Sample Spaces and Events

Probability is the mathematics of uncertainty. Before we can calculate the probability of an event, we need to know all the outcomes that could possibly occur.

The complete collection of possible outcomes is called the sample space.

For example, when rolling a standard six-sided die:

S = {1, 2, 3, 4, 5, 6}

If we are interested in rolling an even number, the event is:

E = {2, 4, 6}

The sample space describes everything that could happen, while an event describes the outcomes we are interested in.

Outcomes, Sample Spaces, and Events

These three ideas form the foundation of probability.

Outcome

An outcome is one possible result of an experiment.

Examples:

• rolling a 4
• flipping heads
• drawing the queen of hearts
• selecting a red marble

Sample Space

The sample space is the set of all possible outcomes.

It is commonly represented by:

S

Event

An event is a collection of one or more outcomes from the sample space.

For a die:

S = {1, 2, 3, 4, 5, 6}

An event A representing "roll an odd number" is:

A = {1, 3, 5}

An event B representing "roll a number greater than 4" is:

B = {5, 6}

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Constructing a Sample Space

A sample space should include:

• every possible outcome
• each outcome only once
• clearly distinguishable outcomes

For a coin toss:

S = {H, T}

where:

H = heads
T = tails

For choosing a day of the weekend:

S = {Saturday, Sunday}

For selecting a letter from the word MATH:

S = {M, A, T, H}

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Sample Spaces and Set Notation

Because a sample space is a set, we normally write its outcomes inside braces:

{ }

For example:

S = {1, 2, 3, 4, 5, 6}

Individual outcomes are separated by commas.

If:

A = {2, 4, 6}

then A represents an event containing three outcomes.

The notation:

2 ∈ A

means:

"2 is an element of A."

The notation:

3 ∉ A

means:

"3 is not an element of A."

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Simple Events

A simple event contains only one outcome.

Suppose:

S = {1, 2, 3, 4, 5, 6}

The event:

A = {4}

is a simple event because it contains exactly one outcome.

Examples include:

• rolling exactly 5
• drawing the ace of spades
• flipping heads once
• selecting one specific person

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Compound Events

A compound event contains more than one outcome.

For a die:

A = {2, 4, 6}

represents the event:

"roll an even number."

This is a compound event because three different outcomes satisfy the condition.

Another event:

B = {4, 5, 6}

could represent:

"roll a number greater than 3."

Compound events are extremely common in probability.

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Counting Outcomes

The notation:

n(S)

means the number of outcomes in the sample space.

For:

S = {1, 2, 3, 4, 5, 6}

we have:

n(S) = 6

If:

A = {2, 4, 6}

then:

n(A) = 3

When all outcomes are equally likely:

P(A) = n(A)/n(S)

Therefore:

P(A) = 3/6

= 1/2

This connects sample spaces directly to theoretical probability.

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Two Coin Tosses

Suppose we toss a coin twice.

Simply writing:

{H, T}

is no longer enough because we need to record the result of both tosses.

The sample space is:

S = {HH, HT, TH, TT}

There are four possible outcomes.

Notice that:

HT ≠ TH

because the order is different.

Events with Two Coins

Using:

S = {HH, HT, TH, TT}

consider the event:

A = exactly one head

Then:

A = {HT, TH}

So:

n(A) = 2

and:

n(S) = 4

Therefore:

P(A) = 2/4

= 1/2

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At Least and At Most

Probability questions often use phrases such as at least and at most.

At Least

"At least one head" means:

one or more heads

For two tosses:

A = {HH, HT, TH}

The only excluded outcome is TT.

Therefore:

P(A) = 3/4

At Most

"At most one head" means:

zero or one head

So:

B = {HT, TH, TT}

Again:

P(B) = 3/4

Understanding this language is essential when constructing events correctly.

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Using a Table to Construct a Sample Space

Suppose we roll two six-sided dice.

There are too many outcomes to list casually, so a table is useful.

