.
When every first-stage result can be followed by every second-stage result, the sample space consists of all the corresponding ordered pairs. This is a way to list outcomes, not a probability calculation. If some second-stage choices are unavailable after a particular first-stage result, only the permitted sequences belong in the sample space.
Worked Example: Tossing a Coin Twice
A hypothetical experiment tosses a coin twice and records the result of each toss in order. List the sample space.
The first stage has two possible recorded results, \(H\) and \(T\), and the second stage also has two. Every result of the first toss can be followed by either result of the second toss. A complete outcome must therefore record both tosses in order:
To check that the list is complete, group the possibilities by the first toss. If the first toss is \(H\), the second toss can be \(H\) or \(T\), giving \((H,H)\) and \((H,T)\). If the first toss is \(T\), the second can again be \(H\) or \(T\), giving \((T,H)\) and \((T,T)\). These are all possible two-toss sequences.
The outcomes \((H,T)\) and \((T,H)\) are distinct because they record different results at the first stage. Writing only “one head and one tail” would combine those sequences and would use a different recording rule.
A Multi-Stage Campus-Choice Situation
The same approach applies when the stages are choices rather than physical trials. Before listing outcomes, state which choices are available and whether every combination is allowed. The sample space depends on those details.
Worked Example: Two Choices on Campus
For a hypothetical campus-planning exercise, a student first chooses a study location, either the library or the student center, and then chooses a drink, either tea or coffee. The exercise permits every drink choice at either location. List the sample space, using the location first and the drink second.
There are two possible first-stage choices: library (\(L\)) and student center (\(C\)). There are two possible second-stage choices: tea (\(T\)) and coffee (\(K\)). Because each drink can be chosen at either location, form every permitted location–drink pair:
The list is complete: after choosing the library, the student can choose tea or coffee; after choosing the student center, the student can also choose tea or coffee. Each outcome records both stages, and the stated rules allow all four pairs. For example, \((C,K)\) means student center first and coffee second.
The campus example also shows why the procedure must be specified before its sample space is written. If coffee were unavailable at the student center, then \((C,K)\) would not be a possible outcome. A list that included it would no longer describe the stated procedure. In other situations, a later choice may depend on an earlier one; in that case, list only the sequences that are actually allowed.
Why the Recording Rule Matters
The same physical process can lead to different sample spaces when the recorded information differs. In two coin tosses, recording the faces in order gives \(S=\{(H,H),(H,T),(T,H),(T,T)\}\). If instead the procedure records only the number of heads, its possible recorded outcomes are \(S=\{0,1,2\}\). The second sample space answers a different question: it does not distinguish the two sequences with one head.
Neither description is automatically better. The right one is the sample space that matches the specified experiment and the information needed. If the order or identity of stage results matters, preserve it in the outcome. If the procedure records only a summary, the outcome should be that summary, not extra details that were not recorded.
A Reliable Check for Completeness
A sample space is complete when it satisfies two checks: every possible result of the specified procedure appears in the list, and every listed result can actually occur under that procedure. For a small experiment, these checks can be made by listing results stage by stage, as in the two-toss and campus examples.
For a multi-stage procedure, first describe the possible results at each stage. Then follow each allowed first-stage result through the permitted later choices, writing down a full ordered outcome each time. This step-by-step inspection justifies the list: it shows that no permitted sequence was skipped and that no impossible sequence was added. It does not require assigning numerical probabilities to the outcomes.
State how many stages there are and what is recorded at each stage.
Decide whether an outcome is a single value or an ordered sequence of stage results.
Include every result allowed by the procedure and leave out any combination that is impossible.
Confirm that each possible procedure result is represented and that the listed outcomes match the recording rule.
Common Mistakes
- Listing only the result that occurred. If a coin actually lands heads, \(H\) is the observed outcome; the sample space for one toss is still \(\{H,T\}\).
- Leaving out stage order. For two tosses, \((H,T)\) and \((T,H)\) are different ordered outcomes when each toss is recorded separately.
- Listing summaries instead of recorded details. “One head” is a summary of two sequences; it is not the outcome description when the experiment records each toss in order.
- Adding impossible combinations. A pair belongs in a multi-stage sample space only if the procedure allows that first-stage and second-stage combination.
- Being vague about the procedure. “Choose a campus option” does not say what is chosen or what is recorded. Name the stages and their available choices before listing outcomes.
Check Your Understanding
For each question, state the procedure or reasoning needed, not just a final list.
- A standard coin is tossed once and the face showing is recorded. What are the experiment and its sample space?
- A standard six-sided die is rolled once. What sample space applies if the recorded result is the number on the upper face?
- A coin is tossed twice, with each face recorded in order. Why are \((H,T)\) and \((T,H)\) distinct outcomes?
- A hypothetical campus exercise first selects a building, East Hall or West Hall, then a meeting room, Room 1 or Room 2. Every room is available in either building. Write the sample space, with building first.
- In two coin tosses, suppose only the number of heads is recorded. What are the possible recorded outcomes, and how does this recording rule differ from recording both tosses in order?