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GCSE · Probability

5.23D Venn Diagrams: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 55 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

The universal set is E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 2, 3, 4, 5}. B = {4, 5, 6, 7}.
Write down the elements of A ∩ B.

(Total for Question 1 is 1 mark)

Using the sets given in Question 1, write down the elements of A ∪ B.

(Total for Question 2 is 1 mark)

Using the sets in Question 1, find n(A ∪ B).

(Total for Question 3 is 1 mark)

Using the sets in Question 1, write down the elements of A' (the complement of A within E).

(Total for Question 4 is 1 mark)

Priya says that 8 is a member of set A (using the sets in Question 1). Is she correct? Give a reason for your answer.

(Total for Question 5 is 1 mark)

Ms Nowak's form group has 20 pupils. 9 of the pupils play a musical instrument (M) and 11 pupils are in a sports team (T). 4 pupils do both. Find n(M ∪ T).

(Total for Question 6 is 2 marks)

Using the information in Question 6, find the number of pupils in the form group who do neither.

(Total for Question 7 is 1 mark)

On a two-set Venn diagram showing sets A and B inside E, describe in words the region that represents A ∩ B'.

(Total for Question 8 is 1 mark)

Describe in words the region that represents A' ∩ B'.

(Total for Question 9 is 1 mark)

A class of 24 students was asked whether they study French (F) or Spanish (S). The Venn diagram shows: French only = 6, both French and Spanish = 4, Spanish only = 9, and neither = 5.
Write down the number of students who study both French and Spanish.

(Total for Question 10 is 1 mark)

Using the diagram in Question 10, write down n(F).

(Total for Question 11 is 1 mark)

Using the diagram in Question 10, find the probability that a student, chosen at random, studies French only.

(Total for Question 12 is 1 mark)

Using the diagram in Question 10, find P(F ∩ S).

(Total for Question 13 is 1 mark)

Using the diagram in Question 10, find P(F').

(Total for Question 14 is 1 mark)

A survey of 50 Year 11 students at a school in Leeds asked whether they study Drama (D) and/or Music (M). 28 students study Drama, 19 students study Music, and 8 students study neither.

(a)Find the number of students who study both Drama and Music.(3)

(b)Find the probability that a randomly selected student studies Drama only.(2)

(Total for Question 15 is 5 marks)

The universal set is E = {1, 2, 3, ..., 20}. A = {factors of 20}. B = {multiples of 5}.

(a)List the elements of set A.(1)

(b)List the elements of set B.(1)

(c)Find A ∩ B.(1)

(Total for Question 16 is 3 marks)

Chidinma organises a fun run in Bristol with 90 entrants. Every entrant runs the 5 km race (R), the 10 km race (F), or both; nobody runs neither. n(R) = 63 and n(F) = 47.

(a)Find n(R ∩ F).(2)

(b)Find the probability that a randomly chosen entrant ran only the 10 km race.(2)

(Total for Question 17 is 4 marks)

In a class of 32 students, x study French only, 2x study Spanish only, x study both French and Spanish, and 4 students study neither.

(a)Form an equation in x and solve it to find the value of x.(3)

(b)Hence write down the number of students who study French.(1)

(Total for Question 18 is 4 marks)

Dr Kaur, a vet in Cardiff, surveys 60 pet owners. 35 own a dog (D), 24 own a cat (C), and 10 own both a dog and a cat. One of the dog owners is chosen at random. Find the probability that this dog owner also owns a cat.

(Total for Question 19 is 3 marks)

The Venn diagram shows solar panels (S) and heat pumps (H) for the 48 houses on a street in Manchester. S only = 3y, H only = y, both S and H = 2y, and 12 houses have neither.

(a)Form an equation in y and solve it.(3)

(b)A house is selected at random. Find P(S).(1)

(Total for Question 20 is 4 marks)

60 students were surveyed about which sports they play: football (F), basketball (B) and rugby (R). The three-circle Venn diagram shows: F only = 14, B only = 9, R only = 7, F and B only = 6, F and R only = 5, B and R only = 4, all three sports = 3, and none of the three = 12.

(a)Find the number of students who play at least two of the three sports.(2)

(b)Find the probability that a randomly selected student plays exactly one sport.(2)

(Total for Question 21 is 4 marks)

A gym in Glasgow, run by manager Callum, has 44 members in total. Of these, x members use the pool only, x² members use both the pool and the gym floor, and (2x + 1) members use the gym floor only. The remaining 3 members use neither facility.

(a)Show that x² + 3x - 40 = 0.(2)

(b)Solve the equation to find the value of x, explaining why the other solution is not valid.(2)

(c)Hence find the probability that a randomly selected member uses the pool (including those who use both facilities).(1)

(Total for Question 22 is 5 marks)

A and B are sets such that A is a subset of B (written A ⊆ B, meaning every element of A is also in B). State, in terms of A and B, what A ∩ B and A ∪ B simplify to, and justify your answer using a numerical example.

