The Central Idea
Probability studies uncertainty by describing a collection of possible results and then measuring how likely selected results are.
The basic structure is
\[ \boxed{\text{experiment} \longrightarrow \text{outcome} \longrightarrow \text{sample space}}. \]Before asking how likely something is, we must first determine exactly what the possible outcomes are.
What Is a Probability Experiment?
A probability experiment is a process or procedure whose result is uncertain before the process is carried out.
Examples include:
- tossing a coin;
- rolling a die;
- drawing a card from a shuffled deck;
- recording the number of customers arriving in an hour;
- measuring the lifetime of a component;
- selecting an individual from a population according to a stated rule.
The word experiment does not necessarily mean a laboratory experiment. It means any well-defined process used to generate or observe an uncertain result.
A Trial Is One Performance of the Experiment
A trial is one performance of a probability experiment.
If the experiment is rolling a six-sided die, then one roll is one trial. Rolling the die \(100\) times produces \(100\) trials of the same experiment.
The distinction is useful:
\[ \boxed{\text{experiment}=\text{the procedure},\qquad \text{trial}=\text{one execution of the procedure}}. \]What Is an Outcome?
An outcome is one possible result of a probability experiment.
For a coin toss, the outcomes may be written
\[ H \]and
\[ T, \]where \(H\) means heads and \(T\) means tails.
If the coin is tossed and lands heads, then the observed outcome of that trial is \(H\).
What Is a Sample Space?
The sample space is the set of all outcomes being considered for an experiment.
A common symbol for the sample space is \(S\). Another widely used symbol is the Greek letter \(\Omega\).
For one coin toss,
\[ \boxed{S=\{H,T\}}. \]Every outcome in the model belongs to the sample space.
Outcomes Are Elements of the Sample Space
Because a sample space is a set, set notation applies naturally.
If
\[ S=\{1,2,3,4,5,6\}, \]then
\[ 4\in S \]means that \(4\) is one possible outcome, while
\[ 9\notin S \]means that \(9\) is not an outcome in that model.
Example: Rolling One Die
Suppose the experiment is:
Roll one standard six-sided die and record the number showing.
The possible outcomes are
\[ 1,\;2,\;3,\;4,\;5,\;6. \]Therefore,
\[ \boxed{S=\{1,2,3,4,5,6\}}. \]The sample space records the possible results, not yet their probabilities.
The Description of the Experiment Matters
A sample space is determined by what the experiment says to observe.
Consider rolling one standard die.
If we record the exact face, then
\[ S=\{1,2,3,4,5,6\}. \]If instead we record only whether the result is odd or even, then
\[ S=\{\text{odd},\text{even}\}. \]The physical action is the same, but the recorded outcome is different.
Sample Spaces Are Models
A sample space is not merely a list of everything that could physically occur. It is a mathematical model of the distinctions relevant to the question.
For a coin toss, a simple model might be
\[ S=\{H,T\}. \]A more detailed physical model could distinguish a coin landing on its edge, bouncing away, or being interrupted. Whether such outcomes belong in the model depends on the assumptions and purpose of the experiment.
Good probability work states the model clearly enough that the outcomes are understood.
Outcomes Should Cover Every Modeled Possibility
A valid sample space must be exhaustive: it must include every outcome allowed by the model.
If a six-sided die is rolled and the face is recorded, then
\[ \{1,2,3,4,5\} \]is not a complete sample space because the outcome \(6\) has been omitted.
The complete sample space is
\[ \boxed{\{1,2,3,4,5,6\}}. \]Elementary Outcomes Should Not Overlap
When a trial is completed, exactly one elementary outcome should describe the observed result at the chosen level of detail.
For example,
\[ S=\{1,2,3,4,5,6\} \]works for the exact face of a die because one roll cannot simultaneously produce both \(2\) and \(5\).
This non-overlapping structure allows the sample space to partition the possible results into distinct cases.
Finite Sample Spaces
A finite sample space contains finitely many outcomes.
