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Venn diagrams & set notation
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35 questions at your level
Difficulty
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Level 1
Level 2
Level 3
Level 4
Level 5
Venn diagrams
13 questions
Lesson
Not started
Mark as done
30
30
30
students were asked whether they play football (F) or tennis (T).
ξ
F
T
12
5
8
Work out how many students play neither sport.
●
●●●●
Level 1
2 marks
Start
→
Mark as done
Here is a Venn diagram.
ξ
4
A
B
7
3
6
How many elements are in set
A
A
A
?
●
●●●●
Level 1
1 mark
Start
→
Mark as done
In a class of
25
25
25
students,
15
15
15
have a cat,
9
9
9
have a dog and
4
4
4
have both. Draw a Venn diagram (or reason it out) and find how many have a cat but not a dog.
●
●●●●
Level 1
1 mark
Start
→
Mark as done
There are
30
30
30
students in a class. Each student was asked whether they have a sister and whether they have a brother.
The Venn diagram shows the results.
S
S
S
is the group who have a sister and
B
B
B
is the group who have a brother.
ξ
5
S
B
11
6
8
(a)
Write down the number of students who have a sister.
(b)
Write down the number of students who have a brother but not a sister.
(c)
Work out the number of students who have at least one sibling.
●●
●●●
Level 2
4 marks
Start
→
Mark as done
50
50
50
people were asked whether they had visited France (
F
F
F
) or Spain (
S
S
S
).
28
28
28
had visited France,
24
24
24
had visited Spain, and
6
6
6
had visited neither.
(a)
Work out how many people had visited both France and Spain. (2 marks)
(b)
One of the
50
50
50
people is chosen at random. Work out the probability that they had visited France but not Spain. (1 mark)
●●●
●●
Level 3
3 marks
Start
→
Mark as done
85
85
85
students were asked whether they study Spanish and whether they study German.
47
47
47
study Spanish,
33
33
33
study German,
15
15
15
study both Spanish and German.
(a)
Draw a Venn diagram to show this information.
(b)
Work out how many of the
85
85
85
students study exactly one of the two languages.
●●●
●●
Level 3
5 marks
Start
→
Mark as done
A sports club has
50
50
50
members.
26
26
26
of the members play badminton,
19
19
19
play squash and
8
8
8
play both badminton and squash.
(a)
Work out how many members play neither badminton nor squash.
(b)
Sam says: "
26
+
19
=
45
26 + 19 = 45
26
+
19
=
45
, and
50
−
45
=
5
50 - 45 = 5
50
−
45
=
5
, so
5
5
5
members play neither."
Explain why Sam's method is wrong.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
Owen carried out a survey of
90
90
90
people.
He asked each person if they like apples (
A
A
A
), bananas (
B
B
B
) or grapes (
G
G
G
).
9
9
9
people like all three fruits.
23
23
23
people like apples and bananas.
11
11
11
people like apples and grapes but not bananas.
16
16
16
people like bananas and grapes.
44
44
44
people like grapes.
54
54
54
people like apples.
6
6
6
people like only bananas.
(a)
Complete the Venn diagram for this information.
ξ
A
B
G
(4 marks)
One of the
90
90
90
people is chosen at random.
Given that this person likes grapes,
(b)
find the probability that this person also likes bananas.
(2 marks)
●●●●
●
Level 4
6 marks
Start
→
Mark as done
40
40
40
people were asked whether they like tea and whether they like coffee.
24
24
24
of them like tea and
21
21
21
of them like coffee.
The Venn diagram shows some of this information.
T
T
T
is the group who like tea,
C
C
C
is the group who like coffee, and
x
x
x
is the number of people who like both.
ξ
3
T
C
x
(a)
Work out the value of
x
x
x
.
(b)
Write down how many of the
40
40
40
people like tea but not coffee.
●●●●
●
Level 4
4 marks
Start
→
Mark as done
50
50
50
students were asked which of the art club (
A
A
A
), the drama club (
D
D
D
) and the music club (
M
M
M
) they attend.
The Venn diagram shows the results.
ξ
14
A
D
M
9
7
6
5
4
3
2
(a)
Work out how many students attend the art club.
(b)
Work out how many students attend exactly two of the three clubs.
(c)
Work out how many students attend at least one club.
●●●●
●
Level 4
6 marks
Start
→
Mark as done
60
60
60
pupils were asked which of football, netball and rounders they play.
28
28
28
play football,
25
25
25
play netball,
22
22
22
play rounders,
10
10
10
play football and netball,
8
8
8
play football and rounders,
9
9
9
play netball and rounders,
4
4
4
play all three sports.
Here is an empty Venn diagram, with the circles labelled
F
F
F
(football),
N
N
N
(netball) and
R
R
R
(rounders).