(Total for Question 23 is 3 marks)

Mark scheme · 5.23D Venn Diagrams: Fluency and Exam Drill

Question 1

B1 {4, 5} oe (any order)
Answer: A ∩ B = {4, 5}

Question 2

B1 {1, 2, 3, 4, 5, 6, 7} cao
Answer: A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Question 3

B1 7 cao
Answer: 7

Question 4

B1 {6, 7, 8, 9, 10} oe
Answer: A' = {6, 7, 8, 9, 10}

Question 5

B1 No, with correct reason, e.g. A = {1,2,3,4,5} does not contain 8, so 8 is not a member of A
Answer: No; 8 is not a member of A

Question 6

M1 9 + 11 - 4 oe (uses n(M) + n(T) - n(M ∩ T))
A1 16 cao
Answer: 16

Question 7

B1 4 cao, ft from Question 6
Answer: 4

Question 8

B1 correct description, e.g. the elements in A but not in B (the 'A only' region)
Answer: The elements that are in A but not in B (the A only region).

Question 9

B1 correct description, e.g. the elements outside both circles, in neither A nor B
Answer: The elements outside both circles, i.e. in neither A nor B.

Question 10

B1 4 cao
Answer: 4

Question 11

B1 10 cao
Answer: 10

Question 12

B1 6/24 oe (= 1/4)
Answer: 6/24 = 1/4

Question 13

B1 4/24 oe (= 1/6)
Answer: 4/24 = 1/6

Question 14

B1 14/24 oe (= 7/12), ft from n(F)
Answer: 14/24 = 7/12

Question 15

(a) M1 50 - 8 (= 42) used for n(D ∪ M)
(a) M1 28 + 19 - 42 (oe) to find the overlap
(a) A1 5 cao
(a) Answer: 5
(b) M1 28 - 5 (= 23) oe, ft from part (a)
(b) A1 23/50 cao
(b) Answer: 23/50

Question 16

(a) B1 {1, 2, 4, 5, 10, 20} cao (any order)
(a) Answer: A = {1, 2, 4, 5, 10, 20}
(b) B1 {5, 10, 15, 20} cao
(b) Answer: B = {5, 10, 15, 20}
(c) B1 {5, 10, 20} cao, ft from parts (a) and (b)
(c) Answer: A ∩ B = {5, 10, 20}

Question 17

(a) M1 63 + 47 - 90 oe
(a) A1 20 cao
(a) Answer: 20
(b) M1 47 - 20 (= 27) oe, ft from part (a)
(b) A1 27/90 oe (= 3/10)
(b) Answer: 27/90 = 3/10

Question 18

(a) M1 x + 2x + x + 4 = 32 oe (correct equation formed)
(a) M1 4x = 28 (oe, correct simplification)
(a) A1 x = 7 cao
(a) Answer: x = 7
(b) B1 14 cao, ft from part (a)
(b) Answer: 14

Question 19

M1 identifies 35 (the dog owners) as the reduced sample space
M1 10 identified as the number who own both a dog and a cat
A1 10/35 oe (= 2/7)
Answer: 10/35 = 2/7

Question 20

(a) M1 3y + y + 2y + 12 = 48 oe (correct equation formed)
(a) M1 6y = 36 (oe, correct simplification)
(a) A1 y = 6 cao
(a) Answer: y = 6
(b) B1 30/48 oe (= 5/8), ft from part (a)
(b) Answer: 30/48 = 5/8

Question 21

(a) M1 6 + 5 + 4 + 3 (identifies the four relevant regions)
(a) A1 18 cao
(a) Answer: 18
(b) M1 14 + 9 + 7 (= 30) oe
(b) A1 30/60 oe (= 1/2)
(b) Answer: 30/60 = 1/2

Question 22

(a) M1 x + x² + (2x + 1) + 3 = 44 oe (correct sum set equal to 44)
(a) A1 cso, correctly rearranged to x² + 3x - 40 = 0
(a) Answer: x² + 3x - 40 = 0 (shown)
(b) M1 (x + 8)(x - 5) = 0 oe (correct factorisation, or correct use of the quadratic formula)
(b) A1 x = 5, rejecting x = -8 since a number of members cannot be negative
(b) Answer: x = 5
(c) B1 30/44 oe (= 15/22), ft from part (b)
(c) Answer: 30/44 = 15/22

Question 23

B1 A ∩ B = A oe
B1 A ∪ B = B oe
B1 correct justification with a valid example, e.g. every element of A is already in B, so the overlap is all of A and the union adds nothing new
Answer: A ∩ B = A and A ∪ B = B

Venn Diagrams: Fluency and Exam Drill - Worksheets, Questions and Revision