Each outcome can be represented as an ordered pair:

(first die, second die)

For example:

(2, 5)

means:

first die = 2
second die = 5

A systematic sample space begins:

(1,1), (1,2), (1,3), ..., (1,6)

(2,1), (2,2), (2,3), ..., (2,6)

and continues through:

(6,1), (6,2), ..., (6,6)

There are:

6 × 6 = 36 outcomes

Why Ordered Pairs Matter

When rolling two dice:

(2, 5)

and:

(5, 2)

are different outcomes.

They produce the same total, but the individual dice show different results.

This distinction is important.

There are 36 equally likely ordered outcomes, but there are only 11 possible sums:

2, 3, 4, ..., 12

These sums are not equally likely.

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Example: Sum of 7

Let event A be:

"the sum of two dice is 7."

Then:

A = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

Therefore:

n(A) = 6

Since:

n(S) = 36

we have:

P(A) = 6/36

= 1/6

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Why Some Dice Sums Are More Likely

Consider a sum of 2.

Only one outcome produces it:

(1,1)

So:

P(sum of 2) = 1/36

Now consider a sum of 7.

Six outcomes produce it.

Therefore:

P(sum of 7) = 6/36

= 1/6

So although the numbers 2 and 7 are both possible sums, they are not equally likely.

This illustrates why constructing the complete sample space can be so useful.

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Tree Diagrams

A tree diagram is another systematic method for constructing sample spaces.

Suppose we toss a coin and then roll a three-sided spinner labelled:

1, 2, 3

The first stage has:

H or T

Each of these branches then has:

1, 2, or 3

The outcomes are:

S = {H1, H2, H3, T1, T2, T3}

Therefore:

n(S) = 6

Tree diagrams are especially useful for experiments that occur in stages.

The Multiplication Principle

If an experiment has several stages, the number of possible outcomes can often be found by multiplication.

Suppose:

• Stage 1 has 3 possible outcomes.
• Stage 2 has 4 possible outcomes.

Then:

total outcomes = 3 × 4 = 12

If a meal includes:

3 starters

4 main courses

2 desserts

then the number of possible meals is:

3 × 4 × 2 = 24

This is sometimes called the fundamental counting principle.

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Events as Sets

Events are sets, so we can use set notation to describe relationships between them.

Suppose:

S = {1, 2, 3, 4, 5, 6}

Let:

A = {2, 4, 6}

represent even numbers.

Let:

B = {4, 5, 6}

represent numbers greater than 3.

These events overlap.

The outcome 4 belongs to both.

The outcome 6 also belongs to both.

This leads to important set operations.

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Intersection: A AND B

The intersection of two events contains outcomes that belong to both events.

Notation:

A ∩ B

Using:

A = {2, 4, 6}

B = {4, 5, 6}

we get:

A ∩ B = {4, 6}

This represents:

"even AND greater than 3."

Union: A OR B

The union contains outcomes belonging to A, B, or both.

Notation:

A ∪ B

Using:

A = {2, 4, 6}

B = {4, 5, 6}

we obtain:

A ∪ B = {2, 4, 5, 6}

Notice that 4 and 6 are written only once.

Sets do not list duplicate elements.

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Complement: NOT A

The complement of an event contains all outcomes in the sample space that are not in the event.

It can be written as:

Aᶜ

Suppose:

S = {1, 2, 3, 4, 5, 6}

and:

A = {2, 4, 6}

Then:

Aᶜ = {1, 3, 5}

If A means:

"roll an even number"

then Aᶜ means:

"do not roll an even number."

In this case:

"roll an odd number."

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Complement Probability

Because an event either occurs or does not occur:

P(A) + P(Aᶜ) = 1

Therefore:

P(Aᶜ) = 1 − P(A)

This is called the complement rule.

For example, if:

P(A) = 0.7

then:

P(Aᶜ) = 1 − 0.7

= 0.3

This becomes extremely useful for problems involving phrases such as:

"at least one"

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Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time.

For a single die roll:

A = "roll a 2"

B = "roll a 5"

A and B cannot both occur on the same roll.