ξ
F
N
R
(a)
Copy the diagram and write the correct number of pupils in each of the seven regions inside the circles.
(b)
Work out how many of the
60
60
60
pupils play none of the three sports.
●●●●
●
Level 4
6 marks
Start
→
Mark as done
100
100
100
people were asked which of apples, bananas and cherries they had eaten in the last week.
52
52
52
had eaten apples,
41
41
41
had eaten bananas,
39
39
39
had eaten cherries,
20
20
20
had eaten apples and bananas,
17
17
17
had eaten apples and cherries,
15
15
15
had eaten bananas and cherries,
9
9
9
had eaten none of the three fruits.
(a)
Work out how many of the
100
100
100
people had eaten all three fruits.
(b)
Work out how many of the
100
100
100
people had eaten exactly one of the three fruits.
●●●●●
Level 5
6 marks
Start
→
Mark as done
A youth club has
64
64
64
members. Every member does at least one of drama and dance.
28
28
28
members do drama but not dance.
The number of members who do both drama and dance is
3
3
3
times the number who do dance but not drama.
(a)
Work out how many members do dance.
(b)
Work out how many members do drama.
●●●●●
Level 5
5 marks
Start
→
Set notation
11 questions
Lesson
Not started
Mark as done
A
=
{
1
,
2
,
3
,
4
}
A = \{1, 2, 3, 4\}
A
=
{
1
,
2
,
3
,
4
}
and
B
=
{
3
,
4
,
5
,
6
}
B = \{3, 4, 5, 6\}
B
=
{
3
,
4
,
5
,
6
}
. List the elements of
A
∩
B
A \cap B
A
∩
B
.
●
●●●●
Level 1
1 mark
Start
→
Mark as done
A
=
{
1
,
2
,
3
,
4
}
A = \{1, 2, 3, 4\}
A
=
{
1
,
2
,
3
,
4
}
and
B
=
{
3
,
4
,
5
,
6
}
B = \{3, 4, 5, 6\}
B
=
{
3
,
4
,
5
,
6
}
. List the elements of
A
∪
B
A \cup B
A
∪
B
.
●
●●●●
Level 1
1 mark
Start
→
Mark as done
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
\xi = \{1, 2, 3, 4, 5, 6, 7, 8\}
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
and
A
=
{
2
,
4
,
6
,
8
}
A = \{2, 4, 6, 8\}
A
=
{
2
,
4
,
6
,
8
}
. List the elements of
A
′
A'
A
′
.
●
●●●●
Level 1
1 mark
Start
→
Mark as done
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
}
\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
}
A
=
{
prime numbers
}
A = \{\text{prime numbers}\}
A
=
{
prime numbers
}
B
=
{
odd numbers
}
B = \{\text{odd numbers}\}
B
=
{
odd numbers
}
(a)
List the members of
A
∩
B
A \cap B
A
∩
B
.
(b)
List the members of
A
∪
B
A \cup B
A
∪
B
.
(c)
Find
n
(
B
′
)
n(B')
n
(
B
′
)
.
●●
●●●
Level 2
3 marks
Start
→
Mark as done
30
30
30
students were asked whether they play hockey (
H
H
H
) and whether they play netball (
N
N
N
).
The Venn diagram shows the results.
ξ
3
H
N
12
7
8
(a)
Write down
n
(
H
∩
N
)
n(H \cap N)
n
(
H
∩
N
)
.
(b)
Find
n
(
H
∪
N
)
n(H \cup N)
n
(
H
∪
N
)
.
(c)
Write down
n
(
H
∩
N
′
)
n(H \cap N')
n
(
H
∩
N
′
)
.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
ξ
=
{
1
,
2
,
3
,
…
,
15
}
\xi = \{1, 2, 3, \ldots, 15\}
ξ
=
{
1
,
2
,
3
,
…
,
15
}
A
=
{
multiples of
3
}
A = \{\text{multiples of }3\}
A
=
{
multiples of
3
}
B
=
{
factors of
15
}
B = \{\text{factors of }15\}
B
=
{
factors of
15
}
(a)
List the members of
A
∩
B
A \cap B
A
∩
B
.
(b)
Find
n
(
A
∪
B
)
n(A \cup B)
n
(
A
∪
B
)
.
(c)
Ali says that
n
(
A
∪
B
)
n(A \cup B)
n
(
A
∪
B
)
can always be worked out as
n
(
A
)
+
n
(
B
)
n(A) + n(B)
n
(
A
)
+
n
(
B
)
.
Explain why Ali is wrong.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
There are
30
30
30
students in a class.
W
W
W
is the set of students who walk to school and
L
L
L
is the set of students who have a school lunch.
