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2.1 · Describe experiments, outcomes, events, and sample spaces

Learn to describe experiments, outcomes, events, and sample spaces through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Probability

MATH 215 study topic 2.1

Statistics often begins by describing what could happen in a defined situation. Before summarizing results, we need to be clear about the process being observed and the possible results it can produce. This lesson introduces four connected ideas: an experiment, an outcome, an event, and a sample space. They apply to situations such as recording a delivery status, selecting a card, or observing how many customers arrive in a time period. The goal here is to describe possibilities accurately, not to calculate their likelihood. A little algebra is enough for the notation: a set is a collection of items, and braces such as

\{1,2,3\}

show the items in a collection. A letter such as

S

can stand for a set, while a letter such as

A

can stand for a smaller collection drawn from it.

What you will learn

Describe a statistical experiment as a defined process that produces an observed result.
Distinguish a single outcome from an event containing one or more outcomes.
List a sample space that matches the experiment and includes every possible outcome.
Use set notation to describe events as parts of a sample space.

1. Define the experiment and its outcomes

In this topic, an experiment is a clearly described process or observation that produces a result. The word does not mean that someone must conduct a laboratory test. Recording whether a scheduled bus arrives on time can be an experiment in this statistical sense. A single performance of the process is often called a trial. For a useful description, specify what is done or observed and when the trial is considered complete.

An outcome is one possible result of a single trial. If the process is to record whether a package arrives by its promised date, possible outcomes might be “on time” and “late,” provided those categories cover the results being recorded. An outcome is not the process itself: “check the package status” describes the process, while “late” is a possible result.

The level of detail matters. If a trial consists of rolling one standard six-sided die and recording the upper face, an outcome is one face value, such as

4

. If the trial instead records whether that value is odd or even, the outcomes are the categories “odd” and “even.” The experiment’s description tells us what counts as one recorded result.

An experiment is the defined process or observation.
A trial is one performance of that process.
An outcome is one possible result of one trial.

2. Describe the sample space

The sample space is the collection of all possible outcomes for the experiment as it has been defined. It is commonly represented by the letter

S

. To describe it well, first make sure the trial and the recorded result are clear. Then list the possible outcomes at the chosen level of detail.

A good sample space must be complete: it must not leave out a result that the experiment could produce. Its outcomes should also be distinct for the recording task, so one trial is not counted as two different outcomes at once. For example, if a die is rolled and its face value is recorded,

1

through

6

are distinct results. If instead the recorded result is only odd or even, listing both the face value and its parity as separate outcomes would mix two descriptions of the same trial.

Some experiments have a short list of possible outcomes; others have many. If a clinic records the number of people who arrive during a specified hour, the possible values are counts such as

0

1

2

, and higher whole numbers. The end of the hour and the meaning of an arrival should be specified so that the observation is well defined. Do not add possibilities that the experiment cannot produce, and do not assume a result is possible or impossible without considering the process.

S=\{all possible outcomes of the experiment\}

The sample space includes every possible outcome for the defined experiment.
Clear trial rules and clear recording categories help make the sample space accurate.
The appropriate level of detail depends on what the experiment records.

3. Describe events as collections of outcomes

An event is a specified collection of outcomes from the sample space. It can contain one outcome or several. In set notation, an event is written as a set, often using a capital letter such as

A

. For example, when a die’s face value is recorded, the event “the result is greater than four” contains the outcomes

5

and

6

An event happens on a trial when the observed outcome is one of the outcomes listed in that event. If the die shows

6

, the event “greater than four” happens; if it shows

3

, it does not. This description says what it means for an event to happen; it does not assign a chance or calculate a probability.

An event with one outcome is sometimes called a simple event, while an event with more than one outcome is sometimes called a compound event. The key idea is the same: both are collections of outcomes from the sample space. An event containing every outcome is the whole sample space. An event containing no outcomes is the empty set, written

\varnothing

; in a given experiment, it describes a condition that cannot occur.

A\subseteq S

An event is a subset of the sample space: its members are outcomes in that space.
An event occurs when the observed outcome belongs to the event.
Events can contain one outcome, several outcomes, all outcomes, or no outcomes.

4. A reliable description procedure

Start by stating the experiment in one sentence, including what is observed and when one trial ends. Next, name the recorded result precisely. List the possible outcomes and check that the list is complete and that its entries use a consistent level of detail. Finally, translate any condition of interest into an event by collecting the outcomes that satisfy it.

