Part 1 Module 3 Survey Problems and Venn diagrams EXAMPLE 1.3.1 A survey of 64 informed consumers revealed the following information: 45 believe that Elvis.
Download ReportTranscript Part 1 Module 3 Survey Problems and Venn diagrams EXAMPLE 1.3.1 A survey of 64 informed consumers revealed the following information: 45 believe that Elvis.
Slide 1
Part 1 Module 3
Survey Problems and Venn diagrams
EXAMPLE 1.3.1
A survey of 64 informed consumers revealed the
following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
1. How many believe Elvis is still alive or believe that
they have been abducted by space aliens?
2. How many believe neither of these things?
Slide 2
Solution
A survey of 64 informed consumers revealed the following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
We will use a Venn diagram to organize the information.
Let E represent the set of those who believe Elvis is still alive. Let A
represent the set of those who believe they have been abducted by
aliens. The universe (U) in this case is the set of 64 people who were
surveyed.
Slide 3
Solution, page 2
A survey of 64 informed consumers revealed the following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
42 believe both of these things
E
42
A
Slide 4
Solution, page 3
49 believe that they have been abducted by space aliens
49 believe that they have been
abducted by space aliens
E
42
7
A
Slide 5
Solution, page 4
45 believe that Elvis is still alive
45 believe that Elvis is still alive
E
3
42
7
A
Slide 6
Solution, page 5
64 people were surveyed.
64 people were surveyed.
12
E
3
42
7
A
Slide 7
Solution, page 6
1. How many believe Elvis is alive or believe that they have been
abducted by aliens?
The Venn diagram shows that the answer is 3 + 42 + 7 = 52
These are the people who believe Elvis
is still alive or believe they have been
abducted by aliens (or both).
12
E
3
42
7
A
Slide 8
Solution, page 7
2. How many believe neither of these things?
The Venn diagram shows that the answer is 12.
These are the people who believe
neither of the two things.
12
E
3
42
7
A
Slide 9
Exercise #2
99 grumpy old people were surveyed regarding things they
dislike.
66 dislike puppies 34 dislike kitties
32 dislike children
10 dislike puppies and dislike kitties and dislike children
28 dislike puppies and dislike children
8 dislike kitties and don't dislike children and don't dislike
puppies
25 dislike puppies and dislike kitties
How many don't dislike puppies and don't dislike kitties and
don't dislike children?
A. 50
B. 48
C. 16
D. 21
E. None of these
Slide 10
Solution
We will use a three-circle Venn diagram to organize the information.
Let P represent those who dislike puppies, K represent those who
dislike kitties, and C represent those who dislike children.
We will start with this fact:
10 dislike puppies and dislike kitties and dislike children
K
P
10
C
10 dislike puppies and
dislike kitties and dislike
children
Slide 11
Solution, page 2
Next:
8 dislike kitties and don't dislike children and don't dislike puppies
P
10
C
K
8
8 dislike kitties and don't
dislike children and don't
dislike puppies
Slide 12
Solution, page 3
Next:
25 dislike puppies and dislike kitties
Because 25 people dislike both puppies and kitties, the part of the diagram
where those circles intersect must contain a total of 25 people. Ten of the
25 have already been accounted for.
This causes the number 15 to appear in the region of the diagram where P and
K intersect without intersecting C.
25 dislike
(both)
puppies and
kitties
P
15
10
C
K
8
Slide 13
Solution, page 4
Next:
28 dislike puppies and dislike children
Because 28 people dislike both puppies and children, the part of the diagram
where those circles intersect must contain a total of 28 people. Ten of the
28 have already been accounted for.
This causes the number 18 to appear in the region where P and C intersect
without intersecting K.
28 dislike
(both)
puppies and
children
15
P
18
10
C
K
8
Slide 14
Solution, page 5
Next:
66 dislike puppies
Because 66 people dislike puppies, the four regions of circle P must contain a
total of 66 people.
This causes the number 23 to appear in the region that is inside P but outside
the other two circles.
66 dislike
puppies
P
23
18
15
10
C
K
8
Slide 15
Solution, page 6
Next:
34 dislike kitties
Because 34 people dislike kitties, the four regions of circle K must contain a
total of 34 people.
This causes the number 1 to appear in the region where K and C intersect
without intersecting P.
P
23
18
15
10
C
K
8
1
34 dislike
kitties
Slide 16
Solution, page 7
Next:
32 dislike children
Because 32 people dislike children, the four regions of circle C must contain a
total of 32 people. This causes the number 3 to appear in the region that is
inside C but outside the other two circles.
P
23
18
K
8
15
10
C
1
3
32 dislike
children
Slide 17
Solution, page 8
Next:
99 grumpy old people were surveyed.
Because 99 people were surveyed, the eight numbers in the diagram must add
up to 99. This allows us to find the number that occupies the region outside
of all three circles.