The Venn diagram shows the numbers of students inside the circles. The number of students outside both circles has been left off.
ξ
W
L
13
5
8
(a)
Write down
n
(
W
∩
L
′
)
n(W \cap L')
n
(
W
∩
L
′
)
.
(b)
Find
n
(
(
W
∪
L
)
′
)
n\big((W \cup L)'\big)
n
(
(
W
∪
L
)
′
)
.
(c)
Explain why
n
(
W
′
∩
L
′
)
n(W' \cap L')
n
(
W
′
∩
L
′
)
must give the same answer as part
(b)
, without doing a second calculation.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
ξ
\xi
ξ
is the set of the
50
50
50
members of a running club.
A
A
A
is the set of members who have run a marathon.
B
B
B
is the set of members who have run a half marathon.
n
(
A
)
=
26
n(A) = 26
n
(
A
)
=
26
,
n
(
B
)
=
21
n(B) = 21
n
(
B
)
=
21
and
n
(
A
∩
B
)
=
8
n(A \cap B) = 8
n
(
A
∩
B
)
=
8
.
(a)
Find
n
(
A
∪
B
)
n(A \cup B)
n
(
A
∪
B
)
.
(b)
Find
n
(
A
∩
B
′
)
n(A \cap B')
n
(
A
∩
B
′
)
.
(c)
Find
n
(
A
′
∩
B
′
)
n(A' \cap B')
n
(
A
′
∩
B
′
)
.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
40
40
40
members of a sports club were asked which of football (
F
F
F
), netball (
N
N
N
) and tennis (
T
T
T
) they play.
The Venn diagram shows the results.
ξ
11
F
N
T
7
5
6
4
3
2
2
(a)
Write down
n
(
F
∩
N
)
n(F \cap N)
n
(
F
∩
N
)
.
(b)
Find
n
(
F
∪
T
)
n(F \cup T)
n
(
F
∪
T
)
.
(c)
Write down
n
(
F
∩
N
∩
T
′
)
n(F \cap N \cap T')
n
(
F
∩
N
∩
T
′
)
.
(d)
Find
n
(
N
′
)
n(N')
n
(
N
′
)
.
●●●●
●
Level 4
6 marks
Start
→
Mark as done
ξ
\xi
ξ
is the set of the
60
60
60
students in Year 11 at a school.
A
A
A
is the set of students who have a part-time job.
B
B
B
is the set of students who play a musical instrument.
n
(
A
)
=
34
n(A) = 34
n
(
A
)
=
34
,
n
(
B
)
=
29
n(B) = 29
n
(
B
)
=
29
and
n
(
A
′
∩
B
′
)
=
8
n(A' \cap B') = 8
n
(
A
′
∩
B
′
)
=
8
.
(a)
Find
n
(
A
∪
B
)
n(A \cup B)
n
(
A
∪
B
)
.
(b)
Find
n
(
A
∩
B
)
n(A \cap B)
n
(
A
∩
B
)
.
(c)
Find
n
(
A
∩
B
′
)
n(A \cap B')
n
(
A
∩
B
′
)
.
●●●●●
Level 5
5 marks
Start
→
Mark as done
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
}
\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}
ξ
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
}
A
=
{
multiples of
4
}
A = \{\text{multiples of }4\}
A
=
{
multiples of
4
}
B
=
{
even numbers
}
B = \{\text{even numbers}\}
B
=
{
even numbers
}
C
=
{
multiples of
5
}
C = \{\text{multiples of }5\}
C
=
{
multiples of
5
}
(a)
Explain why
A
⊂
B
A \subset B
A
⊂
B
.
(b)
Write down
n
(
A
∩
C
)
n(A \cap C)
n
(
A
∩
C
)
.
(c)
List the members of
(
B
∪
C
)
′
(B \cup C)'
(
B
∪
C
)
′
.
●●●●●
Level 5
5 marks
Start
→
Probability from Venn diagrams
12 questions
Lesson
Not started
Mark as done
Here is a Venn diagram for
40
40
40
students showing who studies French (F) and Spanish (S).
ξ
10
F
S
14
6
10
A student is picked at random. Work out the probability that they study both languages.
●
●●●●
Level 1
1 mark
Start
→
Mark as done
Here is a Venn diagram for
40
40
40
students showing who studies French (F) and Spanish (S).
ξ
10
F
S
14
6
10
A student is picked at random. Work out the probability that they study French.
●
●●●●
Level 1
2 marks
Start
→
Mark as done
40
40
40
adults were asked whether they own a cat (
C
C
C
) and whether they own a dog (
D
D
D
). The Venn diagram shows the results.
ξ
9
C
D
14
6
11
One of the
40
40
40
adults is chosen at random.
(a)
Find the probability that this adult owns a cat.