These checks are important because a sample space depends on the experiment’s definition. “Select a customer and record their order size” differs from “select a customer and record whether they placed an order.” The first records a size; the second records a category. Neither description is complete until the possible recorded results are made clear.

This topic is about describing possibilities, not measuring how likely they are. A list of outcomes does not claim that each outcome is equally likely. For example, a sample space can identify possible service statuses without saying how frequently each status occurs. Keeping that distinction clear prevents a description of the experiment from being mistaken for a numerical conclusion.

Define the trial before listing outcomes.
Check that outcomes are complete and stated consistently.
Describe an event by selecting the outcomes that meet its condition.
A sample space alone does not say how likely its outcomes are.

Worked example

Record a two-step inspection result

A worker inspects two items, one after the other, and records each item as either acceptable (A) or needs rework (R). Describe the experiment, one outcome, the sample space, and the event that exactly one item needs rework.

Define the experiment
One trial consists of inspecting two items in order and recording the result for each. Because the order is recorded, the first position refers to the first item inspected and the second position to the second item.
List the possible outcomes
Each position can be A or R. Pairing the first item’s result with the second item’s result gives four distinct ordered outcomes. This list is complete because each of the two positions has exactly the two stated categories.
$S=\{AA,AR,RA,RR\}$
Identify an outcome
The outcome $AR$ means the first item is acceptable and the second needs rework. It is one possible result of the whole two-item trial, not an event containing multiple results.
$AR$
Describe the event
Exactly one item needs rework when the recorded pair has one R and one A. Both possible orders meet that condition, so the event contains $AR$ and $RA$ .
$E=\{AR,RA\}$

Answer: The experiment is the ordered inspection of two items. Its sample space is

\{AA,AR,RA,RR\}

. One outcome is

AR

, and the event that exactly one item needs rework is

\{AR,RA\}

Check: Each event member is in the sample space, and both orders with exactly one R are included. The description does not make a claim about how often any outcome occurs.

Common mistakes and how to avoid them

Calling a condition such as “exactly one item needs rework” an outcome.

Correction: That condition usually describes an event containing the outcomes that satisfy it. In the example, the event contains

AR

and

RA

Leaving out a possible result when listing the sample space.

Correction: Check every recorded position or category against the experiment’s rules. For two ordered A/R results, all four pairs must be considered.

Treating the sample space as a statement about likelihood.

Correction: The sample space lists what can happen. It does not tell us how likely each outcome is.

Ignoring order when the experiment records order.

Correction: If the first and second positions refer to different inspection times,

AR

and

RA

are different outcomes.

Lesson summary

An experiment is a defined process or observation; a trial is one performance of it.
An outcome is one possible result of a single trial.
The sample space is the complete collection of possible outcomes.
An event is a collection of outcomes from the sample space, and it occurs when the observed result belongs to that collection.
Describe the experiment and recorded result before listing outcomes; a sample space does not assign likelihoods.

Check your understanding

Question 1

A single standard six-sided die is rolled, and its face value is recorded. Which collection is the sample space?

$\{1,2,3,4,5,6\}$
$\{\text{odd},\text{even}\}$
$\{4,5,6\}$
The action of rolling the die

Show answer and explanation

\{1,2,3,4,5,6\}

The experiment records the face value, so the sample space includes all six possible face values. Odd and even are categories that would fit a different recording rule.

Question 2

For the two-item inspection in the worked example, which event describes the first item needing rework?

$\{AR,RA\}$
$\{RA,RR\}$
$\{AA,AR,RA,RR\}$
$\{A,R\}$

Show answer and explanation

\{RA,RR\}

The first position is the first item. Outcomes beginning with R are

RA

and

RR

, so the event is

\{RA,RR\}

Question 3

What does a sample space tell you?

Which outcomes are possible under the stated experiment
How often each outcome will occur
Which outcome is most likely
Whether the experiment has been repeated enough times

Show answer and explanation

Which outcomes are possible under the stated experiment

A sample space lists possible outcomes. By itself, it does not describe their frequency or likelihood.

Key terms

Experiment: A clearly described process or observation that produces a result.
Trial: One performance of a defined experiment.
Outcome: One possible result of a single trial.
Sample space: The complete collection of possible outcomes for an experiment.
Event: A specified collection of outcomes from a sample space.
Set: A collection of distinct items, such as outcomes, written between braces.

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