Uses of Venn Diagrams
Venn Diagrams
and Counting
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In this section we look at how Venn diagrams
are used to solve counting problems.
Venn diagrams are especially useful in
analyzing survey data.
An Illustration
How They Are Useful
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Each Venn diagram divides the universal
set U into a certain number of regions.
Each of these regions is called a basic
region of the Venn diagram.
Knowing the number of elements in each
basic region is of great use in many
applied problems.
A survey of 200 people yielded the following
information: 94 owned a VCR, 127 owned a
microwave oven, and 78 owned both. How
many people owned the following?
a.) a VCR or a microwave oven
b.) a VCR but not a microwave
c.) a microwave oven but not a VCR
d.) neither a VCR nor a microwave oven
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A survey of 200 people yielded the following
information: 94 owned a VCR, 127 owned a
microwave oven, and 78 owned both.
A survey of 200 people yielded the following information:
94 owned a VCR, 127 owned a microwave oven, and 78
owned both. How many people owned…
a.) a VCR or microwave oven?
n (V ∪ M) = 16 + 78 + 49 = 143
n(U) = 200
V
16
n(V) = 94
57
n(M) = 127
n(V ∩ M) = 78
M
78
49
V
b.) a VCR, but not a microwave oven?
n (V ∩ M′) = 16
c.) a microwave oven, but not a VCR?
16
M
78
49
57
n (M ∩ V′) = 49
d.) neither a VCR nor a microwave oven?
n (M′ ∩ V′) = 57
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Example
Example
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A survey of 100 college faculty who exercise
regularly found that 45 jog, 30 swim, 20
cycle, 6 jog and swim, 1 jogs and cycles, 5
swim and cycle, and 1 does all three.
How many of the faculty members do not do
any of these three activities?
How many just jog?
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Example: Small Businesses
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A survey of the characteristics of 100 small businesses that
had failed revealed that 95 of them either were
undercapitalized, had inexperienced management, or had a
poor location.
Four of the businesses had all three of these characteristics.
Forty businesses were undercapitalized but had experienced
management and good location.
Fifteen businesses had inexperienced management but
sufficient capitalization and good location.
Seven were undercapitalized and had inexperienced
management.
Nine were undercapitalized and had poor location.
Ten had inexperienced management and poor location.
A group of applicants for training as airtraffic controllers consisted of 35 pilots, 20
veterans, 30 pilots who were not veterans,
and 50 people who were neither veterans
nor pilots.
How large was the group?
Example: Small Businesses
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Construct a Venn diagram to represent this
situation.
How many of the businesses had poor location?
Which of the three characteristics was most
prevalent in the failed businesses?
Challenge Example!
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A company that specializes in language tutoring
lists the following information concerning its
English-speaking employees: 23 speak
German, 25 speak French, 31 speak Spanish, 43
speak Spanish or French, 38 speak French or
German, 46 speak German or Spanish, 8 speak
Spanish, French, and German, and 7 office
workers and secretaries speak English only.
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a.) What percent of the employees speak at least one
language other than English?
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b.) What percent of the employees speak at least two
languages other than English?
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