Browse Probability

The number of members who play both the piano and the guitar is $6$ .
The number of members who play neither the piano nor the guitar is $9$ .
The number of members who play the piano is twice the number of members who play the guitar.

(a) • Show this information on a Venn diagram.

Determine the probability that a member selected at random plays the guitar.

(4 marks)

Two different members of the society are chosen at random.
(b) Given that exactly one of the two members plays the guitar, determine the probability that exactly one of the two members plays the piano.

(6 marks)

●●●●●Level 510 marksStart

Tariq has a box containing 4 blue pens and 5 black pens only.

He takes two pens at random from the box, one at a time without replacement.

The tree diagram below, where

r

s

t

and

u

are probabilities, shows this information.(a) Complete the tree diagram by finding the values of

r

s

t

and

u

(3 marks)
(b) Find the probability that Tariq takes exactly one blue pen.

(2 marks)
(c) Find the probability that Tariq takes at least one blue pen.

(2 marks)
(d) Given that Tariq takes at least one blue pen, find the probability that he takes two blue pens.

(2 marks)

Tariq's teacher has a box containing

n

pens. The pens are blue, black and red in the ratio

2 : 3 : 1

Four pens are taken at random from the teacher's box, one at a time without replacement.

The probability that the first pen is blue, the second pen is black, the third pen is red and the fourth pen is black is

\frac{25}{1566}

(e) Show that

n

satisfies the equation

13n^2 + bn + c = 0

where

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 513 marksStart

Tomás has two fair six-sided dice, one green and one white.

The green die has faces numbered 1, 2, 3, 4, 5 and 6
The white die has faces numbered 1, 3, 5, 7, 9 and 11

The discrete random variable

G

is the number on the green die when it is rolled once. The discrete random variable

W

is the number on the white die when it is rolled once.
(a) Write down the name of the probability distribution of

G

(1 mark)
(b) Find

\mathrm{P}(G < 5)

(1 mark)
(c) Write down the value of

(i)

\mathrm{E}(G)

(ii)

\mathrm{E}(W)

(2 marks)
(d) Showing your working, find

\mathrm{Var}(W)

(3 marks)

Tomás rolls each die once.
(e) Find

\mathrm{P}(G + W \leqslant 7)

(2 marks)
(f) Find

\mathrm{P}(G > W)

(3 marks)

The random variable

L

is the smaller of the number on the green die and the number on the white die. If the two numbers are equal,

L

is that number.

The table below shows the cumulative distribution function of

L

\begin{array}{c|cccccc} l & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathrm{F}(l) & \dfrac{11}{36} & \dfrac{4}{9} & \dfrac{2}{3} & p & \dfrac{11}{12} & 1 \end{array}

(g) Showing your working, find the value of

p

(3 marks)

●●●●●Level 515 marksStart

Venn diagrams

23 questions

LessonNot started

A library service has 960 staff.

The staff are classified as librarian, assistant or technician.

The following table shows

the number of staff in each classification
the two sites, A or B, where the staff are based

	A	B
Librarian	240	120
Assistant	210	150
Technician	180	60

A member of staff is chosen at random.

Find the probability that this member of staff
(a) is an assistant,

(1 mark)
(b) is based at site B and is not a librarian.

(1 mark)

Some classifications of staff are more likely to have completed first-aid training.

70% of librarians in both site A and site B have completed first-aid training
30% of assistants in both site A and site B have completed first-aid training
10% of technicians in both site A and site B have completed first-aid training
Event $X$ is that the member of staff is a librarian
Event $Y$ is that the member of staff has completed first-aid training
Event $Z$ is that the member of staff is based at site A

A Venn diagram for the events

X

Y

and

Z

shows the number of staff in each region. Four of the regions have already been completed:

$X \cap Y' \cap Z$ contains 72 staff
$X \cap Y' \cap Z'$ contains 36 staff
$X' \cap Y' \cap Z$ contains 309 staff
$X' \cap Y' \cap Z'$ contains 159 staff

(c) Using this information, find the number of staff in each of the four regions

X \cap Y \cap Z

X \cap Y \cap Z'

X' \cap Y \cap Z

and

X' \cap Y \cap Z'

(4 marks)
(d) Find

\mathrm{P}(Z' \cap X)

(1 mark)
(e) Find

\mathrm{P}([Y \cup Z]')

(1 mark)
(f) Find

\mathrm{P}(X \mid Y)

(2 marks)

●●●●●Level 510 marksStart

The Venn diagram, where

p

q

and

r

are probabilities, shows the events

A

B

C

and

D

and associated probabilities.

