Figure — understand the maths, don't just answer it

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Calculating probabilities and sample space

14 questions

LessonNot started

Two fair six-sided dice are rolled. Find the probability that the sum of the scores is

7

.

●●●●●Level 22 marksStart

Venn diagrams

23 questions

LessonNot started

The Venn diagram, where

p

and

q

are probabilities, shows the two events

A

and

B

and their associated probabilities.

The diagram shows that

$\mathrm{P}(A \cap B') = 0.45$
$\mathrm{P}(A \cap B) = p$
$\mathrm{P}(A' \cap B) = q$
$\mathrm{P}([A \cup B]') = 0.3$

The events

A

and

B

are independent.
(a) Find the value of

p

and the value of

q

(3 marks)
(b) Find

\mathrm{P}(A' \mid B)

(2 marks)

●●●●●Level 25 marksStart

The Venn diagram shows probabilities for events

A

and

B

. Find

P(A \cup B)

.

●●●●●Level 22 marksStart

A sample of

90

people who visited a museum were asked which, if any, of the following they used during their visit:

the café ( $C$ )
the gift shop ( $G$ )
the audio guide ( $A$ )

The results are shown in the table below.

Used	Number of people
$C$	$46$
$G$	$32$
$A$	$22$
$C$ and $G$	$15$
$C$ and $A$	$10$
$G$ and $A$	$7$
$C$ , $G$ and $A$	$4$

(a) Represent this information by fully completing the Venn diagram below.(3 marks)
(b) A person is chosen at random from the sample.

Find the probability that the person used the gift shop only.

(1 mark)

●●●●●Level 24 marksStart

Mutually exclusive and independent events

30 questions

LessonNot started

The Venn diagram, where

p

and

q

are probabilities, shows the two events

A

and

B

and their associated probabilities.

The diagram shows that

$\mathrm{P}(A \cap B') = 0.45$
$\mathrm{P}(A \cap B) = p$
$\mathrm{P}(A' \cap B) = q$
$\mathrm{P}([A \cup B]') = 0.3$

The events

A

and

B

are independent.
(a) Find the value of

p

and the value of

q

(3 marks)
(b) Find

\mathrm{P}(A' \mid B)

(2 marks)

●●●●●Level 25 marksStart

The mutually exclusive events

A

and

B

have

P(A) = 0.3

and

P(B) = 0.4

. Find

P(A \cup B)

.

●●●●●Level 22 marksStart

SECTION B: STATISTICS

Two events

A

and

B

are such that

\mathrm{P}(A) = 0.4

,

\quad \mathrm{P}(B) = 0.55

,

\quad \mathrm{P}(A \cup B) = 0.73

.
(a) Find

\mathrm{P}(A \cap B)

. (2 marks)
(b) Determine whether the events

A

and

B

are independent. (2 marks)
(c) Find

\mathrm{P}(A' \cap B')

. (2 marks)

●●●●●Level 26 marksStart

Tree diagrams and sequential events

16 questions

LessonNot started

A bag contains

5

red and

3

blue counters. Two are drawn at random, with replacement. Find the probability that both are red.

●●●●●Level 22 marksStart

Set notation for events

29 questions

LessonNot started

For events

A

and

B

,

P(A) = 0.5

,

P(B) = 0.4

and

P(A \cap B) = 0.2

. Find

P(A \cap B')

.

●●●●●Level 22 marksStart

For events

A

and

B

,

P(A) = 0.5

,

P(B) = 0.4

and

P(A \cap B) = 0.2

. Find

P(A \cup B)

.

●●●●●Level 22 marksStart

Events

A

and

B

are independent, with

P(A) = 0.4

and

P(B) = 0.5

.
(a) Find

P(A \cap B)

. (2 marks)
(b) Find

P(A \cup B)

. (2 marks)

●●●●●Level 24 marksStart

For events

A

and

B

,

\; P(A) = 0.62

and

P(A \cap B) = 0.25

.

Find

P(A \cap B')

. (2 marks)

●●●●●Level 22 marksStart

SECTION B: STATISTICS

Two events

A

and

B

are such that

\mathrm{P}(A) = 0.4

,

\quad \mathrm{P}(B) = 0.55

,

\quad \mathrm{P}(A \cup B) = 0.73

.
(a) Find

\mathrm{P}(A \cap B)

. (2 marks)
(b) Determine whether the events

A

and

B

are independent. (2 marks)
(c) Find

\mathrm{P}(A' \cap B')

. (2 marks)

●●●●●Level 26 marksStart

Conditional probability

61 questions

LessonNot started

A cycling club has 480 members.

The members are classified as junior, senior or veteran.

The following table shows

the number of members in each classification
the two venues, A or B, where the members train

	A	B
Junior	110	70
Senior	130	90
Veteran	50	30

A member is chosen at random.

Find the probability that this member
(a) is a senior,

(1 mark)
(b) trains at venue B and is not a junior.

(1 mark)
(c) Given that the member trains at venue A, find the probability that the member is a veteran.

(2 marks)

●●●●●Level 24 marksStart

A bag contains a large number of counters. Each counter is numbered 1, 2, 3 or 4

A counter is taken at random from the bag.

The random variable

X

represents the number on the counter.

The probability distribution of

X

is given in the following table

$x$	1	2	3	4
$\mathrm{P}(X = x)$	0.15	$k$	$2k$	0.25

where

k

is a constant.
(a) Find the value of

k

(2 marks)
(b) Find

\mathrm{P}(X \geqslant 2 \mid X \leqslant 3)

(2 marks)

●●●●●Level 24 marksStart

For events

A

and

B

,

P(A \cap B) = 0.2

and

P(B) = 0.5

. Find

P(A \mid B)

.

●●●●●Level 22 marksStart

For events

A

and

B

,

P(A \cap B) = 0.2

and

P(A) = 0.5

. Find

P(B \mid A)

.

●●●●●Level 22 marksStart

For events

A

and

B

,

\; P(A \cap B) = 0.12

and

P(B) = 0.4

.

Find

P(A \mid B)

. (2 marks)

●●●●●Level 22 marksStart

For events

A

and

B

,

\; P(A \mid B) = 0.4

and

P(B) = 0.35

.

Find

P(A \cap B)

. (2 marks)

●●●●●Level 22 marksStart

Events

A

and

B

are mutually exclusive, with

P(A) = 0.3

and

P(B) = 0.45

.

Write down

P(A \mid B)

, justifying your answer. (2 marks)

●●●●●Level 22 marksStart

A tray holds

6

cupcakes. Four of the cupcakes cost

£2

each and two of the cupcakes cost

£5

each.

Two cupcakes are taken at random from the tray, one after the other, without replacement.

Find the probability that the mean price of the two cupcakes is

£3.50

(3 marks)

●●●●●Level 23 marksStart

Conditional probability and tree diagrams

26 questions

LessonNot started

A bag contains

3

red and

2

blue counters. Two are drawn at random without replacement. Find the probability that both are red.

●●●●●Level 22 marksStart