Venn diagrams

Use Venn diagrams to find probabilities for intersecting events, apply the addition rule P(A or B) = P(A) + P(B) - P(A and B), and solve problems involving two overlapping events.

Worked examples

Reading a Venn diagram

Straightforward

Problem

A Venn diagram shows events

A

and

B

within a sample space of

20

equally likely outcomes. There are

6

outcomes in

A

only,

4

outcomes in

B

only,

3

outcomes in both

A

and

B

, and

7

outcomes in neither. Find

P(A \text{ or } B)

Filling in a Venn diagram

Moderate

Problem

In a class of

30

students,

18

play sport,

14

study music, and

7

do both. A student is chosen at random. Find

P(\text{neither sport nor music})

Using the addition rule to find P(A and B)

Challenging

Problem

For two events

A

and

B

P(A) = \dfrac{3}{5}

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

, and

P(A \text{ or } B) = \dfrac{4}{5}

. Find

P(A \text{ and } B)

Practise

Q1·Straightforward

A Venn diagram shows events

A

and

B

within a sample space of

20

equally likely outcomes. There are

6

outcomes in

A

only,

4

outcomes in

B

only,

3

outcomes in both

A

and

B

, and

7

outcomes in neither. Find

P(A)

Q2·Straightforward

A Venn diagram shows events

A

and

B

within a sample space of

20

equally likely outcomes. There are

6

outcomes in

A

only,

4

outcomes in

B

only,

3

outcomes in both

A

and

B

, and

7

outcomes in neither. Find

P(A \text{ and } B)

Q3·Straightforward

A Venn diagram shows events

C

and

D

within a sample space of

30

equally likely outcomes. There are

10

outcomes in

C

only,

8

outcomes in

D

only,

5

outcomes in both

C

and

D

, and

7

outcomes in neither. Find

P(C \text{ or } D)

Q4·Straightforward

A Venn diagram shows events

E

and

F

within a sample space of

24

equally likely outcomes. There are

7

outcomes in

E

only,

5

outcomes in

F

only,

4

outcomes in both

E

and

F

, and

8

outcomes in neither. Find

P(F)

Q5·Moderate

In a class of

30

students,

18

play sport,

14

study music, and

7

do both. A student is chosen at random. Find

P(\text{sport and music})

Q6·Moderate

In a survey of

28

people,

16

own a smartphone,

12

own a tablet, and

5

own both. A person is chosen at random. Find

P(\text{smartphone only})

Q7·Moderate

In a class of

30

students,

18

play sport,

14

study music, and

7

do both. A student is chosen at random. Find

P(\text{neither sport nor music})

Q8·Moderate

In a group of

40

people,

25

prefer tea,

20

prefer coffee, and

10

prefer both. A person is chosen at random. Find

P(\text{tea or coffee})

using the addition rule.

Q9·Challenging

In a class of

35

students, every student studies at least one of English or Maths.

22

study English and

20

study Maths. Find

P(\text{English only})

Q10·Challenging

For two events

A

and

B

P(A) = \dfrac{3}{5}

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

, and

P(A \text{ or } B) = \dfrac{4}{5}

. Find

P(A \text{ and } B)

Q11·Challenging

A Venn diagram shows events

A

and

B

within a sample space of

40

equally likely outcomes. There are

14

outcomes in

A

only,

11

outcomes in

B

only, and

9

outcomes in both

A

and

B

. Find

P(A' \text{ and } B')

, the probability of neither

A

nor

B

Q12·Challenging

In a group of

60

people,

35

own a car,

28

own a bike, and

8

own neither. A person is chosen at random. Find

P(\text{owns both a car and a bike})