(b)
Find the probability that this adult owns neither a cat nor a dog.
●●
●●●
Level 2
3 marks
Start
→
Mark as done
50
50
50
people were asked whether they had visited France (
F
F
F
) or Spain (
S
S
S
).
28
28
28
had visited France,
24
24
24
had visited Spain, and
6
6
6
had visited neither.
(a)
Work out how many people had visited both France and Spain. (2 marks)
(b)
One of the
50
50
50
people is chosen at random. Work out the probability that they had visited France but not Spain. (1 mark)
●●●
●●
Level 3
3 marks
Start
→
Mark as done
40
40
40
people were asked whether they drink tea (
T
T
T
) and whether they drink coffee (
C
C
C
).
The Venn diagram shows the results.
ξ
9
T
C
13
9
9
One of the
40
40
40
people is chosen at random.
(a)
Work out the probability that this person drinks tea or coffee or both. (1 mark)
(b)
Work out the probability that this person drinks neither. (1 mark)
One of the people who drinks tea is chosen at random.
(c)
Work out the probability that this person also drinks coffee. (2 marks)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
50
50
50
people were asked whether they read a newspaper (
N
N
N
) and whether they listen to the radio (
R
R
R
). The Venn diagram shows the results.
ξ
9
N
R
22
8
11
One of the
50
50
50
people is chosen at random.
(a)
Find
P
(
N
∪
R
)
P(N \cup R)
P
(
N
∪
R
)
.
(b)
Find
P
(
N
′
)
P(N')
P
(
N
′
)
.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
The Venn diagram shows the probabilities for two events
A
A
A
and
B
B
B
.
ξ
0.3
A
B
0.35
0.2
0.15
(a)
Find
P
(
A
)
P(A)
P
(
A
)
.
(b)
Find
P
(
A
∪
B
)
P(A \cup B)
P
(
A
∪
B
)
.
(c)
Find
P
(
B
′
)
P(B')
P
(
B
′
)
.
●●●
●●
Level 3
6 marks
Start
→
Mark as done
80
80
80
students were asked whether they play a musical instrument (
M
M
M
) and whether they are in a sports team (
S
S
S
). The Venn diagram shows the results.
ξ
21
M
S
27
18
14
One of the
80
80
80
students is chosen at random.
(a)
Find
P
(
M
∪
S
)
P(M \cup S)
P
(
M
∪
S
)
.
(b)
Kai works out
P
(
M
∪
S
)
P(M \cup S)
P
(
M
∪
S
)
by adding
P
(
M
)
P(M)
P
(
M
)
and
P
(
S
)
P(S)
P
(
S
)
together. Explain why Kai's method gives an answer that is too large.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
30
30
30
members of a music club were asked whether they play the guitar (
G
G
G
) and whether they play the violin (
V
V
V
). The Venn diagram shows the results.
ξ
7
G
V
9
6
8
One member is chosen at random.
(a)
Find the probability that this member plays the violin.
(b)
Given that the chosen member plays the violin, find the probability that they also play the guitar.
●●●●
●
Level 4
4 marks
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→
Mark as done
60
60
60
students were asked which of swimming (
S
S
S
), tennis (
T
T
T
) and running (
R
R
R
) they take part in. The Venn diagram shows the results.
ξ
15
S
T
R
11
9
7
6
5
4
3
One of the
60
60
60
students is chosen at random.
(a)
Find the probability that the student takes part in all three activities.
(b)
Find the probability that the student swims.
(c)
Find the probability that the student takes part in exactly one of the three activities.
●●●●
●
Level 4
5 marks
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→
Mark as done
120
120
120
people were asked whether they own a bicycle (
A
A
A
) and whether they have a gym membership (
B
B
B
). The Venn diagram shows the results.
ξ
48
A
B
42
18
12
One of the
120
120
120
people is chosen at random.
(a)
Find
P
(
A
∩
B
)
P(A \cap B)
P
(
A
∩
B
)
.
(b)
Determine whether the events
A
A
A
and
B
B
B
are independent. You must show your working.
(c)
Ben says that
A
A
A
and
B
B
B
are mutually exclusive. Is Ben correct? Give a reason for your answer.
●●●●●
Level 5
6 marks
Start
→
Mark as done
The Venn diagram shows the probabilities for two events
A
A
A
and
B
B
B
. The probability that neither
A
A
A
nor
B
B
B
occurs is
x
x
x
.
ξ
x
A
B
0.25
0.18
0.27
(a)
Find the value of
x
x
x
.
(b)
Find
P
(
A
′
)
P(A')
P
(
A
′
)
.
(c)
Given that
B
B
B
occurs, find the probability that
A
A
A
also occurs.
●●●●●
Level 5
6 marks
Start
→