Examples include:
\[ \{H,T\}, \] \[ \{1,2,3,4,5,6\}, \]and the possible suits of a playing card,
\[ \{\text{clubs},\text{diamonds},\text{hearts},\text{spades}\}. \]Finite sample spaces are especially convenient because their outcomes can often be listed explicitly.
Example: Tossing Two Coins
Suppose two coins are tossed and the result of each coin is recorded in order.
The possible outcomes are
\[ HH,\qquad HT,\qquad TH,\qquad TT. \]Therefore,
\[ \boxed{S=\{HH,HT,TH,TT\}}. \]The outcomes \(HT\) and \(TH\) are distinct because the first and second tosses are distinguished.
Order Can Matter
In a multistage experiment, ask whether the order of results is part of the recorded information.
For two coin tosses,
\[ HT\neq TH \]when the first and second tosses are recorded separately.
If the experiment records only the number of heads, however, the sample space becomes
\[ \boxed{S=\{0,1,2\}}. \]Once again, the sample space depends on what the experiment records.
Reduced Outcomes Need Not Be Equally Likely
When detailed outcomes are combined into broader categories, the new outcomes may have different probabilities.
For two fair coin tosses,
\[ \{HH,HT,TH,TT\} \]contains four equally likely ordered outcomes.
If we record only the number of heads, then
\[ S=\{0,1,2\}. \]But the result \(1\) head can occur through either \(HT\) or \(TH\), while \(0\) heads and \(2\) heads each have only one corresponding ordered result.
Therefore, merely having three listed outcomes does not imply that each has probability \(1/3\).
Example: Rolling Two Dice
Suppose two dice are rolled and the face on each die is recorded.
An outcome is naturally represented as an ordered pair
\[ (i,j), \]where \(i\) is the first die and \(j\) is the second die.
The sample space is
\[ \boxed{S=\{(i,j):i,j\in\{1,2,3,4,5,6\}\}}. \]There are
\[ 6\cdot 6=36 \]ordered outcomes.
Listing a Two-Dice Sample Space
The same two-dice sample space can be displayed systematically:
\[ \begin{aligned} S=\{& (1,1),(1,2),(1,3),(1,4),(1,5),(1,6),\\ &(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\ &(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),\\ &(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\ &(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),\\ &(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) \}. \end{aligned} \]A structured listing helps prevent omissions and duplicates.
If We Record Only the Sum
If two dice are rolled but only their sum is recorded, then the possible recorded values are
\[ \boxed{S=\{2,3,4,5,6,7,8,9,10,11,12\}}. \]These eleven outcomes are not equally likely.
For example, the sum \(2\) occurs only through
\[ (1,1), \]while the sum \(7\) occurs through
\[ (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). \]The choice of sample space must be kept separate from any assumption that its listed outcomes have equal probabilities.
Tables Can Organize Multistage Outcomes
When two stages each have a manageable number of possibilities, a table can display their combinations.
If the first stage has outcomes
\[ A,\;B \]and the second has outcomes
\[ 1,\;2,\;3, \]then the combined sample space is
\[ \boxed{S=\{(A,1),(A,2),(A,3),(B,1),(B,2),(B,3)\}}. \]The product structure gives
\[ 2\cdot 3=6 \]possible ordered outcomes.
Tree Diagrams Can Organize Sequential Outcomes
A tree diagram represents an experiment stage by stage.
For two coin tosses, the first toss branches into
\[ H\quad\text{or}\quad T. \]From each of those branches, the second toss again branches into
\[ H\quad\text{or}\quad T. \]Reading each complete path gives
\[ HH,\qquad HT,\qquad TH,\qquad TT. \]Tree diagrams become especially useful when later stages depend on earlier results.
Counting Outcomes Without Listing All of Them
Some sample spaces are too large to list conveniently.
If an experiment consists of three independent recording stages with \(4\), \(5\), and \(2\) possible labels respectively, then the number of complete outcome strings is
\[ 4\cdot5\cdot2=40. \]The multiplication principle counts the possible outcomes even if we do not write all forty of them.
Later probability calculations often combine counting methods with a carefully specified sample space.