Therefore:

A ∩ B = ∅

The symbol:

∅

means the empty set.

It contains no outcomes.

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Events That Are Not Mutually Exclusive

Suppose:

A = "roll an even number"

B = "roll a number greater than 3"

Then:

A = {2, 4, 6}

B = {4, 5, 6}

Since:

A ∩ B = {4, 6}

the events can occur together.

Therefore, they are not mutually exclusive.

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Certain and Impossible Events

The sample space itself represents a certain event.

Therefore:

P(S) = 1

The empty set represents an impossible event.

Therefore:

P(∅) = 0

All probabilities must satisfy:

0 ≤ P(A) ≤ 1

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Venn Diagrams

A Venn diagram provides a visual representation of events.

Usually:

• the rectangle represents the sample space
• circles represent events
• overlapping regions represent intersections
• everything inside either circle represents a union
• everything outside an event represents its complement

Venn diagrams become especially useful when several events overlap.

Example Using a Venn Diagram

Suppose a group of 30 students includes:

18 who play football

12 who play basketball

5 who play both

Let:

F = football

B = basketball

Since 5 students belong to both groups:

n(F ∩ B) = 5

Students who play only football:

18 − 5 = 13

Students who play only basketball:

12 − 5 = 7

Students who play at least one sport:

13 + 5 + 7

= 25

Therefore, students who play neither:

30 − 25

= 5

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The Addition Rule

When events overlap:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Why subtract the intersection?

Because outcomes belonging to both A and B would otherwise be counted twice.

If events are mutually exclusive:

P(A ∩ B) = 0

so the rule simplifies to:

P(A ∪ B) = P(A) + P(B)

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Example: Playing Sports

Using the previous example:

30 students total

18 play football

12 play basketball

5 play both

Probability that a randomly selected student plays football or basketball:

P(F ∪ B) = 18/30 + 12/30 − 5/30

= 25/30

= 5/6

The subtraction prevents the five students who play both sports from being counted twice.

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Systematic Listing

When constructing a sample space, it is important to work systematically.

Suppose a café offers:

Bread:

white, wholegrain

Filling:

cheese, chicken, tuna

The possible sandwiches are:

white-cheese

white-chicken

white-tuna

wholegrain-cheese

wholegrain-chicken

wholegrain-tuna

There are:

2 × 3 = 6 possibilities

A systematic approach helps prevent missing or repeating outcomes.

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Three Coin Tosses

Suppose a fair coin is tossed three times.

The sample space is:

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

Therefore:

n(S) = 8

This also follows from:

2 × 2 × 2 = 8

or:

2³ = 8

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Example: Exactly Two Heads

Let A represent:

"exactly two heads."

From the sample space:

A = {HHT, HTH, THH}

Therefore:

n(A) = 3

and:

P(A) = 3/8

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Example: At Least One Head

Using three coin tosses:

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

The event "at least one head" includes seven outcomes.

We could count all seven.

But there is an easier method.

The complement of "at least one head" is:

no heads

which is:

TTT

Therefore:

P(at least one H)

= 1 − P(TTT)

= 1 − 1/8

= 7/8

This is a powerful use of the complement rule.

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Simple and Compound Experiments

A simple experiment involves one stage.

Examples:

• roll one die
• flip one coin
• draw one card

A compound experiment involves two or more stages.

Examples:

• flip two coins
• roll two dice
• select a card and then spin a spinner
• choose a shirt and trousers
• toss a coin three times

Compound experiments usually have larger sample spaces and benefit from tables or tree diagrams.

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Ordered and Unordered Outcomes

Sometimes order matters.

Suppose two students are selected for:

captain and vice-captain

Choosing:

Alice as captain and Ben as vice-captain

is different from:

Ben as captain and Alice as vice-captain.

Order matters because the roles are different.

But if two students are simply selected to form a team, then:

Alice and Ben

is the same pair as:

Ben and Alice.

Before constructing a sample space, always ask:

Does order matter?

