Browse Combined events & tree diagrams

44 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Tree diagrams & independent events

17 questions

LessonNot started

3

red and

2

blue counters. A counter is taken, its colour noted, and it is put back. A second counter is taken. Write down the probability that the second counter is blue.

●●●●●Level 11 markStart

The probability that Ana is late on any day is

0.2

, independently of other days. Work out the probability that she is late on both Monday and Tuesday.

●●●●●Level 11 markStart

A fair coin is flipped twice. Use a tree diagram or a list to find the probability of getting a head then a tail.

●●●●●Level 12 marksStart

Each time Ravi plays a game on his phone, the probability that he wins is

0.4

.

Ravi plays the game twice. The results of the two games are independent.

(a) Work out the probability that Ravi wins both games.

(b) Work out the probability that Ravi loses both games.

●●●●●Level 24 marksStart

Two events

A

and

B

are independent.

P(A) = 0.3

and

P(B) = 0.6

.

(a) Work out the probability that both

A

and

B

happen. [1]

(b) Work out the probability that neither

A

nor

B

happens. [2]

●●●●●Level 23 marksStart

Bea rolls a fair six-sided dice numbered

1

6

. She also spins a spinner with

5

equal sections,

2

of which are green.

The roll of the dice and the spin of the spinner are independent.

(a) Work out the probability that the dice lands on

6

and the spinner lands on green.

(b) Work out the probability that the dice does not land on

6

and the spinner does not land on green.

●●●●●Level 35 marksStart

The probability that Nia's train is on time on any morning is

0.85

. Whether the train is on time on one morning is independent of every other morning.

Nia catches the train on Monday morning and on Tuesday morning.

(a) Work out the probability that the train is on time on both mornings.

(b) Work out the probability that the train is on time on exactly one of the two mornings.

●●●●●Level 35 marksStart

A bag contains

3

red counters and

2

blue counters.

A counter is taken at random and replaced. A second counter is then taken at random.

Work out the probability that the two counters are different colours. (3 marks)

●●●●●Level 33 marksStart

The probability that Sam is late for school on any given day is

0.15

Whether Sam is late on one day is independent of whether he is late on any other day.
(a) Work out the probability that Sam is late on both of the next two days. (2 marks)
(b) Work out the probability that Sam is late on exactly one of the next two days. (2 marks)

●●●●●Level 34 marksStart

A spinner can land on red, blue or yellow.

P(\text{red}) = 0.5

and

P(\text{blue}) = 0.2

.

The spinner is spun twice. The two spins are independent.

(a) Work out the probability that the spinner lands on yellow on any one spin.

(b) Work out the probability that the spinner lands on the same colour on both spins.

●●●●●Level 44 marksStart

A bag contains

3

red counters and

7

white counters.

A counter is taken at random from the bag, its colour is recorded, and the counter is put back in the bag. This is done three times.

(a) Work out the probability that all three counters taken are red.

(b) Work out the probability that exactly two of the three counters taken are red.

●●●●●Level 45 marksStart

On any day, the probability that it rains in a town is

0.2

. Whether it rains on one day is independent of whether it rains on any other day.

Kofi wants the probability that it rains on at least one of the next two days. He writes:

P(\text{at least one rainy day}) = 0.2 + 0.2 = 0.4

(a) Explain why Kofi's method is wrong.

(b) Work out the correct probability that it rains on at least one of the next two days.

●●●●●Level 44 marksStart

A biased spinner has five sections numbered

1

2

3

4

and

5

Leah spins the spinner

80

times.

The table gives the number of times the spinner lands on each number.

Number	$1$	$2$	$3$	$4$	$5$
Frequency	$14$	$20$	$11$	$19$	$16$

Leah is going to spin the spinner two more times.
(a) Work out an estimate for the probability that the spinner will land on

4

both times.

(3 marks)

Marcus will spin the same spinner

n

times and use his results to get a better estimate of this probability.
(b) What can you say about the value of

n

?

(1 mark)

●●●●●Level 44 marksStart

There are only

3

purple pens and

5

orange pens in a box.

Jay takes at random a pen from the box.

He records its colour and puts the pen back into the box.

Jay does this two more times.

Show that the probability that Jay takes at least two purple pens is

\frac{81}{256}

(4 marks)

●●●●●Level 44 marksStart

A machine makes components. The probability that any component it makes is faulty is

0.04

, independently of every other component.

An inspector takes a sample of

3

components at random from a large batch.

(a) Work out the probability that at least one of the

3

components in the sample is faulty. Give your answer correct to

3

decimal places.

(b) The inspector takes

500

such samples of

3

components. Work out an estimate for the number of these

500

samples that contain at least one faulty component.

●●●●●Level 55 marksStart

Each time Salma takes a shot at a basketball hoop, the probability that she scores is

p

. The results of her shots are independent.

Salma takes two shots. The probability that she misses both shots is

0.09

.

(a) Work out the value of

p

.

(b) Work out the probability that Salma scores with exactly one of her two shots.

●●●●●Level 56 marksStart

There are only green pens and black pens in a box.

The number of green pens is

3

times the number of black pens.