In the diagram, the region for

A

overlaps only the region for

B

, the region for

B

also overlaps the region for

C

, and the region for

D

overlaps only the region for

C

.

The diagram shows that

the probability that $A$ occurs but $B$ does not occur is $q$
$\mathrm{P}(A \cap B) = 0.06$
the probability that $B$ occurs but neither $A$ nor $C$ occurs is $p$
$\mathrm{P}(B \cap C) = 0.18$
the probability that $C$ occurs but none of $A$ , $B$ or $D$ occurs is $0.22$
$\mathrm{P}(C \cap D) = 0.05$
the probability that $D$ occurs but $C$ does not occur is $r$
$\mathrm{P}([A \cup B \cup C \cup D]') = 0.08$

(a) State any pair of mutually exclusive events from

A

B

C

and

D

(1 mark)

The events

B

and

C

are independent.
(b) Find the value of

p

(2 marks)
(c) Find the greatest possible value of

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

(3 marks)

Given that

\mathrm{P}(B \mid A') = 0.425

(d) find the value of

q

and the value of

r

(3 marks)
(e) Find

\mathrm{P}([A \cup B]' \cap C)

(1 mark)
(f) Use set notation to write an expression for the event with probability

p

(1 mark)

●●●●●Level 511 marksStart

There are

30

members in a music society.

The number of members who play both the piano and the guitar is $6$ .
The number of members who play neither the piano nor the guitar is $9$ .
The number of members who play the piano is twice the number of members who play the guitar.

(a) • Show this information on a Venn diagram.

Determine the probability that a member selected at random plays the guitar.

●●●●●Level 510 marksStart

The Venn diagram below shows the probabilities that a randomly selected member of an allotment society grows tomatoes (

T

), beans (

B

) and courgettes (

C

), where

p

and

q

are probabilities.(a) Find the probability that a randomly selected member of the society grows tomatoes but does not grow courgettes.

(1 mark)

Given that the events

T

and

B

are independent,
(b) find the value of

p

(4 marks)
(c) Hence find the value of

q

(2 marks)
(d) Find

(i)

\mathrm{P}((T \cup B) \cap C')

(ii)

\mathrm{P}(T \mid (B \cap C))

(3 marks)

A second allotment society has a large number of members. In this society, 56 members grow both tomatoes and courgettes.
(e) Assuming that the probabilities in the Venn diagram also apply to the second society, estimate the number of members of the second society who do not grow beans.

(3 marks)

●●●●●Level 513 marksStart

Mutually exclusive and independent events

30 questions

LessonNot started

The Venn diagram, where

p

q

and

r

are probabilities, shows the events

A

B

C

and

D

and associated probabilities.

In the diagram, the region for

A

overlaps only the region for

B

, the region for

B

also overlaps the region for

C

, and the region for

D

overlaps only the region for

C

.

The diagram shows that

the probability that $A$ occurs but $B$ does not occur is $q$
$\mathrm{P}(A \cap B) = 0.06$
the probability that $B$ occurs but neither $A$ nor $C$ occurs is $p$
$\mathrm{P}(B \cap C) = 0.18$
the probability that $C$ occurs but none of $A$ , $B$ or $D$ occurs is $0.22$
$\mathrm{P}(C \cap D) = 0.05$
the probability that $D$ occurs but $C$ does not occur is $r$
$\mathrm{P}([A \cup B \cup C \cup D]') = 0.08$

(a) State any pair of mutually exclusive events from

A

B

C

and

D

(1 mark)

The events

B

and

C

are independent.
(b) Find the value of

p

(2 marks)
(c) Find the greatest possible value of

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

(3 marks)

Given that

\mathrm{P}(B \mid A') = 0.425

(d) find the value of

q

and the value of

r

(3 marks)
(e) Find

\mathrm{P}([A \cup B]' \cap C)

(1 mark)
(f) Use set notation to write an expression for the event with probability

p

(1 mark)

●●●●●Level 511 marksStart

The Venn diagram below shows the probabilities that a randomly selected member of an allotment society grows tomatoes (

T

), beans (

B

) and courgettes (

C

), where

p

and

q

are probabilities.(a) Find the probability that a randomly selected member of the society grows tomatoes but does not grow courgettes.

(1 mark)

Given that the events

T

and

B

are independent,
(b) find the value of

p

(4 marks)
(c) Hence find the value of

q

(2 marks)
(d) Find

(i)

\mathrm{P}((T \cup B) \cap C')

(ii)

\mathrm{P}(T \mid (B \cap C))

●●●●●Level 513 marksStart

A biased four-sided die has faces numbered 1, 2, 3 and 4. The die is rolled once. The discrete random variable

X

represents the number on the face on which the die lands. The cumulative distribution function of

X

is shown in the table below.