Example: A Three-Digit Code
Suppose a code consists of three decimal digits, repetition is allowed, and leading zeroes are permitted.
Each position has \(10\) possible outcomes:
\[ 0,1,2,\ldots,9. \]Therefore, the sample space contains
\[ 10^3=1000 \]outcome strings.
A typical outcome is
\[ 407. \]Simple Outcomes and Events
An individual member of the sample space is sometimes called a simple outcome or sample point.
A collection of outcomes is called an event.
If
\[ S=\{1,2,3,4,5,6\}, \]then the event “roll an even number” is
\[ \{2,4,6\}. \]The next lesson develops events and their set operations in detail. For now, the important point is that events are built from outcomes in the sample space.
Sample Spaces Can Be Written with Set-Builder Notation
When listing every outcome would be inconvenient, set-builder notation can describe the sample space compactly.
For the roll of two six-sided dice,
\[ S=\{(i,j):i,j\in\{1,2,3,4,5,6\}\}. \]For a measurement \(x\) that can take any value from \(0\) through \(10\), including the endpoints,
\[ \boxed{S=\{x\in\mathbb{R}:0\le x\le10\}}. \]This is the interval
\[ [0,10]. \]Countably Infinite Sample Spaces
A sample space does not have to be finite.
Suppose a coin is tossed repeatedly until the first head appears, and the outcome is the number of tosses required.
The sample space is
\[ \boxed{S=\{1,2,3,4,\ldots\}}. \]There is no largest possible number of tosses, so the sample space is infinite.
Because its outcomes can be placed in a sequence, it is countably infinite.
Continuous Sample Spaces
Some experiments produce measurements that can vary continuously.
Suppose the experiment records a waiting time \(t\) in minutes, and any nonnegative real value is possible.
Then
\[ \boxed{S=[0,\infty)}. \]Unlike a finite sample space, this sample space cannot be listed outcome by outcome.
Discrete and Continuous Outcomes
A useful first distinction is between discrete and continuous outcomes.
Discrete outcomes are separated into distinct countable possibilities, such as
\[ 0,1,2,3,\ldots \]customers arriving during an interval.
Continuous outcomes vary across intervals of real numbers, such as a length, mass, time, or temperature measured to arbitrary precision in the model.
The distinction affects how probabilities are assigned later.
Example: Number of Arrivals
Suppose the experiment records the number of customers who enter a store during the next ten minutes.
The possible counts are
\[ 0,1,2,3,\ldots \]so a natural sample space is
\[ \boxed{S=\{0,1,2,3,\ldots\}}. \]This is a discrete sample space even though it contains infinitely many possible values.
Example: Lifetime of a Component
Suppose \(T\) represents the lifetime, in hours, of a component and the model permits any nonnegative real lifetime.
Then
\[ \boxed{S=\{t\in\mathbb{R}:t\ge0\}}. \]Equivalently,
\[ S=[0,\infty). \]This is a continuous sample space.
The Sample Space Depends on the Level of Detail
Consider drawing one card from a standard deck.
If the exact card is recorded, the sample space has \(52\) outcomes.
If only the suit is recorded, then
\[ S=\{\text{clubs},\text{diamonds},\text{hearts},\text{spades}\}. \]If only the color is recorded, then
\[ S=\{\text{red},\text{black}\}. \]Each is a valid sample space for a different recorded variable.
Outcomes Must Match the Question Being Asked
A useful sample space contains enough detail to answer the probability question of interest.
If we want the probability that a drawn card is the queen of hearts, the sample space
\[ \{\text{red},\text{black}\} \]is too coarse by itself because it does not distinguish the queen of hearts from other red cards.
The model should preserve the distinctions needed for the analysis.
Random Does Not Mean Patternless
In elementary probability, an experiment is called random when its specific outcome is not known in advance under the model.
Randomness does not mean that every outcome must be equally likely, and it does not mean that the process has no structure.
A weighted spinner, for example, can have clearly defined outcomes with unequal probabilities.
The sample space identifies what can happen; probability values describe how likely those possibilities are.