Zara takes at random one pen from the box, records its colour and puts it back in the box.

She does this

n

times (

n \geqslant 2

).

Write down an expression in

n

for the probability that she takes at least one green pen and at least one black pen.

(2 marks)

●●●●●Level 52 marksStart

Probability without replacement

17 questions

LessonNot started

A box has

4

red and

6

green pens. One pen is taken and not put back. It is red. Write down the probability that the next pen taken is also red.

●●●●●Level 11 markStart

There are

10

cards numbered

1

10

. A card is taken and not replaced; it is the

7

. Write down the probability that the next card taken is an even number.

●●●●●Level 11 markStart

A bag contains 4 mint sweets and 6 toffee sweets.

Priya takes a sweet at random and eats it. She then takes a second sweet at random.

(a) Work out the probability that both sweets are mint.

(b) Work out the probability that both sweets are toffee.

●●●●●Level 24 marksStart

A bag has

5

sweets:

2

orange and

3

lemon. Two sweets are taken without replacement. Work out the probability that both are lemon.

●●●●●Level 22 marksStart

A box contains 7 red pens and 5 blue pens.

Two pens are taken from the box at random, without replacement.

(a) Work out the probability that both pens are red.

(b) Work out the probability that one pen is red and the other pen is blue.

●●●●●Level 35 marksStart

A pencil case contains 9 pencils.

4 of the pencils are blue and the other 5 are black.

Kofi takes two pencils at random from the pencil case, without replacement.

Work out the probability that at least one of the two pencils is blue.

●●●●●Level 33 marksStart

A bag contains

4

red counters and

6

blue counters. Two counters are taken at random without replacement.

(a) Work out the probability that both counters are blue. [2]

(b) Work out the probability that the counters are different colours. [2]

●●●●●Level 34 marksStart

A bag contains 5 white counters and 4 black counters.

Three counters are taken from the bag at random, without replacement.

(a) Work out the probability that all three counters are white.

(b) Work out the probability that all three counters are the same colour.

●●●●●Level 46 marksStart

A bag contains 5 green counters and 7 yellow counters.

Two counters are taken from the bag at random, without replacement.

Jack writes:

P(\text{both green}) = \frac{5}{12}\times\frac{5}{11} = \frac{25}{132}

(a) Explain what is wrong with the second fraction in Jack's working.

(b) Work out the correct probability that both counters are green.

●●●●●Level 43 marksStart

A tin contains 10 biscuits.

3 of the biscuits are chocolate and the other 7 are plain.

Three biscuits are taken from the tin at random, without replacement.

Work out the probability that at least one of the three biscuits is chocolate.

●●●●●Level 44 marksStart

A bag contains only green counters and white counters.

The probability that a counter taken at random from the bag is green is

\frac{3}{8}

.

There are 15 white counters in the bag.

(a) Show that there are 24 counters in the bag.

(b) Two counters are taken from the bag at random, without replacement.

Work out the probability that the two counters are the same colour.

●●●●●Level 46 marksStart

A bag contains

5

red counters and

3

blue counters and nothing else.

Kim takes at random two counters from the bag. She does not replace the first counter before taking the second.

Work out the probability that the two counters are different colours.

Give your answer as a fraction in its simplest form. (4 marks)

●●●●●Level 44 marksStart

A bag contains only red counters and blue counters.

There are

n

red counters and 4 blue counters in the bag.

Nia takes a counter at random from the bag and does not replace it. She then takes a second counter at random.

The probability that both of Nia's counters are red is

\frac{1}{3}

.

(a) Show that

n^2 - 5n - 6 = 0

.

(b) Work out the probability that Nia's two counters are different colours.

●●●●●Level 56 marksStart

There are only

5

white counters,

3

purple counters and

2

black counters in a bag.

Omar takes at random three counters from the bag.

Work out the probability that the bag is then left with more black counters than purple counters.

You must show all your working.

(5 marks)

●●●●●Level 55 marksStart

There are

12

discs numbered

1

12

in a tin.

Hana will take at random two discs from the tin.

Hana says,

"The probability that the sum of the two numbers is odd is twice the probability that their product is odd."

Is Hana correct?

You must show how you get your answer.

(5 marks)

●●●●●Level 55 marksStart

There are only

n

green sweets and

1

red sweet in a bag.

Leo takes at random a sweet from the bag and eats the sweet.

He then takes at random a second sweet from the bag and eats it.

Show that P(Leo eats one sweet of each colour)

= \dfrac{2}{n + 1}

(2 marks)

●●●●●Level 52 marksStart

A box holds

y

counters:

x

are red,

3

are white and the others are yellow.

x : y = 1 : 4

Seb takes at random two counters from the box.

Find an expression, in terms of

x

, for the probability that both counters are the same colour.

Give your answer as a fraction in the form

\dfrac{ax^2 + bx + c}{dx^2 + ex}

where

a

b

c

d

and

e

are integers.

(5 marks)

●●●●●Level 55 marksStart

Conditional probability

10 questions

LessonNot started

The two-way table shows information about the

80

members of a swimming club.

