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

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

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 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

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

12

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

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}

●●●●●Level 52 marksStart

y

x

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}

a

b

c

d

e

are integers.

(5 marks)

●●●●●Level 55 marksStart

Conditional probability

10 questions

LessonNot started

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

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

250

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