Events (Formal) Examples: 45 Problems with Answers
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 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(Ac), and verify the complement rule.
Answer
P(A)=21; P(Ac)=21; sum = 1. ✓
First step
1
Sample space: S={1,2,3,4,5,6}
Full solution
2
Event A (even): {2,4,6}; P(A)=63=21
3
Complement Ac (odd): {1,3,5}; P(Ac)=63=21
4
Verify: P(A)+P(Ac)=21+21=1 ✓
The complement rule states P(Ac)=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(Ac) if the complement is simpler.
Example 2
medium
At least one approach: Find P(at least one head in 3 coin flips) using the complement rule.
Example 3
medium
Two dice are rolled. Find P(sum=7).
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
If P(rain tomorrow)=0.35, find P(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(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 ∅ 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(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∩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∪B).
Example 13
medium
Two events on a die are mutually exclusive: A={1,2}, B={5,6}. Find P(A∪B).
Example 14
medium
The complement of event A ('roll a 6', P(A)=1/6) is 'not 6'. Find P(Ac).
Example 15
medium
Tossing two coins, list the sample space and find P(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∪B and P(A∪B).
Example 19
medium
For a die, event A={2,4,6}. Describe its complement Ac and find P(Ac).
Example 20
challenge
From a deck, A={heart}, B={face card: J,Q,K}. Using inclusion-exclusion, find P(A∪B).
Example 21
challenge
Rolling two dice, find P(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∪B)=0.7. Find P(A∩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(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(red or blue).
Example 27
easy
Drawing a card from a 52-card deck, find P(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(red).
Example 30
medium
Two events have P(A)=0.4, P(B)=0.5, and they are mutually exclusive. Find P(A∪B).
Example 31
medium
Two events have P(A)=0.6, P(B)=0.5, and P(A∩B)=0.2. Find P(A∪B).
Example 32
medium
Rolling a die, A={odd} and B={prime}. List A∩B and find P(A∩B).
Example 33
medium
Three coins are flipped. Find P(exactly two heads).
Example 34
medium
A spinner has sectors {1,2,3,4,5,6,7,8} equally likely. Find P(even).
Example 35
medium
Two dice are rolled. Find P(sum≥10).
Example 36
medium
From a deck, draw one card. Find P(not a face card).
Example 37
hard
A box has 4 red, 3 green, 5 blue balls. Two are drawn without replacement. Find P(both red).
Example 38
hard
Four coins are flipped. Find P(at least one head).
Example 39
hard
In a group of 30 students, 18 take Spanish, 14 take French, 6 take both. Find P(Spanish or French).
Example 40
hard
Two dice are rolled. Find P(at least one 6).
Example 41
challenge
A 5-card hand is drawn from a deck. Find P(all hearts).