Venn Diagrams - Worksheets, Questions and Revision

16 original exam-style questions - 17 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Statistics

S26 Venn Diagrams

EDEXCEL 1ST0 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
30 employees at a company were asked whether they walk (W) or cycle (C) to work. The Venn diagram shows the results.
ξ = 30 employees W C 8 5 7 10
(a)Write down the number of employees who walk and cycle to work.(1)
(b)Write down the number of employees who neither walk nor cycle to work.(1)
(c)Work out the total number of employees who walk to work (whether or not they also cycle).(2)
(Total for Question 1 is 4 marks)
2
The universal set is ξ = {1, 2, 3, ..., 12}.
A = {multiples of 3}
B = {multiples of 4}
The Venn diagram shows the elements of ξ placed into sets A and B.
ξ = {1, 2, 3, ..., 12} A B 3 6 9 12 4 8 1 2 5 7 10 11
(a)List the elements of A ∩ B.(1)
(b)List the elements that are in neither A nor B.(1)
(c)Write down n(A).(1)
(d)Work out n(A ∪ B).(2)
(Total for Question 2 is 5 marks)
3
The Venn diagram shows two sets, A and B, inside the universal set ξ. The four regions are labelled with the letters W, X, Y and Z.
ξ A B W X Y Z
(a)Write down the letter that represents A ∩ B.(1)
(b)Write down the letter that represents the elements in neither A nor B.(1)
(c)Write down the letter that represents the elements that are in A but not in B.(1)
(Total for Question 3 is 3 marks)
4
25 people were asked whether they drink tea (T) or coffee (C). The Venn diagram shows the results.
ξ = 25 people T C 9 6 4 6
(a)Write down the number of people who drink both tea and coffee.(1)
(b)Work out the total number of people who drink tea (whether or not they also drink coffee).(2)
(c)Write down the number of people who drink coffee but not tea.(1)
(Total for Question 4 is 4 marks)
5
40 students were asked whether they own a cat (C) or a dog (D). The Venn diagram shows the results.
ξ = 40 students C D 11 7 9 13
(a)Work out the probability that a randomly selected student owns neither a cat nor a dog.(2)
(b)Work out the probability that a randomly selected student owns a dog.(2)
(c)Work out the probability that a randomly selected student owns both a cat and a dog.(1)
(Total for Question 5 is 5 marks)
6
For each statement about Venn diagrams, write down whether it is True or False.
(i)The rectangle in a Venn diagram represents the universal set, containing all the elements under consideration.(1)
(ii)If two circles in a Venn diagram overlap, sets A and B must contain exactly the same elements.(1)
(iii)The region outside both circles but inside the rectangle represents n((A ∪ B)'), the elements in neither set.(1)
(iv)For any two sets A and B, n(A) + n(B) is always equal to n(A ∪ B).(1)
(Total for Question 6 is 4 marks)
7
In a class of 28 students, 16 students study French (F), 11 students study Spanish (S), and 5 students study both French and Spanish.
(a)Work out the number of students who study French only (and not Spanish).(2)
(b)Work out the number of students who study Spanish only (and not French).(2)
(c)Work out the number of students who study neither French nor Spanish.(2)
(d)Draw a Venn diagram to show this information. Label each of the four regions with the correct number of students.(2)
ξ = 28 students F S
(Total for Question 7 is 8 marks)
8
40 students were asked whether they like Maths (M) or Science (S). The Venn diagram shows the results, where x is unknown.
ξ = 40 students M S 12 x 9 7
(a)Form an equation in x for the total number of students, and solve it to find the value of x.(3)
(b)Hence work out the probability that a randomly selected student likes both Maths and Science.(2)
(Total for Question 8 is 5 marks)
9
32 students were asked whether they play the guitar (G) or the piano (P). The Venn diagram shows the results.
ξ = 32 students G P 9 4 6 13
(a)Write down n(G ∩ P).(1)
(b)Work out n(G ∪ P).(2)
(c)Work out n(G'), the number of students who do not play the guitar.(2)
(d)Write down n((G ∪ P)').(1)
(Total for Question 9 is 6 marks)
10
The Venn diagram shows two sets, P and Q, inside the universal set ξ. A region has been shaded.
