18 Magda carried out a survey to find out the drinks that people like.

She asked 100 people if they like coffee (\(C\)) or tea (\(T\)) or hot chocolate (\(H\)).

32 people like all three drinks.
15 people like coffee and hot chocolate but not tea.
43 people like coffee and tea.
39 people like tea and hot chocolate.
67 people like coffee.
58 people like hot chocolate.
5 people like only tea.

(a) Complete the Venn diagram for this information.

(4)

One of the people is chosen at random.

Given that this person likes coffee,

4 \(\mathscr{E} = \{\)odd numbers between 0 and 20\(\}\)
\(A = \{3, 9, 15, 17\}\)
\(B = \{5, 7, 11, 13, 17, 19\}\)

Tom was asked to draw a Venn diagram for this information.
Here is his answer.

Write down two things Tom should do to make his answer fully correct. (2)

4 Here is a Venn diagram.

A number is chosen at random from the universal set, \(\mathscr{E}\)

21 Vicky has a collection of medals.

The Venn diagram gives information about the number of medals in her collection where

\(\mathscr{E}\) = {all medals}
\(A\) = {English medals}
\(B\) = {gold medals}

Vicky is going to take at random a medal from her collection.

Given that the medal is gold, the probability that the medal is English is \(\dfrac{2}{11}\)

Work out the number of medals in Vicky’s collection. (4)

5 \(\mathscr{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
\(A = \{\)odd numbers\(\}\)
\(B = \{\)square numbers\(\}\)

(a) Complete the Venn diagram for this information.

(3)

A number is chosen at random from the universal set \(\mathscr{E}\)

16 A shop manager wants to advertise special offers on social media platforms.

The manager asks 100 customers which of type \(A\), type \(B\) or type \(C\) they use.

Of these customers,
4 use all three types
16 do not use any of type \(A\), type \(B\) or type \(C\)
8 use both type \(A\) and type \(B\), but not type \(C\)
14 use both type \(B\) and type \(C\)
62 in total use type \(A\)
all 20 who use type \(C\) also use at least one of type \(A\) and type \(B\).

(a) Complete the Venn diagram for this information.

(4)

One of the customers is chosen at random.

Given that this customer uses type \(A\),

2 \(\mathscr{E} = \{\)even numbers less than 19\(\}\)
\(A = \{6, 12, 18\}\)
\(B = \{2, 6, 14, 18\}\)

Complete the Venn diagram for this information. (3)

1 \(\mathscr{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\)
\(A = \{1, 5, 6, 8, 9\}\)
\(B = \{2, 6, 9\}\)

A number is chosen at random from the universal set \(\mathscr{E}\).

1 \(\mathscr{E} = \{\text{even numbers between 1 and 25}\}\)
\(A = \{2, 8, 10, 14\}\)
\(B = \{6, 8, 20\}\)
\(C = \{8, 18, 20, 22\}\)

(a) Complete the Venn diagram for this information.

(4)

A number is chosen at random from \(\mathscr{E}\).

20 50 people were asked if they speak French or German or Spanish.

Of these people,

31 speak French
2 speak French, German and Spanish
4 speak French and Spanish but not German
7 speak German and Spanish
8 do not speak any of the languages
all 10 people who speak German speak at least one other language

Two of the 50 people are chosen at random.

Work out the probability that they both only speak Spanish. (5)

8 60 people were asked if they prefer to go on holiday in Britain or in Spain or in Italy.

38 of the people were male.
11 of the 32 people who said Britain were female.
8 males said Italy.
12 people said Spain.

One of the females is chosen at random.

What is the probability that this female said Spain? (4)

1 \(\mathscr{E} = \{\)odd numbers less than 30\(\}\)
\(A = \{3, 9, 15, 21, 27\}\)
\(B = \{5, 15, 25\}\)

(a) Complete the Venn diagram to represent this information.

(4)

A number is chosen at random from the universal set, \(\mathscr{E}\).

16 A and B are two events.

Match each box on the left to the correct shaded area on the Venn diagrams.