Do Not Assign Equal Probability Automatically
One of the most important early probability habits is to separate the number of outcomes from their probabilities.
If
\[ S=\{A,B,C\}, \]it does not follow automatically that
\[ P(A)=P(B)=P(C)=\frac13. \]Equal probabilities require additional justification.
The lesson on equally likely outcomes will develop that issue directly.
Repeated Trials Produce Outcome Sequences
When an experiment is repeated, the combined outcome records the result of each trial.
If a coin is tossed three times, the sample space is
\[ \boxed{ S=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\} }. \]There are
\[ 2^3=8 \]ordered sequences.
General Repeated-Trial Structure
Suppose each trial has \(m\) possible recorded outcomes and the experiment is repeated \(n\) times.
If every stage permits all \(m\) labels, then the number of length-\(n\) outcome sequences is
\[ \boxed{m^n}. \]For example, four coin tosses produce
\[ 2^4=16 \]possible \(H/T\) sequences.
This counts possible outcome sequences; it does not by itself establish their probabilities.
Experiments with Changing Possibilities
Not every stage of a multistage experiment has the same available outcomes.
Suppose two objects are selected from a set of five distinct objects without replacement and order is recorded.
There are \(5\) possibilities for the first selection and \(4\) remaining possibilities for the second.
Therefore, there are
\[ 5\cdot4=20 \]ordered outcomes.
The sample-space structure must reflect the rules of the experiment.
With Replacement and Without Replacement
The phrase with replacement means a selected object is returned before the next selection.
The phrase without replacement means it is not returned.
If two selections are made from \(5\) distinct objects and order matters, then:
- with replacement, there are \(5\cdot5=25\) ordered outcomes;
- without replacement, there are \(5\cdot4=20\) ordered outcomes.
A small change in procedure can change the sample space.
Impossible Results Are Not Outcomes
Outcomes must satisfy the rules of the experiment.
If two cards are drawn without replacement from a deck and exact cards are recorded, an outcome cannot contain the same physical card twice.
Likewise, if a standard die is rolled, \(7\) is not an outcome when the face value is recorded.
Building a sample space requires respecting all stated restrictions.
Labels Should Be Unambiguous
Outcome notation should make clear what each result means.
If two distinguishable dice are rolled, writing
\[ (2,5) \]can mean “first die \(2\), second die \(5\).”
If the dice are called red and blue, we can state instead that the first coordinate is red and the second is blue.
Clear labeling prevents accidental loss of information.
Sample-Space Size
For a finite sample space, the number of outcomes is often written
\[ |S|. \]If
\[ S=\{1,2,3,4,5,6\}, \]then
\[ \boxed{|S|=6}. \]For two ordered six-sided dice,
\[ |S|=36. \]This notation will be useful when probability is computed by counting.
Sample Spaces and Counting-Based Probability
Later, when all elementary outcomes are known to be equally likely, the probability of an event \(A\) can be found by comparing counts:
\[ P(A)=\frac{|A|}{|S|}. \]But this formula depends on the assumption of equally likely elementary outcomes.
The first responsibility is therefore to construct an appropriate sample space and understand how its outcomes are generated.
Sample Spaces and Simulation
A simulation imitates a probability experiment using another controlled process such as random digits, software-generated random values, or repeated physical trials.
To simulate an experiment correctly, each simulated result must correspond to an outcome in the intended sample space.
A poor mapping between simulation outputs and modeled outcomes can produce a distorted probability model.
Later lessons will use simulations to estimate probabilities empirically.
Model Assumptions Begin Here
Even the sample space can depend on assumptions.
For a die-roll model, we may assume:
- the die has six labeled faces;
- exactly one face is recorded after each valid roll;
- rolls that do not produce a valid resting face are repeated;
- the recorded value is one of \(1,2,3,4,5,6\).
These assumptions define the mathematical experiment before probabilities are assigned.
A Reliable Procedure for Building a Sample Space
- State the experiment precisely.
- Identify what is being recorded.
- Determine whether order matters.
