using venn diagrams to solve probability problems · complement of an event •is the subset of all...
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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 ?
AB
Venn Diagram Example 2
• A = Cars with Sunroofs
B = Cars with Air conditioning
• What does the shaded area represent ?
BA
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 • denoted by the symbol
• is the event containing all elements that belong to
A, B, or both.
• This is an OR probability problem!
A B
A B
Example:
If A = band members
and B = club
members, find the
probability of AUB in
the school.
160
Lewis High School
530
35
475
Intersection of 2 Events A and B
• denoted by the symbol
• is the event containing all elements that are
COMMON to A and B
• This is an AND probability problem!
A B
A B
Example:
If A = drink coffee
and B = drink soda,
find the probability
that a person will
drink both.
83
Survey of Office Workers
31 25 12
Complement of an Event
• is the subset of all elements of sample
space that are not in the event
• Denoted as A' or Ac
A
B
Example:
If A = plays
volleyball and B =
plays softball, find
the probability that
a person will not
play volleyball.
Grayesville High Female Students
395
33
22 4
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?
4 13 1( ) , ( ) , ( )
52 52 52P Q P H P Q H
( ) ( ) ( ) ( )P Q H P Q P H P Q H
4 13 1
52 52 52
16 4
52 13
Q = Queen and H = Heart
Mutually Exclusive Events
Two events are mutually exclusive if
This means that A and B have no
elements in common.
Draw a Venn Diagram that depicts two
mutually exclusive events.
A B
Mutually Exclusive Events P A B P A P B
A: Birthday in
Summer: 38
B: Birthday in
Winter: 56
Birthday in
Spring or
Fall: 116
( )P A B 38 56( )
210 210P A B
94 47
210 105
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.
Draw each Venn diagram (and label!).
State whether the events are
mutually exclusive: