Venn Diagrams - Worksheets, Questions and Revision

17 original exam-style questions - 14 pages of questions with a full mark scheme - free printable PDF.

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GCSE · Probability

5.23 Venn Diagrams

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
The universal set is E = {1, 2, 3, ..., 15}.
A is the set of multiples of 3 in E.
B is the set of multiples of 5 in E.
(a)Write down the elements of set A.(1)
(b)Write down the elements of A ∩ B.(1)
(c)List the elements of A ∪ B.(2)
(Total for Question 1 is 4 marks)
2
30 students were asked whether they like football (F) or netball (N). The results are shown in the Venn diagram: 8 students are in F only, 5 students are in the overlap of F and N, 10 students are in N only, and 7 students are outside both circles but inside E.
E F N 8 5 10 7
(a)Write down n(F).(1)
(b)Write down n(F ∩ N).(1)
(c)Write down n((F ∪ N)').(1)
(d)A student is chosen at random from the 30 students. Find the probability that the student likes netball but not football.(2)
(Total for Question 2 is 5 marks)
3
20 people were asked whether they drink tea (T) or coffee (C). The Venn diagram shows: 6 people in T only, 4 people in the overlap of T and C, 8 people in C only, and 2 people outside both circles.
E T C 6 4 8 2
(a)Write down n(T ∩ C').(1)
(b)A person is chosen at random from the 20 people. Find P(C).(2)
(Total for Question 3 is 3 marks)
4
A ball is drawn at random from a bag containing only red, blue and green balls. Event R is that the ball is red. Event Bl is that the ball is blue.
(a)Explain why events R and Bl are mutually exclusive.(1)
(b)On a Venn diagram, state how you would show that two events are mutually exclusive.(1)
(Total for Question 4 is 2 marks)
5
Using the Venn diagram from Question 2, a student is chosen at random from the 30 students.
(a)What is n(F ∪ N)?
A) 5 B) 13 C) 23 D) 30(1)
  • A) 5
  • B) 13
  • C) 23
  • D) 30
(Total for Question 5 is 1 mark)
6
60 employees at a company were asked whether they cycle (C) or walk (W) to work. The Venn diagram shows: 20 employees in C only, 6 employees in the overlap of C and W, 10 employees in W only, and 24 employees outside both circles.
E C W 20 6 10 24
(a)An employee is chosen at random. Find P(C).(2)
(b)Find the probability that an employee neither cycles nor walks to work.(1)
(Total for Question 6 is 3 marks)
7
A class of 24 students was asked whether they play piano (P) or guitar (G). The Venn diagram shows: 6 students in P only, 4 students in the overlap of P and G, 9 students in G only, and 5 students outside both circles.
Venn diagram for parts (a)-(c) E P G 6 4 9 5 Part (d): which diagram below has the correct region shaded? E P G P ∩ G A E P G P ∪ G B E P G P' C E P G (P ∪ G)' D
(a)Write down n(E).(1)
(b)Write down n(P').(1)
(c)A student is picked at random. Find P(P ∩ G').(2)
(d)Which shaded region below represents students who play at least one instrument?
A) P ∩ G B) P ∪ G C) P' D) (P ∪ G)'(1)
  • A) P ∩ G
  • B) P ∪ G
  • C) P'
  • D) (P ∪ G)'
(Total for Question 7 is 5 marks)
8
A survey of 50 people asked whether they own a cat (C) or a dog (D). 22 people own a cat, 19 people own a dog, and 8 people own both a cat and a dog.
E C D
(a)Complete a Venn diagram to show this information. State the number of people in each of the four regions: C only, both C and D, D only, and neither.(3)
(b)Using your Venn diagram, find the probability that a randomly selected person owns a dog but not a cat.(2)
(c)Find P(C ∪ D).(2)
(Total for Question 8 is 7 marks)
9
There are 32 students in a class. 6 students study both Art and Drama.
(a)Show that the probability that a randomly selected student studies both Art and Drama is 3/16.(2)
(Total for Question 9 is 2 marks)
10
The universal set is E = {1, 2, 3, ..., 20}.
A is the set of multiples of 4 in E.
B is the set of multiples of 6 in E.
(a)List the elements of A ∩ B.(1)