One has been done for you. [3 marks]

10 The Venn diagram shows some of the numbers of items in each set.

P(A) \(= \dfrac{1}{2}\)

Work out the value of \(x\). [2 marks]

10 Clive owns two coffee shops, A and B.

Out of 150 people

• 18 people use both shops
• 46% use shop A
• \(\dfrac{2}{5}\) use shop B only.

Complete the Venn diagram to represent the information. [4 marks]

ξ = 150 people
A = people who use shop A
B = people who use shop B

9 On the Venn diagram, shade the section represented by \(\quad P \cap Q\) [1 mark]

5 A school year has 78 students.

• 28 wear glasses.
• \(\dfrac{1}{4}\) of the students who wear glasses are left-handed.
• 30% of the students who do not wear glasses are left-handed.

(a) \(\xi =\) students in the school year

G \(=\) wears glasses

L \(=\) left-handed

Complete the Venn diagram. [3 marks]

16 The Venn diagram shows information about 80 people who visited an online shop.

\(\xi =\) 80 people
T = people who bought trainers
H = people who bought a hoodie

10

Which of these represents the shaded region?

Circle your answer. [1 mark]

• B
• \(A^{\prime} \cup B\)
• \(A^{\prime} \cap B\)
• \(A^{\prime}\)

11 The Venn diagram represents 100 items.

10 Here is some information about 120 people who visit a shop.

\(\dfrac{3}{4}\) of the people buy neither a coat nor a dress.
19 people buy a coat.
14 people buy a dress.

Complete this Venn diagram to represent the information. [3 marks]

\(\xi\) = 120 people who visit the shop
C = people who buy a coat
D = people who buy a dress

7 In a group of 98 students

• 25 study both Art and French
• 10 study Art but do not study French
• 41 study French.

Joel draws this Venn diagram to represent the information.

\(\xi\) = the group of 98 students
\(A\) = the students who study Art
\(F\) = the students who study French

Make two criticisms of his diagram. [2 marks]

2 Circle the expression that means    the probability of A and not B. [1 mark]

• \(\text{P}(A' \cup B)\)
• \(\text{P}(A \cup B')\)
• \(\text{P}(A' \cap B)\)
• \(\text{P}(A \cap B')\)

7 Here is some information about 80 people who play in bands.

12 are singers but not guitar players.
30% are neither a singer nor a guitar player.
\(\dfrac{1}{4}\) of the guitar players are also singers.

Complete this Venn diagram to represent the information. [4 marks]

\(\xi\) = 80 people who play in bands
S = singers
G = guitar players

1 What does \(\quad A \cup B \quad\) represent in \(\quad P(A \cup B)\)?

Circle your answer. [1 mark]

• A or B or both
• A but not B
• not A and not B
• A and B

20 The Venn diagram shows information about some houses.

G = houses with a garage
S = houses with a shed

A house is chosen at random.

22 \(\xi\) = 29 students in a class

C = students who own a cat

D = students who own a dog

4

Which of these represents the shaded region?

Circle your answer. [1 mark]

• \(A \cup B\)
• \((A \cap B)'\)
• \(A \cap B\)
• \(A' \cup B'\)

14 A and B are two events.

Some probabilities are shown on the Venn diagram.

Work out \(\quad \text{P}(A' \cup B)\) [2 marks]

1 What does (A ∩ B) represent in  P(A ∩ B)?

Circle your answer. [1 mark]

• A or B or both
• A but not B
• not A and not B
• A and B

25 The Venn diagram shows some information about 150 students.

\(\xi\) = 150 students
C = students who study Chemistry
P = students who study Physics

The probability that a Physics student, chosen at random, also studies Chemistry is \(\dfrac{5}{12}\)

One of the 150 students is chosen at random.

Work out the probability that the student does not study either Chemistry or Physics. [4 marks]

22

Which of these represents the shaded region?

Circle your answer. [1 mark]

• \(A \cap B^{\prime}\)
• \(B^{\prime}\)
• \(A \cup B^{\prime}\)
• \(A^{\prime} \cup B^{\prime}\)

18

(a)

Which of these represents the shaded region?