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Equally Likely Outcomes

The simple probability formula:

P(A) = n(A)/n(S)

works directly when the outcomes in S are equally likely.

For a fair die:

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

are equally likely.

But consider a spinner where half the circle is red and the remaining half is divided between blue and green.

Writing:

S = {red, blue, green}

does not mean each colour has probability 1/3.

The outcomes listed by name are not necessarily equally likely.

This distinction is important when solving probability problems.

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Sample Spaces and Real-World Decisions

Sample spaces are not limited to coins and dice.

They can be used to analyse:

• medical testing outcomes
• weather possibilities
• manufacturing defects
• sports results
• genetics
• insurance risks
• survey responses
• transportation choices
• quality control
• computer algorithms

The central idea remains the same:

identify all possible outcomes before calculating probabilities.

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A Systematic Problem-Solving Strategy

When solving a sample-space problem:

Identify the Experiment

What process is taking place?

Identify Each Stage

Is there one action or several?

Decide Whether Order Matters

Would changing the order create a different outcome?

Construct the Sample Space

Use:

• a list
• a table
• a tree diagram
• ordered pairs
• set notation

Define the Event

Identify exactly which outcomes satisfy the condition.

Count the Outcomes

Determine:

n(S) and n(A)

Calculate Probability

If outcomes are equally likely:

P(A) = n(A)/n(S)

Check

Make sure:

0 ≤ P(A) ≤ 1

and that no possible outcomes were missed.

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Common Mistakes

Confusing an outcome with an event

An outcome is one possible result.

An event may contain one or many outcomes.

Leaving outcomes out of the sample space

A sample space must include every possible outcome.

Counting an outcome twice

Each distinct outcome should appear only once.

Assuming order never matters

HT and TH are different outcomes when recording two sequential coin tosses.

Assuming order always matters

If two people are simply selected as a pair, AB and BA may represent the same outcome.

Confusing AND with OR

A ∩ B means A AND B.

A ∪ B means A OR B.

Forgetting the overlap

When finding A ∪ B, outcomes in A ∩ B should not be counted twice.

Misinterpreting "at least"

"At least 2" means:

2 or more

not exactly 2.

Misinterpreting "at most"

"At most 2" means:

2 or fewer

Assuming listed outcomes are equally likely

The way outcomes are named does not guarantee equal probability.

Confusing mutually exclusive with independent

Mutually exclusive events cannot occur together.

Independent events are events where one does not change the probability of the other.

These are different ideas.

Using n(A)/n(S) when outcomes are not equally likely

This formula requires equally likely outcomes.

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Key Terms

Probability experiment — A process that produces an uncertain outcome.

Outcome — One possible result of a probability experiment.

Sample space — The set containing all possible outcomes of an experiment.

Event — A set of outcomes satisfying a particular condition.

Simple event — An event containing exactly one outcome.

Compound event — An event containing more than one outcome.

Set — A collection of distinct objects or outcomes.

Element — An individual member of a set.

Set notation — Mathematical notation used to describe sets and their relationships.

n(S) — The number of outcomes in sample space S.

n(A) — The number of outcomes belonging to event A.

Intersection — Outcomes belonging to both events.

A ∩ B — The intersection of events A and B; A AND B.

Union — Outcomes belonging to at least one of two events.

A ∪ B — The union of events A and B; A OR B.

Complement — All outcomes in the sample space that do not belong to an event.

Aᶜ — The complement of event A.

Empty set — A set containing no elements, written ∅.

Mutually exclusive events — Events that cannot occur simultaneously.

Certain event — An event with probability 1.

Impossible event — An event with probability 0.

Equally likely outcomes — Outcomes that have the same probability of occurring.

Ordered pair — Two values written in a particular order, such as (2,5).

Tree diagram — A branching diagram showing possible outcomes at successive stages.

Venn diagram — A diagram using regions to represent sets and their relationships.

Fundamental counting principle — A method of counting outcomes by multiplying the number of possibilities at successive stages.

Complement rule — The relationship P(Aᶜ) = 1 − P(A).

