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

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Discrete random variables and probability distributions

39 questions

LessonNot started

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

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

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

is

4

.

(7 marks)

●●●●●Level 512 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

The binomial distribution as a model

32 questions

LessonNot started

A company makes smoke alarms. It knows that

1.5\%

of the alarms it makes are faulty. A batch of

200

alarms is selected at random.
(a) Write down a suitable distribution to model the number of faulty alarms in the batch.

(1 mark)
(b) Using a suitable approximation, find the probability that the batch contains

(i) exactly

4

faulty alarms,

(ii) more than

5

faulty alarms.

(4 marks)
(c) Justify why a Poisson approximation is suitable in this case.

(1 mark)

After a change to the production process, the company claims that the proportion of faulty alarms has decreased. A random sample of

500

alarms made after the change contains

3

faulty alarms.
(d) Using a suitable approximation, test, at the

5\%

level of significance, whether or not there is evidence to support the company's claim. State your hypotheses clearly.

(5 marks)

The company plans to test a random sample of

n

alarms made after the change, and to accept that its claim is supported, at the

5\%

level of significance, only if none of the

n

alarms is faulty.
(e) Using a Poisson approximation, show that a sample with no faulty alarms supports the claim when

n \geqslant \frac{200}{3}\ln 20

(2 marks)
(f) Hence find the smallest possible value of

n

.

(1 mark)

●●●●●Level 514 marksStart

The weights of apples from an orchard,

W

grams, are modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

.

It is known that

10\%

of the apples weigh less than

120

g and that

5\%

of the apples weigh more than

190

g.
(a) Find the value of

\mu

and the value of

\sigma

, giving each answer to

3

significant figures. (6 marks)

An apple is classed as large if it weighs more than

160

g.
(b) Show that the probability that a randomly chosen apple is large is approximately

0.35

. (2 marks)

A wholesaler buys a random sample of

200

apples from the orchard.
(c) Using a suitable approximation, which you should justify, estimate the probability that at least

80

of these apples are large. Use

\mathrm{P}(\text{large}) = 0.35

. (5 marks)

●●●●●Level 513 marksStart

The continuous random variable

X

is uniformly distributed over the interval

[-2k, 3k]

, where

k

is a positive constant.
(a) Write down

\mathrm{E}(X)

in terms of

k

and show that

\mathrm{Var}(X) = \dfrac{25k^2}{12}

(3 marks)
(b) Find

\mathrm{E}(X^2)

in terms of

k

(2 marks)

The random variable

Y

is defined by

Y = |X|

(c) Show that

\mathrm{P}(Y \leqslant y) = \dfrac{2y}{5k}

for

0 \leqslant y \leqslant 2k

(3 marks)
(d) Find

\mathrm{P}(Y \leqslant y)

for

2k < y \leqslant 3k

, giving your answer in terms of

y

and

k

(2 marks)
(e) Find the median of

Y

in terms of

k

(2 marks)

Given that

k = 2

, a random sample of

50

independent observations of

X

is taken.
(f) Using a suitable approximation, find the probability that more than

25

of these observations satisfy

|X| < 3

(4 marks)

●●●●●Level 516 marksStart

Orla is practising archery.

Each time Orla shoots an arrow at a target, the probability that the arrow lands in the gold ring is

0.24

, independently of all other arrows.

In a round, Orla shoots

12

arrows at the target.

The random variable

X

represents the number of arrows that land in the gold ring in a round.
(a) Using a suitable distribution for

X

, find

(i)

\mathrm{P}(X = 3)

(ii)

\mathrm{P}(X \geqslant 5)

(3 marks)

Orla shoots one round each day for

25

days and records the number of arrows that land in the gold ring.
(b) Find the probability that there were at least

8

days when exactly

3

arrows landed in the gold ring.

(3 marks)
(c) Find an estimate for the total number of arrows that landed in the gold ring during these

25

days.

(1 mark)

Orla now shoots

150

arrows at the target.
(d) Use a normal approximation to estimate the probability that more than

42

of these arrows land in the gold ring.

(4 marks)

●●●●●Level 511 marksStart

In this question you must show detailed reasoning.

The probability that a seed of a particular variety of sunflower fails to germinate is 0.24, independently of other seeds.
(a) A grower plants 400 of these seeds. The number of seeds that fail to germinate is denoted by the random variable

Y

. Determine the smallest value of

a

such that

\mathrm{P}(Y > a) < 0.05

.

(3 marks)

In the expansion of

(0.24 + 0.76)^{40}

, the terms involving

0.24^r

and

0.24^{r+1}

are denoted by

T_r

and

T_{r+1}

respectively.
(b) Show that

\dfrac{T_{r+1}}{T_r} = \dfrac{6(40 - r)}{19(r + 1)}

.

(3 marks)

The number of seeds that fail to germinate in a packet of 40 of these seeds is modelled by the random variable

X

.
(c) (i) Determine the values of

r

for which

\mathrm{P}(X = r) \leqslant \mathrm{P}(X = r + 1)

.

(ii) Hence determine the most likely number of seeds in a packet of 40 that fail to germinate.

(6 marks)

●●●●●Level 512 marksStart

A cinema chain prints a reward code on every ticket it sells. Each code is worth either

1

point or

5

points, and

1

-point codes and

5

-point codes are in the ratio

9 : 1

respectively.