(b)List the elements of (A ∪ B)'.(2)
(c)Write down another way of expressing (A ∪ B)' using set notation.(1)
(Total for Question 10 is 4 marks)
11
A gym with 80 members recorded whether members attend spin classes (S) and yoga classes (Y) in the two-way table below.
Yoga No Yoga Total
Spin 12 28 40
No Spin 18 22 40
Total 30 50 80
80 members S Y
(a)Draw a Venn diagram to represent this information, using circles S and Y. State the number of members in each region.(3)
(b)A member is selected at random. Find the probability they attend yoga but not spin.(2)
(c)Given that a member attends yoga, find the probability that they also attend spin.(2)
(Total for Question 11 is 7 marks)
12
In a school of 40 students, 3/5 of the students like Maths, 1/2 like Science, and 1/4 like both Maths and Science.
(a)Find the number of students who like Maths.(1)
(b)Find the number of students who like Science.(1)
(c)Find the number of students who like both Maths and Science.(1)
(d)Work out the number of students who like neither Maths nor Science.(3)
(Total for Question 12 is 6 marks)
13
100 students were surveyed about whether they attend a Maths club (M) or a Science club (S). On a Venn diagram, the number in M only is 2x, the number in both is x, the number in S only is 3x, and the number in neither is 10.
E M S 2x x 3x 10
(a)Write down an equation in x and show that it simplifies to 6x + 10 = 100.(2)
(b)Solve the equation to find the value of x.(2)
(c)Work out the number of students who attend the Science club.(2)
(Total for Question 13 is 6 marks)
14
40 people at a gym were asked whether they do yoga (Y) or pilates (P). The Venn diagram shows: 10 people in Y only, 6 people in the overlap of Y and P, 14 people in P only, and 10 people outside both circles.
E Y P 10 6 14 10
(a)Find P(Y).(1)
(b)Given that a person does pilates, find the probability that they also do yoga.(2)
(c)Two people are chosen at random from all 40 people, without replacement. Find the probability that both people do neither yoga nor pilates.(3)
(Total for Question 14 is 6 marks)
15
50 students were asked which of French (F), German (G) and Spanish (S) they study. 28 study French, 20 study German, 22 study Spanish, 10 study French and German, 12 study French and Spanish, 8 study German and Spanish, and 5 study all three languages.
E F G S
(a)Complete a three-set Venn diagram to show this information, and find the number of students who study none of the three languages.(4)
(b)Find the probability that a randomly selected student studies exactly two of the three languages.(2)
(c)A student is chosen at random from those who study French. Find the probability that they also study at least one other language.(2)
(Total for Question 15 is 8 marks)
16
There are 40 students in a group split by whether they are in set A, set B, both, or neither. On a Venn diagram, the number in A only is x + 2, the number in B only is 2x - 4, the number in both A and B is x, and the number in neither is 6.
E A B x + 2 x 2x - 4 6
(a)Show that the total number of students gives the equation 4x + 4 = 40.(2)
(b)Solve the equation to find the value of x.(2)
(c)Work out n(A).(2)
(d)Find the probability that a randomly selected student is in B but not A.(2)
(Total for Question 16 is 8 marks)
17
In a school of 200 students, R is the event that a student plays rugby and H is the event that a student plays hockey. 90 students play rugby, 70 students play hockey, and 50 students play neither sport.
E R H
(a)Find n(R ∩ H).(2)
(b)A student who plays hockey is selected at random. Find the probability that they also play rugby.(2)
(c)Two students are selected at random from all 200 students, without replacement. Find the probability that both students play rugby only (rugby but not hockey).(3)
(Total for Question 17 is 7 marks)
Mark scheme · 5.23 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

Question 17