	Under 16	16 or over	Total
Male	$18$	$22$	$40$
Female	$27$	$13$	$40$
Total	$45$	$35$	$80$

(a) One of the members aged under 16 is chosen at random.

Work out the probability that this member is female.

(b) One of the female members is chosen at random.

Work out the probability that this member is aged under 16.

●●●●●Level 24 marksStart

A cycling club has

120

members.

45

of the members are aged under

18

and the rest are aged

18

or over.

27

of the members aged under

18

own a racing bike.

30

of the members aged

18

or over own a racing bike.

(a) One of the members aged under

18

is chosen at random.

Work out the probability that this member owns a racing bike.

(b) One of the members who owns a racing bike is chosen at random.

Work out the probability that this member is aged

18

or over.

●●●●●Level 35 marksStart

The two-way table gives information about how the

150

pupils in Years 10 and 11 travel to school. One cell has not been filled in.

	Walk	Cycle	Bus	Total
Year 10	$24$	$15$	$36$	$75$
Year 11	$31$	$9$		$75$
Total	$55$	$24$	$71$	$150$

(a) Work out the number of Year 11 pupils who travel by bus.

(b) One of the Year 11 pupils is chosen at random.

Work out the probability that this pupil cycles to school.

(c) One of the pupils who cycles to school is chosen at random.

Work out the probability that this pupil is in Year 10.

●●●●●Level 35 marksStart

400

adults in a town were asked whether they hold a driving licence and whether they own a car. The two-way table shows the results.

	Owns a car	Does not own a car	Total
Holds a licence	$210$	$70$	$280$
Does not hold a licence	$14$	$106$	$120$
Total	$224$	$176$	$400$

(a) One of the

400

adults is chosen at random.

Work out the probability that this adult owns a car and holds a licence.

(b) One of the adults who owns a car is chosen at random.

Work out the probability that this adult holds a licence.

(c) One of the adults who holds a licence is chosen at random.

Work out the probability that this adult owns a car.

●●●●●Level 36 marksStart

40

people were asked whether they drink tea (

T

) and whether they drink coffee (

C

).

The Venn diagram shows the results.One of the

40

people is chosen at random.
(a) Work out the probability that this person drinks tea or coffee or both. (1 mark)
(b) Work out the probability that this person drinks neither. (1 mark)

One of the people who drinks tea is chosen at random.
(c) Work out the probability that this person also drinks coffee. (2 marks)

●●●●●Level 34 marksStart

A theatre sold

240

tickets for a play.

90

of the tickets were sold online and the rest were sold at the box office.

Every ticket was either a concession ticket or a full-price ticket. Altogether

44

of the

240

tickets were concession tickets.

Given that a ticket was sold online, the probability that it was a concession ticket is

\frac{2}{15}

.

(a) Work out the number of concession tickets that were sold online.

(b) One of the concession tickets is chosen at random.

Work out the probability that it was sold at the box office.

●●●●●Level 45 marksStart

The two-way table shows information about

200

people who work in an office and how they travel to work.

	Cycles to work	Does not cycle to work	Total
Aged under 30	$24$	$56$	$80$
Aged 30 or over	$36$	$84$	$120$
Total	$60$	$140$	$200$

(a) One of the

200

people is chosen at random.

Write down the probability that this person cycles to work.

(b) One of the people aged under

30

is chosen at random.

Work out the probability that this person cycles to work.

(c) Ellie says that whether a person cycles to work is independent of their age group.

Is Ellie correct? You must justify your answer.

●●●●●Level 45 marksStart

Owen carried out a survey of

90

people.

He asked each person if they like apples (

A

), bananas (

B

) or grapes (

G

9

people like all three fruits.

23

people like apples and bananas.

11

people like apples and grapes but not bananas.

16

people like bananas and grapes.

44

people like grapes.

54

people like apples.

6

people like only bananas.
(a) Complete the Venn diagram for this information.(4 marks)

One of the

90

people is chosen at random.

Given that this person likes grapes,
(b) find the probability that this person also likes bananas.

(2 marks)

●●●●●Level 46 marksStart

A supermarket recorded the

300

customers who used the self-service tills one morning. The two-way table shows how each customer paid and whether they needed help from an assistant.

	Card	Cash	Phone	Total
Needed help	$21$	$18$	$9$	$48$
Did not need help	$129$	$42$	$81$	$252$
Total	$150$	$60$	$90$	$300$

(a) One of the customers who paid by cash is chosen at random.

Work out the probability that this customer needed help.

(b) One of the customers who needed help is chosen at random.

Work out the probability that this customer did not pay by cash.

(c) One of the customers who paid by card or by phone is chosen at random.

Work out the probability that this customer needed help.

●●●●●Level 58 marksStart

A gym has

400

members. Every member is on either the standard plan or the premium plan, and

240

of the members are on the standard plan.

Given that a member is on the premium plan, the probability that they attend a class is

0.65

.

Altogether

250

of the

400

members attend a class.

(a) Work out the number of premium members who attend a class.

(b) One of the members who attends a class is chosen at random.

Work out the probability that this member is on the standard plan.

(c) One of the members who does not attend a class is chosen at random.

Work out the probability that this member is on the premium plan.

●●●●●Level 58 marksStart