\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline \mathrm{F}(x) & k & 0.35 & 4k & 5k \end{array}

(a) Find the value of the constant

k

(1 mark)
(b) Find the probability distribution of

X

(3 marks)

A bag contains a large number of tokens, each labelled 0, 1, 2, 3 or 4. A token is taken at random from the bag. The discrete random variable

Y

represents the label on the token. The probability distribution of

Y

is shown in the table below, where

a

and

b

are constants.

\begin{array}{c|ccccc} y & 0 & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(Y = y) & a & 0.05 & b & a & b \end{array}

Given that

\mathrm{E}(Y) = 2.3

(c) form and solve two equations in

a

and

b

to show that

a = 0.2

You must show your working.

(Solutions relying on calculator technology are not acceptable.)

(3 marks)
(d) Show that

\mathrm{E}(Y^2) = 7.35

(2 marks)
(e) Find

\mathrm{Var}(7 - 3Y)

(3 marks)

The die is rolled once and a token is taken at random from the bag. The number on the die and the label on the token are independent.
(f) Find the probability that the sum of the number on the die and the label on the token is 4

(2 marks)

●●●●●Level 514 marksStart

Tomás has two fair six-sided dice, one green and one white.

The green die has faces numbered 1, 2, 3, 4, 5 and 6
The white die has faces numbered 1, 3, 5, 7, 9 and 11

The discrete random variable

G

is the number on the green die when it is rolled once. The discrete random variable

W

is the number on the white die when it is rolled once.
(a) Write down the name of the probability distribution of

G

(1 mark)
(b) Find

\mathrm{P}(G < 5)

(1 mark)
(c) Write down the value of

(i)

\mathrm{E}(G)

(ii)

\mathrm{E}(W)

(2 marks)
(d) Showing your working, find

\mathrm{Var}(W)

(3 marks)

Tomás rolls each die once.
(e) Find

\mathrm{P}(G + W \leqslant 7)

(2 marks)
(f) Find

\mathrm{P}(G > W)

(3 marks)

The random variable

L

is the smaller of the number on the green die and the number on the white die. If the two numbers are equal,

L

is that number.

The table below shows the cumulative distribution function of

L

\begin{array}{c|cccccc} l & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathrm{F}(l) & \dfrac{11}{36} & \dfrac{4}{9} & \dfrac{2}{3} & p & \dfrac{11}{12} & 1 \end{array}

(g) Showing your working, find the value of

p

(3 marks)

●●●●●Level 515 marksStart

Tree diagrams and sequential events

16 questions

LessonNot started

Jess and Rhea are playing a game which consists of several points.

The probability that Jess wins the first point is $0.4$ ; otherwise Rhea wins it.
For each point after the first, the player who won the previous point wins with probability $0.6$ .

The first player to win two points wins the game.

The diagram shows the first stage of a probability tree diagram for the game.(a) Write down the greatest possible number of points in a game.

(1 mark)
(b) Complete the probability tree diagram, showing all the possibilities for the game.

(3 marks)
(c) Determine the probability that Rhea wins the game.

(3 marks)
(d) Determine the probability that the game consists of exactly three points.

(2 marks)
(e) Given that Rhea wins the game, determine the probability that the game consists of exactly three points.

(4 marks)

●●●●●Level 513 marksStart

Tariq has a box containing 4 blue pens and 5 black pens only.

He takes two pens at random from the box, one at a time without replacement.

The tree diagram below, where

r

s

t

and

u

are probabilities, shows this information.(a) Complete the tree diagram by finding the values of

r

s

t

and

u

n

pens. The pens are blue, black and red in the ratio

2 : 3 : 1

\frac{25}{1566}

(e) Show that

n

satisfies the equation

13n^2 + bn + c = 0

where

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 513 marksStart

Set notation for events

29 questions

LessonNot started

A library service has 960 staff.

The staff are classified as librarian, assistant or technician.