Call this number x. We add the seven numbers on the diagram, then subtract
the sum from 99.
x = 99 – (23 + 15 + 8 + 18 + 10 + 3)
x = 99 – 77 = 21.
P
23
x = 21
18
x
K
8
15
10
C
1
3
Slide 18
Solution, page 9
Now that the diagram is filled in, we use it to read the
answer to the question “How many don't dislike puppies
and don't dislike kitties and don't dislike children?”
The correct choice is D. 21.
P
23
21 dislike
none of the
three things.
18
21
K
8
15
10
C
1
3
Slide 19
Guidelines for 3-circle survey problems
1. If a description names all three categories, then that description is
pointing at exactly one of the eight regions of the diagram, so the
accompanying number does not get “split up.”
This is illustrated in the diagram below. Assume that a number of
Neanderthals were surveyed regarding their favorite possessions.
Set S represents those who own a stick, R represents those who
own a rock, and B represents those who own a bone.
3 own a rock and
a bone but not a stick
R
B
3
5
S
7
5 own a rock and
a bone and a stick
7 own a stick but
don’t own a rock and
don’t own a bone
Slide 20
Guidelines for 3-circle survey problems
2. If a description names the conjunction of two categories, then that
description is pointing at the intersection of two circles, so the
accompanying number gets “split up” among two regions of the
diagram.
9 own a stick
and rock
R
B
3
4 5 6
S
7
11 own a stick and
a bone
Slide 21
Guidelines for 3-circle survey problems
3. If a description names just one category, then that description is
pointing at an entire circle, so the accompanying number gets
“split up” among four regions of that circle.
20 own a
rock
R
8
3
1 B
4 5 6
S
7
15 own
a bone
Slide 22
Guidelines for 3-circle survey problems
4. “And” means “intersection,” or means “union.”
8 own a
rock
AND a
bone
R
8
3
1 B
R
4 5 6
S
7
8
26 own a stick OR
a bone
(3+1+4+5+6+7 = 26)
3 1 B
4 5 6
6
S
7
6
Slide 23
Exercise #2
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have friends on MyFace.com and
BookFace.com?
A. 19 B. 21 C. 57 D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 24
Exercise #3
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have exactly one of the three kinds of
friends?
A. 23 B. 33 C. 2 D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 25
Exercise #4
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have actual, real friends, but don’t have
friends on Bookface.com?
A. 1
B. 5
C. 2
D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 26
Exercise #5
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have actual real friends, or have friends
on MyFace.com?
A. 38
B. 16
C. 3
D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Part 1 Module 3
Survey Problems and Venn diagrams
EXAMPLE 1.3.1
A survey of 64 informed consumers revealed the
following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
1. How many believe Elvis is still alive or believe that
they have been abducted by space aliens?
2. How many believe neither of these things?
Slide 2
Solution
A survey of 64 informed consumers revealed the following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
We will use a Venn diagram to organize the information.
Let E represent the set of those who believe Elvis is still alive. Let A
represent the set of those who believe they have been abducted by
aliens. The universe (U) in this case is the set of 64 people who were
surveyed.
Slide 3
Solution, page 2
A survey of 64 informed consumers revealed the following information:
45 believe that Elvis is still alive
49 believe that they have been abducted by space aliens
42 believe both of these things
42 believe both of these things
E
42
A
Slide 4
Solution, page 3
49 believe that they have been abducted by space aliens
49 believe that they have been
abducted by space aliens
E
42
7
A
Slide 5
Solution, page 4
45 believe that Elvis is still alive
45 believe that Elvis is still alive
E
3
42
7
A
Slide 6
Solution, page 5
64 people were surveyed.
64 people were surveyed.
12
E
3
42
7
A
Slide 7
Solution, page 6
1. How many believe Elvis is alive or believe that they have been
abducted by aliens?
The Venn diagram shows that the answer is 3 + 42 + 7 = 52
These are the people who believe Elvis
is still alive or believe they have been
abducted by aliens (or both).
12
E
3
42
7
A
Slide 8
Solution, page 7
2. How many believe neither of these things?
The Venn diagram shows that the answer is 12.
These are the people who believe
neither of the two things.
12
E
3
42
7
A
Slide 9
Exercise #2
99 grumpy old people were surveyed regarding things they
dislike.
66 dislike puppies 34 dislike kitties
32 dislike children
10 dislike puppies and dislike kitties and dislike children
28 dislike puppies and dislike children
8 dislike kitties and don't dislike children and don't dislike
puppies
25 dislike puppies and dislike kitties
How many don't dislike puppies and don't dislike kitties and
don't dislike children?
A. 50
B. 48
C. 16
D. 21
E. None of these
Slide 10
Solution
We will use a three-circle Venn diagram to organize the information.
Let P represent those who dislike puppies, K represent those who
dislike kitties, and C represent those who dislike children.