Addition rule — A rule used to find the probability of A or B while accounting for overlap.

At least — A stated amount or more.

At most — A stated amount or less.

Compound experiment — A probability experiment involving multiple stages.

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Key Takeaways

• An outcome is one possible result.
• The sample space contains every possible outcome.
• An event contains the outcomes that satisfy a particular condition.
• Sample spaces are commonly written using set notation.
• A simple event contains one outcome.
• A compound event contains multiple outcomes.
• The number of outcomes in S is written n(S).
• When outcomes are equally likely:

P(A) = n(A)/n(S)

• Sample spaces should be constructed systematically so outcomes are not missed or duplicated.
• Tables are useful for experiments involving two variables, such as rolling two dice.
• Tree diagrams are useful for experiments involving several stages.
• The fundamental counting principle can determine the number of possible outcomes without listing all of them.
• Order sometimes matters and sometimes does not.
• For two coin tosses, HT and TH are distinct outcomes.
• A ∩ B means A AND B.
• A ∪ B means A OR B.
• Aᶜ means NOT A.
• Mutually exclusive events have no outcomes in common.
• The empty set ∅ represents an impossible event.
• The sample space itself represents a certain event.
• The complement rule is:

P(Aᶜ) = 1 − P(A)

• Complement reasoning is especially useful for "at least one" problems.
• The addition rule is:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

• Listed outcomes are not automatically equally likely.
• A complete sample space often makes a difficult probability problem much easier to understand.
• Sample spaces and events provide the foundation for more advanced probability.

A useful strategy is:

Experiment → Sample Space → Event → Count → Probability

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Check Your Understanding

1. Define an outcome.

2. Define a sample space.

3. Define an event.

4. Write the sample space for tossing one coin.

5. Write the sample space for rolling one standard die.

6. For a die roll, write the event A representing an even number.

7. Is A in Question 6 a simple or compound event?

8. Write the sample space for tossing a coin twice.

9. Write the event representing exactly one head.

10. Calculate the probability of exactly one head.

11. Write the event representing at least one head.

12. Calculate its probability.

13. A coin is tossed three times. How many outcomes are in the sample space?

14. List the complete sample space for three coin tosses.

15. Write the event representing exactly two heads.

16. Calculate its probability.

17. Two standard dice are rolled. How many ordered outcomes are possible?

18. List all outcomes that produce a sum of 8.

19. Calculate the probability of rolling a sum of 8.

20. Explain why a sum of 7 is more likely than a sum of 2 when rolling two fair dice.

21. Let:

A = {1, 2, 3, 4}

B = {3, 4, 5, 6}

Find:

A ∩ B

22. Find:

A ∪ B

23. If:

S = {1, 2, 3, 4, 5, 6}

and:

A = {2, 4, 6}

find:

Aᶜ

24. Explain what it means for two events to be mutually exclusive.

25. If:

P(A) = 0.35

find:

P(Aᶜ)

26. A restaurant offers 4 starters, 6 main courses, and 3 desserts. How many different three-course meals are possible?

27. Explain why a tree diagram can be useful for a compound experiment.

28. Explain the difference between "at least 2" and "at most 2."

29. Explain why the formula P(A) = n(A)/n(S) cannot automatically be used when outcomes are not equally likely.

30. Challenge: A student rolls a standard six-sided die and then tosses a fair coin.

a. Construct the complete sample space using outcomes such as (1,H).
b. Determine n(S).
c. Let A be the event "the die shows an even number." List A.
d. Let B be the event "the coin shows heads." List B.
e. Find A ∩ B.
f. Find A ∪ B.
g. Find Aᶜ.
h. Calculate P(A).
i. Calculate P(B).
j. Calculate P(A ∩ B).
k. Calculate P(A ∪ B).
l. Determine whether A and B are mutually exclusive and explain your answer.
m. Find the probability that the die shows an odd number OR the coin shows tails.
n. Explain how a tree diagram could represent the same sample space.
o. Explain why constructing the complete sample space helps reduce errors in this problem.