Events (Formal) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Events (Formal).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formal event is a subset of the sample space — a collection of outcomes to which a probability is assigned; events can be simple (one outcome) or compound (many outcomes).

An event is a question like 'Did I roll higher than 3?' that has yes/no answer.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An event is any collection of outcomes from the sample space to which a probability is assigned.

Common stuck point: The procedure for events (formal) is the easy part; the trap is treating an event as one outcome. Asking "Am I naming a set of outcomes that make a yes/no question true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I naming a set of outcomes that make a yes/no question true?

Worked Examples

Example 1

easy

Rolling a fair die: Event

A

= rolling an even number. Find

P(A)

and

P(A^c)

, and verify the complement rule.

Answer

P(A) = \frac{1}{2}

;

P(A^c) = \frac{1}{2}

; sum = 1. ✓

First step

Sample space:

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

Full solution

2
Event $A$ (even): $\{2,4,6\}$ ; $P(A) = \frac{3}{6} = \frac{1}{2}$
3
Complement $A^c$ (odd): $\{1,3,5\}$ ; $P(A^c) = \frac{3}{6} = \frac{1}{2}$
4
Verify: $P(A) + P(A^c) = \frac{1}{2} + \frac{1}{2} = 1$ ✓

The complement rule states

P(A^c) = 1 - P(A)

. An event and its complement are mutually exclusive and exhaustive — together they cover all possible outcomes. Often it's easier to compute

P(A) = 1 - P(A^c)

if the complement is simpler.

Example 2

medium

At least one approach: Find

P(\text{at least one head in 3 coin flips})

using the complement rule.

Example 3

medium

Two dice are rolled. Find

P(\text{sum} = 7)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

P(\text{rain tomorrow}) = 0.35

, find

P(\text{no rain tomorrow})

and explain the complement rule.

Example 2

hard

A system has 3 independent components, each failing with probability 0.1. The system fails if at least one component fails. Find

P(\text{system fails})

using the complement rule.

Example 3

easy

Rolling a die, the event 'even number' is which subset of the sample space?

Example 4

easy

How many outcomes are in the sample space of a single die roll?

Example 5

easy

Is the empty set

\emptyset

a valid event, and what is its probability?

Example 6

easy

For a die, is the event 'roll a 4' a simple or compound event?

Example 7

easy

For a die, find

P(\text{rolling more than 4})

Example 8

easy

Is the entire sample space itself an event, and what is its probability?

Example 9

easy

Drawing a card, the event 'a heart' has how many outcomes in a 52-card deck?

Example 10

easy

Is 'rolling an even number OR a number greater than 3' on a die a simple or compound event?

Example 11

medium

A die is rolled.

A=\{

even

\}=\{2,4,6\}

B=\{

greater than 3

\}=\{4,5,6\}

. Find

A\cap B

and its probability.

Example 12

medium

With

A=\{2,4,6\}

B=\{4,5,6\}

on a die, use inclusion-exclusion to find

P(A\cup B)

Example 13

medium

Two events on a die are mutually exclusive:

A=\{1,2\}

B=\{5,6\}

. Find

P(A\cup B)

Example 14

medium

The complement of event

A

('roll a 6',

P(A)=1/6

) is 'not 6'. Find

P(A^c)

Example 15

medium

Tossing two coins, list the sample space and find

P(\text{at least one head})

Example 16

medium

Why is 'A or B' not always

P(A)+P(B)

for two events on a die?

Example 17

medium

A spinner has 8 equal sectors numbered 1–8. Event

A

is 'multiple of 3'. List

A

and find

P(A)

Example 18

medium

A die is rolled. Event

A=\{1,2,3\}

and event

B=\{3,4\}

. Find

A\cup B

and

P(A\cup B)

Example 19

medium

For a die, event

A=\{2,4,6\}

. Describe its complement

A^c

and find

P(A^c)

Example 20

challenge

From a deck,

A=\{

heart

\}

B=\{

face card: J,Q,K

\}

. Using inclusion-exclusion, find

P(A\cup B)

Example 21

challenge

Rolling two dice, find

P(\text{sum}=8)

by listing the favorable outcomes in the 36-outcome sample space.

Example 22

challenge

Events

A

and

B

satisfy

P(A)=0.5

P(B)=0.4

P(A\cup B)=0.7

. Find

P(A\cap B)

and state whether

A,B

are mutually exclusive.

Example 23

easy

A die is rolled. List the event 'roll a number less than 3'.

Example 24

easy

A die is rolled. Find

P(\text{roll a number less than 3})

Example 25

easy

A coin is flipped. List the sample space.

Example 26

easy

A spinner has 4 equal colors: red, blue, green, yellow. Find

P(\text{red or blue})

Example 27

easy

Drawing a card from a 52-card deck, find

P(\text{ace})

Example 28

easy

For a die, is the event 'roll a 7' possible? What is its probability?

Example 29

medium

A bag has 5 red and 3 blue marbles. Find

P(\text{red})

Example 30

medium

Two events have

P(A) = 0.4

P(B) = 0.5

, and they are mutually exclusive. Find

P(A \cup B)

Example 31

medium

Two events have

P(A) = 0.6

P(B) = 0.5

, and

P(A \cap B) = 0.2

. Find

P(A \cup B)

Example 32

medium

Rolling a die,

A = \{\text{odd}\}

and

B = \{\text{prime}\}

. List

A \cap B

and find

P(A \cap B)

Example 33

medium

Three coins are flipped. Find

P(\text{exactly two heads})

Example 34

medium

A spinner has sectors

\{1,2,3,4,5,6,7,8\}

equally likely. Find

P(\text{even})

Example 35

medium

Two dice are rolled. Find

P(\text{sum} \ge 10)

Example 36

medium

From a deck, draw one card. Find

P(\text{not a face card})

Example 37

hard

A box has 4 red, 3 green, 5 blue balls. Two are drawn without replacement. Find

P(\text{both red})

Example 38

hard

Four coins are flipped. Find

P(\text{at least one head})

Example 39

hard

In a group of 30 students, 18 take Spanish, 14 take French, 6 take both. Find

P(\text{Spanish or French})

Example 40

hard

Two dice are rolled. Find

P(\text{at least one 6})

Example 41

challenge

A 5-card hand is drawn from a deck. Find

P(\text{all hearts})

Example 42

challenge

Two dice are rolled. Find

P(\text{product is odd})

← Back to Events (Formal) Formula Explained Practice Mode

Related Concepts

Sample Space Independent Events Conditional Probability

Background Knowledge

These ideas may be useful before you work through the harder examples.

sample space