- Apply all restrictions, such as replacement rules or allowable values.
- List or describe every possible recorded outcome.
- Check that no modeled possibility has been omitted.
- Check that elementary outcomes are distinct at the chosen level of detail.
- Use a list, table, tree, product description, interval, or set-builder notation as appropriate.
- Do not assume equal probability unless the model justifies it.
Common Error: Confusing the Experiment with an Outcome
“Roll a die” is an experiment.
“The die shows \(4\)” is an outcome.
“The set \(\{1,2,3,4,5,6\}\)” is the sample space for the face-value model.
Keeping these three levels distinct prevents many basic probability errors.
Common Error: Omitting an Outcome
Suppose a coin is tossed twice and a student writes
\[ S=\{HH,HT,TT\}. \]The outcome
\[ TH \]has been omitted.
The correct ordered sample space is
\[ \boxed{S=\{HH,HT,TH,TT\}}. \]Common Error: Ignoring Order
When two dice are distinguishable, the outcomes
\[ (2,5) \]and
\[ (5,2) \]are different ordered outcomes.
Combining them without justification changes the sample space and can change later probability calculations.
Common Error: Listing Categories That Overlap
Suppose a die roll is described using the labels
\[ \{\text{even},\text{greater than }3\}. \]This does not form a clean set of elementary outcomes because a roll of \(4\) or \(6\) satisfies both descriptions.
Categories used as elementary outcomes should be defined so that one trial produces exactly one of them.
Common Error: Assuming Listed Outcomes Are Equally Likely
Consider the number of heads in two fair coin tosses:
\[ S=\{0,1,2\}. \]These three values are not equally likely because
\[ 0\leftrightarrow TT, \] \[ 1\leftrightarrow HT\text{ or }TH, \]and
\[ 2\leftrightarrow HH. \]A compact sample space can hide multiplicities in the more detailed outcomes.
Common Error: Using an Overly Detailed Sample Space
More detail is not always better.
If the only recorded result is whether a die roll is odd or even, then
\[ S=\{\text{odd},\text{even}\} \]may be the most direct model.
The six-face sample space can still be useful, but it contains more detail than the recorded variable itself requires.
Common Error: Using a Sample Space That Is Too Coarse
A sample space can also discard information needed later.
If we record only
\[ \{\text{odd},\text{even}\} \]for a die roll, that model cannot distinguish the specific outcome \(6\) from the specific outcome \(2\).
Choose a level of detail that supports the questions the model is intended to answer.
Example: Full Sample-Space Workflow
Suppose a coin is tossed and then a six-sided die is rolled. Record both results in order.
The first stage has
\[ \{H,T\}, \]and the second has
\[ \{1,2,3,4,5,6\}. \]Therefore,
\[ \begin{aligned} S=\{& (H,1),(H,2),(H,3),(H,4),(H,5),(H,6),\\ &(T,1),(T,2),(T,3),(T,4),(T,5),(T,6) \}. \end{aligned} \]The sample space contains
\[ \boxed{|S|=2\cdot6=12} \]ordered outcomes.
Example: A Stopping-Time Experiment
Toss a coin until the first tail appears. Record the number of tosses.
Possible results include:
- \(1\), if the first toss is \(T\);
- \(2\), if the sequence begins \(HT\);
- \(3\), if the sequence begins \(HHT\);
- and so on.
Thus,
\[ \boxed{S=\{1,2,3,\ldots\}}. \]This illustrates a countably infinite sample space generated by a random stopping time.
Example: A Continuous Measurement
Suppose a machine cuts a piece whose modeled length \(L\), in centimeters, can take any real value from \(9.8\) through \(10.2\).
Then
\[ \boxed{S=[9.8,10.2]}. \]An outcome such as
\[ L=10.037 \]is one point in this continuous sample space.
Practice 1: One Coin
A coin is tossed once and the side facing upward is recorded.
Write the sample space.
Practice 2: One Die
A standard six-sided die is rolled and only whether the result is odd or even is recorded.
Write the sample space.