ξ P Q
(a)Write down, using set notation, the shaded region.(2)
(b)On the diagram provided, shade the region (P ∪ Q)'.(2)
ξ P Q
(Total for Question 10 is 4 marks)
11
50 people at a sports club were asked whether they hold a Football (F) season ticket or a Rugby (R) season ticket. The Venn diagram shows the results.
ξ = 50 people F R 18 8 14 10
(a)Work out n(R), the total number of people who hold a Rugby season ticket.(2)
(b)A person is chosen at random from those who hold a Rugby season ticket. Find the probability that this person also holds a Football season ticket.(2)
(c)State, giving a reason, whether the events "holds a Football season ticket" and "holds a Rugby season ticket" are mutually exclusive.(1)
(Total for Question 11 is 5 marks)
12
A gym is planning its opening hours. As part of this, it surveys 60 of its members to find out whether they use the Pool and whether they use the Gym Floor. The results are shown in the table.
Uses PoolDoes not use Pool
Uses Gym Floor2115
Does not use Gym Floor915
(a)State the stage of the statistical enquiry cycle at which the gym decides to survey 60 of its members.(1)
(b)Using the table, draw a Venn diagram to show this information. Label each region with the correct number of members.(3)
ξ = 60 members Floor Pool
(c)Work out the probability that a randomly selected member uses the Pool.(2)
(d)The gym manager says "Most of our members use both the Pool and the Gym Floor." State the stage of the statistical enquiry cycle at which this claim is being made, and comment on whether the data supports it.(2)
(Total for Question 12 is 8 marks)
13
80 customers at a supermarket were asked whether they bought Bread (B) or Milk (M) that day. The Venn diagram shows the results.
ξ = 80 customers B M 20 25 15 20
(a)Find P(B), P(M) and P(B ∩ M), giving each as a fraction in its simplest form.(3)
(b)Two events are independent if P(A ∩ B) = P(A) × P(B). Determine, with justification, whether buying bread and buying milk are independent events for these customers.(2)
(Total for Question 13 is 5 marks)
14
50 students were asked which of Netball (N), Tennis (T) and Hockey (H) they play. The Venn diagram shows the results.
ξ = 50 students N T H 7 8 5 4 3 2 1 20
(a)Write down the number of students who play all three sports.(1)
(b)Work out the number of students who play exactly two of the three sports.(2)
(c)Work out the number of students who play Netball.(2)
(d)Work out the number of students who play none of the three sports.(2)
(e)Work out the probability that a randomly selected student plays exactly one of the three sports.(2)
(Total for Question 14 is 9 marks)
15
60 visitors to an art exhibition were asked which of Painting (P), Sculpture (Sc) and Photography (Ph) displays they visited. The Venn diagram shows the results.
ξ = 60 visitors P Sc Ph 10 8 6 5 4 3 2 22
(a)Write down n(P ∩ Sc ∩ Ph).(1)
(b)Find n(P ∩ Sc), the number of visitors who visited both the Painting and Sculpture displays (whether or not they also visited Photography).(2)
(c)Write down n((P ∪ Sc ∪ Ph)').(1)
(d)A visitor is chosen at random from those who visited the Painting display. Find the probability that this visitor also visited the Sculpture display.(2)
(e)Find the probability that a visitor chosen at random from those who visited the Sculpture display also visited the Painting display. Hence explain why this probability and your answer to part (d) are not equal.(2)
(Total for Question 15 is 8 marks)
16
90 students were asked which of the Drama (D), Art (A) and Music (M) clubs they attend. The Venn diagram shows the results in terms of x.
ξ = 90 students D A M x 2x x + 3 x - 1 4 6 3 10
(a)Form an equation in x for the total number of students, and solve it to find the value of x.(3)
(b)Using your value of x, work out n(D), the total number of students who attend the Drama club.(2)
(c)Work out the probability that a randomly selected student attends exactly one of the three clubs.(2)
(Total for Question 16 is 7 marks)
Mark scheme · S26 Venn Diagrams

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16