Circle your answer. [1 mark]

• A
• B\('\)
• A \(\cap\) B\('\)
• A \(\cup\) B\('\)

(b)

Which of these represents the shaded region?

Circle your answer. [1 mark]

• (A \(\cup\) B)\('\)
• (A \(\cap\) B)\('\)
• A\('\) \(\cap\) B
• A\('\) \(\cup\) B\('\)

14 In the Venn diagram

ξ represents 31 students in a class

\(C\) is students who have a cat

\(D\) is students who have a dog

18 A school has 86 teachers.

• 42 are male and 44 are female.
• \(\dfrac{1}{3}\) of the male teachers have blue eyes.
• \(\dfrac{1}{4}\) of the female teachers have blue eyes.

(a) \(\xi =\) teachers in the school

M = male teachers

B = teachers who have blue eyes

Complete the Venn diagram. [3 marks]

22 A school has 90 students in Year 11.

• 50 of the students study French.
• 59 of the students study Spanish.
• 4 of the students do not study either of these languages.

One the students is chosen at random.

Find the probability that the student studies French given that they study Spanish.
You may use the grid or the Venn diagram to help present your working if you wish. [5]

15 In a survey, some students were asked whether they had travelled to school by bus (B) or by car (C) in the last week.
The Venn diagram shows some of the results.

23 In a survey, 50 people are asked which sports they watch.
The results are shown on the Venn diagram.

12 A cafe owner recorded information about customer orders for coffee.
They recorded whether the customer asked for milk (M) and whether the customer asked for sugar (S).

The results are shown in this Venn diagram.

10 100 people were asked whether they had visited France (F) or Spain (S).

• 55 had visited France
• 60 had visited Spain
• 4 had not visited either country.

(a) Complete the Venn diagram.

[3]

16 A salesroom sells various types of car.
Some cars are electric (E), some are petrol (P), some are both and some are neither.

The Venn diagram below shows the salesroom’s stock of cars.

A petrol car is picked at random.

Find the probability that the car is also electric. [2]

14 In a survey about music, some students were asked whether they like pop (P) and whether they like rap (R).

The Venn diagram shows some of the results.
\(x\) students liked both types of music.

16 In a group of 60 students, 40 own a smartphone, 27 own a tablet and 8 own neither.

A student is chosen at random from those that own a tablet.

Find the probability that they also own a smartphone.
You must show your working. [5]

11 In a class of 30 students

• 17 study French (F)
• 20 study German (G)
• 2 do not study either subject.

(a) Complete the Venn diagram.

[3]

4 In a survey, 60 students were asked whether they have a bank account (B) and whether they have a part-time job (J).

The number of students who had neither a bank account nor a part-time job was \(x\).
The Venn diagram shows the results in terms of \(x\).

One of the 60 students is chosen at random.

Find the probability that they have a bank account.
Show your working. [5]

16 50 people attended an outdoor activity day.

• 40 took part in walking.
• 18 took part in sailing.
• 3 did neither activity.

One of the people who walked is chosen at random.

Find the probability that this person also sailed. [5]

18 21 people travelled to a meeting.

• 12 used a train.
• 6 used a car.
• 7 did not use a train or a car.
• Some used a train and a car.

Two people are chosen at random from those who used a train.

Find the probability that both these people also used a car. [6]

9 72 children are asked whether they have a laptop or an iPad.

• 31 have a laptop.
• 48 have an iPad.
• 12 have both.
• 5 have neither.

(a) Represent this information on a Venn diagram.

[3]

14 The Venn diagram shows the number of students studying Mathematics (M) and the number of students studying Physics (P) in a college.
35 students do not study either subject.

9 Here are the results of a survey of 437 people in a town.

• 62 males speak Spanish.
• 153 females do not speak Spanish.
• 280 people do not speak Spanish.

Jeff says

At least 2 out of every 5 females in the town can speak Spanish.

Is he correct?
Show clearly how you reached your decision. [6]

18 In a group of 120 adults, 85 watch football, 78 play a sport and 20 do neither.

Find the probability that an adult chosen at random from those who watch football does not play a sport. [5]