We will start with this fact:
10 dislike puppies and dislike kitties and dislike children
K
P
10
C
10 dislike puppies and
dislike kitties and dislike
children
Slide 11
Solution, page 2
Next:
8 dislike kitties and don't dislike children and don't dislike puppies
P
10
C
K
8
8 dislike kitties and don't
dislike children and don't
dislike puppies
Slide 12
Solution, page 3
Next:
25 dislike puppies and dislike kitties
Because 25 people dislike both puppies and kitties, the part of the diagram
where those circles intersect must contain a total of 25 people. Ten of the
25 have already been accounted for.
This causes the number 15 to appear in the region of the diagram where P and
K intersect without intersecting C.
25 dislike
(both)
puppies and
kitties
P
15
10
C
K
8
Slide 13
Solution, page 4
Next:
28 dislike puppies and dislike children
Because 28 people dislike both puppies and children, the part of the diagram
where those circles intersect must contain a total of 28 people. Ten of the
28 have already been accounted for.
This causes the number 18 to appear in the region where P and C intersect
without intersecting K.
28 dislike
(both)
puppies and
children
15
P
18
10
C
K
8
Slide 14
Solution, page 5
Next:
66 dislike puppies
Because 66 people dislike puppies, the four regions of circle P must contain a
total of 66 people.
This causes the number 23 to appear in the region that is inside P but outside
the other two circles.
66 dislike
puppies
P
23
18
15
10
C
K
8
Slide 15
Solution, page 6
Next:
34 dislike kitties
Because 34 people dislike kitties, the four regions of circle K must contain a
total of 34 people.
This causes the number 1 to appear in the region where K and C intersect
without intersecting P.
P
23
18
15
10
C
K
8
1
34 dislike
kitties
Slide 16
Solution, page 7
Next:
32 dislike children
Because 32 people dislike children, the four regions of circle C must contain a
total of 32 people. This causes the number 3 to appear in the region that is
inside C but outside the other two circles.
P
23
18
K
8
15
10
C
1
3
32 dislike
children
Slide 17
Solution, page 8
Next:
99 grumpy old people were surveyed.
Because 99 people were surveyed, the eight numbers in the diagram must add
up to 99. This allows us to find the number that occupies the region outside
of all three circles.
Call this number x. We add the seven numbers on the diagram, then subtract
the sum from 99.
x = 99 – (23 + 15 + 8 + 18 + 10 + 3)
x = 99 – 77 = 21.
P
23
x = 21
18
x
K
8
15
10
C
1
3
Slide 18
Solution, page 9
Now that the diagram is filled in, we use it to read the
answer to the question “How many don't dislike puppies
and don't dislike kitties and don't dislike children?”
The correct choice is D. 21.
P
23
21 dislike
none of the
three things.
18
21
K
8
15
10
C
1
3
Slide 19
Guidelines for 3-circle survey problems
1. If a description names all three categories, then that description is
pointing at exactly one of the eight regions of the diagram, so the
accompanying number does not get “split up.”
This is illustrated in the diagram below. Assume that a number of
Neanderthals were surveyed regarding their favorite possessions.
Set S represents those who own a stick, R represents those who
own a rock, and B represents those who own a bone.
3 own a rock and
a bone but not a stick
R
B
3
5
S
7
5 own a rock and
a bone and a stick
7 own a stick but
don’t own a rock and
don’t own a bone
Slide 20
Guidelines for 3-circle survey problems
2. If a description names the conjunction of two categories, then that
description is pointing at the intersection of two circles, so the
accompanying number gets “split up” among two regions of the
diagram.
9 own a stick
and rock
R
B
3
4 5 6
S
7
11 own a stick and
a bone
Slide 21
Guidelines for 3-circle survey problems
3. If a description names just one category, then that description is
pointing at an entire circle, so the accompanying number gets
“split up” among four regions of that circle.
20 own a
rock
R
8
3
1 B
4 5 6
S
7
15 own
a bone
Slide 22
Guidelines for 3-circle survey problems
4. “And” means “intersection,” or means “union.”
8 own a
rock
AND a
bone
R
8
3
1 B
R
4 5 6
S
7
8
26 own a stick OR
a bone
(3+1+4+5+6+7 = 26)
3 1 B
4 5 6
6
S
7
6
Slide 23
Exercise #2
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have friends on MyFace.com and
BookFace.com?
A. 19 B. 21 C. 57 D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 24
Exercise #3
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have exactly one of the three kinds of
friends?
A. 23 B. 33 C. 2 D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 25
Exercise #4
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have actual, real friends, but don’t have
friends on Bookface.com?
A. 1
B. 5
C. 2
D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4
Slide 26
Exercise #5
A number of young people were surveyed about
their friends. The results are summarized in the
Venn diagram below.
How many have actual real friends, or have friends
on MyFace.com?
A. 38
B. 16
C. 3
D. None of these
15
M
19
1 2 3
A
1
M: Have “friends” on MyFace.com
B
17
B: Have “friends” on BookFace.com
A: Have actual, real friends
4