The following table shows

the number of staff in each classification
the two sites, A or B, where the staff are based

	A	B
Librarian	240	120
Assistant	210	150
Technician	180	60

70% of librarians in both site A and site B have completed first-aid training
30% of assistants in both site A and site B have completed first-aid training
10% of technicians in both site A and site B have completed first-aid training
Event $X$ is that the member of staff is a librarian
Event $Y$ is that the member of staff has completed first-aid training
Event $Z$ is that the member of staff is based at site A

A Venn diagram for the events

X

Y

and

Z

shows the number of staff in each region. Four of the regions have already been completed:

$X \cap Y' \cap Z$ contains 72 staff
$X \cap Y' \cap Z'$ contains 36 staff
$X' \cap Y' \cap Z$ contains 309 staff
$X' \cap Y' \cap Z'$ contains 159 staff

(c) Using this information, find the number of staff in each of the four regions

X \cap Y \cap Z

X \cap Y \cap Z'

X' \cap Y \cap Z

and

X' \cap Y \cap Z'

(4 marks)
(d) Find

\mathrm{P}(Z' \cap X)

(1 mark)
(e) Find

\mathrm{P}([Y \cup Z]')

(1 mark)
(f) Find

\mathrm{P}(X \mid Y)

(2 marks)

●●●●●Level 510 marksStart

The Venn diagram, where

p

q

and

r

are probabilities, shows the events

A

B

C

and

D

and associated probabilities.

In the diagram, the region for

A

overlaps only the region for

B

, the region for

B

also overlaps the region for

C

, and the region for

D

overlaps only the region for

C

.

The diagram shows that

the probability that $A$ occurs but $B$ does not occur is $q$
$\mathrm{P}(A \cap B) = 0.06$
the probability that $B$ occurs but neither $A$ nor $C$ occurs is $p$
$\mathrm{P}(B \cap C) = 0.18$
the probability that $C$ occurs but none of $A$ , $B$ or $D$ occurs is $0.22$
$\mathrm{P}(C \cap D) = 0.05$
the probability that $D$ occurs but $C$ does not occur is $r$
$\mathrm{P}([A \cup B \cup C \cup D]') = 0.08$

(a) State any pair of mutually exclusive events from

A

B

C

and

D

(1 mark)

The events

B

and

C

are independent.
(b) Find the value of

p

(2 marks)
(c) Find the greatest possible value of

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

(3 marks)

Given that

\mathrm{P}(B \mid A') = 0.425

(d) find the value of

q

and the value of

r

(3 marks)
(e) Find

\mathrm{P}([A \cup B]' \cap C)

(1 mark)
(f) Use set notation to write an expression for the event with probability

p

(1 mark)

●●●●●Level 511 marksStart

The Venn diagram below shows the probabilities that a randomly selected member of an allotment society grows tomatoes (

T

), beans (

B

) and courgettes (

C

), where

p

and

q

are probabilities.(a) Find the probability that a randomly selected member of the society grows tomatoes but does not grow courgettes.

(1 mark)

Given that the events

T

and

B

are independent,
(b) find the value of

p

(4 marks)
(c) Hence find the value of

q

(2 marks)
(d) Find

(i)

\mathrm{P}((T \cup B) \cap C')

(ii)

\mathrm{P}(T \mid (B \cap C))

●●●●●Level 513 marksStart

Conditional probability

61 questions

LessonNot started

A library service has 960 staff.

The staff are classified as librarian, assistant or technician.

The following table shows

the number of staff in each classification
the two sites, A or B, where the staff are based

	A	B
Librarian	240	120
Assistant	210	150
Technician	180	60

70% of librarians in both site A and site B have completed first-aid training
30% of assistants in both site A and site B have completed first-aid training
10% of technicians in both site A and site B have completed first-aid training
Event $X$ is that the member of staff is a librarian
Event $Y$ is that the member of staff has completed first-aid training
Event $Z$ is that the member of staff is based at site A

A Venn diagram for the events

X

Y

and

Z

shows the number of staff in each region. Four of the regions have already been completed:

$X \cap Y' \cap Z$ contains 72 staff
$X \cap Y' \cap Z'$ contains 36 staff
$X' \cap Y' \cap Z$ contains 309 staff
$X' \cap Y' \cap Z'$ contains 159 staff

(c) Using this information, find the number of staff in each of the four regions

X \cap Y \cap Z

X \cap Y \cap Z'

X' \cap Y \cap Z

and

X' \cap Y \cap Z'

(4 marks)
(d) Find

\mathrm{P}(Z' \cap X)

(1 mark)
(e) Find

\mathrm{P}([Y \cup Z]')

(1 mark)
(f) Find

\mathrm{P}(X \mid Y)

(2 marks)

●●●●●Level 510 marksStart

The Venn diagram, where

p

q

and

r

are probabilities, shows the events

A

B

C

and

D

and associated probabilities.

In the diagram, the region for

A

overlaps only the region for

B

, the region for

B

also overlaps the region for

C

, and the region for

D

overlaps only the region for

C

.