Practice 3: Two Coins
Two coins are tossed and the result of each toss is recorded in order.
List the sample space.
Practice 4: Number of Heads
Three coins are tossed, but only the total number of heads is recorded.
Write the sample space.
Practice 5: Two Dice
Two distinguishable six-sided dice are rolled and both face values are recorded.
Describe the sample space and determine \(|S|\).
Practice 6: Sum of Two Dice
Two six-sided dice are rolled and only the sum is recorded.
Write the sample space. Are the listed outcomes automatically equally likely?
Practice 7: Without Replacement
Two distinct objects are selected in order from a set of \(7\) distinct objects without replacement.
How many ordered outcomes are possible?
Practice 8: Countably Infinite
A die is rolled repeatedly until a \(6\) first appears. The number of rolls is recorded.
Write the sample space.
Practice 9: Continuous Measurement
The experiment records a temperature \(T\) that may take any real value from \(-5\) degrees through \(15\) degrees, inclusive.
Write the sample space using interval notation.
Practice 10: Identify the Error
A student says that for two coin tosses,
\[ S=\{HH,HT,TT\}. \]Identify and correct the error.
Practice 11: Choose the Right Level of Detail
One card is drawn from a standard deck, and the only recorded information is its suit.
Write a direct sample space for the recorded experiment.
Practice 12: Multistage Experiment
A coin is tossed and then a three-color spinner labeled red, blue, and green is spun. Record both outcomes in order.
List the sample space and determine its size.
Practice Answers
Practice 1
\[ \boxed{S=\{H,T\}}. \]Practice 2
\[ \boxed{S=\{\text{odd},\text{even}\}}. \]Practice 3
\[ \boxed{S=\{HH,HT,TH,TT\}}. \]Practice 4
The number of heads can be \(0\), \(1\), \(2\), or \(3\):
\[ \boxed{S=\{0,1,2,3\}}. \]Practice 5
Using ordered pairs,
\[ S=\{(i,j):i,j\in\{1,2,3,4,5,6\}\}. \]Therefore,
\[ \boxed{|S|=36}. \]Practice 6
\[ \boxed{S=\{2,3,4,5,6,7,8,9,10,11,12\}}. \]No. The sums are not automatically equally likely. Different sums can be produced by different numbers of ordered die pairs.
Practice 7
There are \(7\) choices first and \(6\) remaining choices second:
\[ \boxed{7\cdot6=42}. \]Practice 8
\[ \boxed{S=\{1,2,3,\ldots\}}. \]Practice 9
\[ \boxed{S=[-5,15]}. \]Practice 10
The outcome \(TH\) is missing. The correct ordered sample space is
\[ \boxed{S=\{HH,HT,TH,TT\}}. \]Practice 11
\[ \boxed{ S=\{\text{clubs},\text{diamonds},\text{hearts},\text{spades}\} }. \]Practice 12
\[ \boxed{ S=\{(H,R),(H,B),(H,G),(T,R),(T,B),(T,G)\} }. \]Therefore,
\[ \boxed{|S|=6}. \]What to Remember
The basic probability vocabulary is
\[ \boxed{ \begin{aligned} \text{experiment}&=\text{an uncertain process},\\ \text{trial}&=\text{one performance of the process},\\ \text{outcome}&=\text{one possible recorded result},\\ \text{sample space}&=\text{the set of all modeled outcomes}. \end{aligned} } \]- The sample space must include every modeled possibility.
- The experiment must state what is being recorded.
- Order matters whenever the procedure distinguishes positions or stages.
- A sample space may be finite, countably infinite, or continuous.
- The level of detail should match the probability question.
- Listing outcomes does not imply that they are equally likely.
- Tables, trees, counting principles, intervals, and set-builder notation can all describe sample spaces.
The most important habit is: define what can happen before calculating how likely it is.
Where We Go Next
A sample space tells us the possible outcomes of an experiment, but most probability questions ask about collections of outcomes.
The next lesson, Events and Event Operations, treats events as subsets of the sample space and develops the union, intersection, complement, and related set operations used throughout probability.