Using Venn Diagrams to
Solve Probability Problems
Venn Diagram Example 2
• A = Cars with Sunroofs
B = Cars with Air conditioning
• What does the shaded area represent ?
A
B
Venn Diagram Example 2
• A = Cars with Sunroofs
B = Cars with Air conditioning
• What does the shaded area represent ?
A
B
Venn Diagram Example 2
• A = Cars with Sunroofs
B = Cars with Air conditioning
• What does the shaded area represent ?
A
B
Union of 2 Events A and B
A B
• denoted by the symbol
• is the event containing all elements that belong to
A, B, or both.
• This is an OR probability problem!
Example:
If A = band members
and B = club
members, find the
probability of AUB in
the school.
Lewis High School
475
A
B
160
35
530
Intersection of 2 Events A and B
A B
• denoted by the symbol
• is the event containing all elements that are
COMMON to A and B
• This is an AND probability problem!
Example:
If A = drink coffee
and B = drink soda,
find the probability
that a person will
drink both.
Survey of Office Workers
83
A
31
B
12
25
Complement of an Event
• is the subset of all elements of sample
space that are not in the event
• Denoted as A' or Ac
Grayesville High Female Students
Example:
If A = plays
volleyball and B =
plays softball, find
the probability that
a person will not
play volleyball.
395
A
22
B
4
33
Additive rule of probability
Given events A and B, the probability of the union of
events A and B is the sum of the probability of events
A and B minus the probability of the intersection of
events A and B
P  A B   P  A  P  B   P  A B 
P  A B   P  A  P  B   P  A B 
Example:
The probability that a student belongs to a club is
P(C)=0.4. The probability that a student works part
time is P(PT)=0.5. The probability that a student
belongs to a club AND works part time is P(C and
PT)=0.05. What is the probability that a student
belongs to a club OR works part time??
Answer: P(C  PT )  P(C )  P( PT )  P(C  PT )
 0.4  0.5  0.05
 0.85
P  A B   P  A  P  B   P  A B 
Example:
A = owns a car
B = has a pet
P(A) = 0.87
P(B) = 0.57
P(A and B) = 0.53
What is the probability that a student owns a car OR
has a pet??
Answer:
 0.87  0.57  0.53
 0.91
P  A B   P  A  P  B   P  A B 
Example: A survey finds that 56% of people are
married. They ask the same group of people, and
67% have at least one child. If there are 41% that are
married and have at least one child, what is the
probability that a person in the survey is married OR
has a child??
Answer:
 0.56  0.67  0.41
 0.82
Ex. A card is drawn from a well-shuffled
deck of 52 playing cards. What is the
probability that it is a queen or a heart?
Q = Queen and H = Heart
4
13
P(Q)  , P( H )  , P(Q
52
52
1
H) 
52
P(Q H )  P(Q)  P( H )  P(Q H )
4 13 1

 
52 52 52
16 4


52 13
Mutually Exclusive Events
Two events are mutually exclusive if
A B  
This means that A and B have no
elements in common.
Draw a Venn Diagram that depicts two
mutually exclusive events.
Mutually Exclusive Events
P  A B   P  A  P  B 
A: Birthday in
Summer: 38
P( A  B )  
B: Birthday in
Winter: 56
Birthday in
Spring or
Fall: 116
38
56
P( A  B) 

210 210
94
47


210 105
Draw each Venn diagram (and label!).
State whether the events are
mutually exclusive:
A. Rolling a die. A = even, B = odd.
B. Drawing a card from a regular deck.
A = red, B = black.
C. Picking a number from 1-100.
A = even, B = # less than 40.
D. Drawing a card from a regular deck.
A = Jack, B = Ace.
E. Drawing a card from a regular deck.
A = Heart, B = Diamond, C = Queen.