The diagram shows that

the probability that $A$ occurs but $B$ does not occur is $q$
$\mathrm{P}(A \cap B) = 0.06$
the probability that $B$ occurs but neither $A$ nor $C$ occurs is $p$
$\mathrm{P}(B \cap C) = 0.18$
the probability that $C$ occurs but none of $A$ , $B$ or $D$ occurs is $0.22$
$\mathrm{P}(C \cap D) = 0.05$
the probability that $D$ occurs but $C$ does not occur is $r$
$\mathrm{P}([A \cup B \cup C \cup D]') = 0.08$

(a) State any pair of mutually exclusive events from

A

B

C

and

D

(1 mark)

The events

B

and

C

are independent.
(b) Find the value of

p

(2 marks)
(c) Find the greatest possible value of

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

(3 marks)

Given that

\mathrm{P}(B \mid A') = 0.425

(d) find the value of

q

and the value of

r

(3 marks)
(e) Find

\mathrm{P}([A \cup B]' \cap C)

(1 mark)
(f) Use set notation to write an expression for the event with probability

p

(1 mark)

●●●●●Level 511 marksStart

A random variable

X

has probability distribution defined as follows.

\mathrm{P}(X = x) = \begin{cases} kx^2 & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}

where

k

is a constant.
(a) Show that

\mathrm{P}(X = 4) = \dfrac{8}{15}

.

(3 marks)
(b) Show in a table the values of

X

and their probabilities.

(2 marks)
(c) Two independent values of

X

are chosen, and their total

T

is found.

(i) Find

\mathrm{P}(T = 6)

.

(ii) Given that

T = 6

, determine the probability that one of the values of

X

4

.

(7 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

The random variable

X

has probability distribution defined as follows.

\mathrm{P}(X = x) = \begin{cases} k(5 - x) & x = 1, 2, 3, 4, \\ 0 & \text{otherwise,} \end{cases}

where

k

is a constant.
(a) Show that

k = 0.1

.

(2 marks)

The values of three independent observations of

X

are denoted by

X_1

X_2

and

X_3

.
(b) Given that

X_1 + X_2 + X_3 = 7

, determine the probability that at least one of these three values is equal to

1

.

(6 marks)

Maya chooses values of

X

at random until she has obtained

X = 4

exactly twice. She then stops.
(c) Determine the probability that she chooses exactly

8

values of

X

.

(3 marks)

●●●●●Level 511 marksStart

Jess and Rhea are playing a game which consists of several points.

The probability that Jess wins the first point is $0.4$ ; otherwise Rhea wins it.
For each point after the first, the player who won the previous point wins with probability $0.6$ .

●●●●●Level 513 marksStart

There are

30

members in a music society.

The number of members who play both the piano and the guitar is $6$ .
The number of members who play neither the piano nor the guitar is $9$ .
The number of members who play the piano is twice the number of members who play the guitar.

(a) • Show this information on a Venn diagram.

Determine the probability that a member selected at random plays the guitar.

●●●●●Level 510 marksStart

100

people,

40

own a car (

A

30

own a bike (

B

) and

10

own both. A person is chosen at random.
(a) Find

P(A \mid B)

.
(b) Find

P(B \mid A)

.
(c) Determine whether owning a car and owning a bike are independent.

●●●●●Level 54 marksStart

Tariq has a box containing 4 blue pens and 5 black pens only.

He takes two pens at random from the box, one at a time without replacement.

The tree diagram below, where

r

s

t

and

u

are probabilities, shows this information.(a) Complete the tree diagram by finding the values of

r

s

t

and

u

n

pens. The pens are blue, black and red in the ratio

2 : 3 : 1

\frac{25}{1566}

(e) Show that

n

satisfies the equation

13n^2 + bn + c = 0

where

b

and

c

are integers to be found.

(4 marks)

●●●●●Level 513 marksStart

The Venn diagram below shows the probabilities that a randomly selected member of an allotment society grows tomatoes (

T

), beans (

B

) and courgettes (

C

), where

p

and

q

are probabilities.(a) Find the probability that a randomly selected member of the society grows tomatoes but does not grow courgettes.

(1 mark)

Given that the events

T

and

B

are independent,
(b) find the value of

p

(4 marks)
(c) Hence find the value of

q

(2 marks)
(d) Find

(i)

\mathrm{P}((T \cup B) \cap C')

(ii)

\mathrm{P}(T \mid (B \cap C))

●●●●●Level 513 marksStart

Conditional probability and tree diagrams

26 questions

LessonNot started

Jess and Rhea are playing a game which consists of several points.

The probability that Jess wins the first point is $0.4$ ; otherwise Rhea wins it.
For each point after the first, the player who won the previous point wins with probability $0.6$ .