A customer's next

3

tickets may be regarded as a random sample from the very large number of tickets sold. The random variable

T

represents the total number of points on the

3

codes.
(a) Explain what you understand by the sampling distribution of

T

(1 mark)
(b) Show that

P(T = 7) = 0.243

(2 marks)
(c) Find the sampling distribution of

T

(4 marks)

A customer whose

3

codes give a total of

11

points or more wins a free drink.

In one week,

125

customers each buy

3

tickets. The random variable

Y

represents the number of these customers who win a free drink.
(d) Using a suitable approximation, find

P(Y \geqslant 6)

(4 marks)
(e) Find the smallest number of customers,

n

, for which the probability that at least one of them wins a free drink is greater than

0.99

(3 marks)

●●●●●Level 514 marksStart

A factory makes a very large number of floor tiles, and each tile is stamped with a serial number when it is made. Each tile is graded

A

,

B

or

C

and sold for

£3

,

£2

or

£1

respectively. Tiles of grades

A

,

B

and

C

are made in the ratio

1 : 6 : 3

respectively.

A quality inspector checks a random sample of the tiles made each day, rather than inspecting every tile.
(a) (i) Suggest a suitable sampling frame for the inspector to use.

(ii) Identify the sampling units.

(2 marks)

A random sample of

3

tiles is taken. The random variable

M

represents the median price, in pounds, of the

3

tiles.
(b) Show that

P(M = 1) = 0.216

(2 marks)
(c) Find the sampling distribution of

M

(4 marks)

On each of

150

days, a random sample of

3

tiles is taken. The random variable

Y

represents the number of these

150

samples that have a median price of

£1

(d) Using a suitable approximation, find the probability that fewer than

25

of these samples have a median price of

£1

(5 marks)

●●●●●Level 513 marksStart

Calculating binomial probabilities P(X = r)

40 questions

LessonNot started

Signal failures on a stretch of railway line occur at random, at a mean rate of

0.5

per week.
(a) State two assumptions, in context, that are needed for a Poisson distribution to be a suitable model for the number of signal failures.

(2 marks)
(b) Show that the probability of at least one signal failure in a

4

-week period is

0.8647

to

4

decimal places.

(2 marks)

A year is divided into

13

periods, each of

4

weeks.
(c) Using your answer to part (b), find the probability that there is at least one signal failure in exactly

11

of the

13

periods.

(3 marks)

New signalling equipment is fitted along the line. The operator claims that the rate of signal failures has decreased. In the

16

weeks after the equipment is fitted, there are

3

signal failures.
(d) Test, at the

5\%

level of significance, whether or not there is evidence to support the operator's claim. State your hypotheses clearly.

(5 marks)
(e) Find the largest number of signal failures in a

16

-week period that would support the operator's claim at the

1\%

level of significance.

(2 marks)

●●●●●Level 514 marksStart

A manufacturer claims that a treatment is effective for

60\%

of patients. In a trial, the treatment is given to a random sample of

20

patients, and

16

of them improve.

A researcher wishes to test, at the

5\%

level of significance, whether the proportion of patients for whom the treatment is effective is greater than

0.6

.
(a) Carry out the test, stating your hypotheses clearly. (5 marks)
(b) Find the critical region for this test. (4 marks)

●●●●●Level 59 marksStart

Orla is practising archery.

Each time Orla shoots an arrow at a target, the probability that the arrow lands in the gold ring is

0.24

, independently of all other arrows.

In a round, Orla shoots

12

arrows at the target.

The random variable

X

represents the number of arrows that land in the gold ring in a round.
(a) Using a suitable distribution for

X

, find

(i)

\mathrm{P}(X = 3)

(ii)

\mathrm{P}(X \geqslant 5)

(3 marks)

Orla shoots one round each day for

25

days and records the number of arrows that land in the gold ring.
(b) Find the probability that there were at least

8

days when exactly

3

arrows landed in the gold ring.

(3 marks)
(c) Find an estimate for the total number of arrows that landed in the gold ring during these

25

days.

(1 mark)

Orla now shoots

150

arrows at the target.
(d) Use a normal approximation to estimate the probability that more than

42

of these arrows land in the gold ring.

(4 marks)

●●●●●Level 511 marksStart

In this question you must show detailed reasoning.

The probability that a seed of a particular variety of sunflower fails to germinate is 0.24, independently of other seeds.
(a) A grower plants 400 of these seeds. The number of seeds that fail to germinate is denoted by the random variable

Y

. Determine the smallest value of

a

such that

\mathrm{P}(Y > a) < 0.05

.

(3 marks)

In the expansion of

(0.24 + 0.76)^{40}

, the terms involving

0.24^r

and

0.24^{r+1}

are denoted by

T_r

and

T_{r+1}

respectively.
(b) Show that

\dfrac{T_{r+1}}{T_r} = \dfrac{6(40 - r)}{19(r + 1)}

.

(3 marks)

The number of seeds that fail to germinate in a packet of 40 of these seeds is modelled by the random variable

X

.
(c) (i) Determine the values of

r

for which

\mathrm{P}(X = r) \leqslant \mathrm{P}(X = r + 1)

.

(ii) Hence determine the most likely number of seeds in a packet of 40 that fail to germinate.

(6 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

Cumulative binomial probabilities

39 questions

LessonNot started

A council claims that

60\%

of the households in a town put out a recycling bin each week. A campaigner believes that the true proportion is lower than this.

The campaigner observes a random sample of

20

households in the town and records the number,

X

, that put out a recycling bin in a particular week.
(a) Using a

5\%

level of significance, find the critical region for a test of the campaigner's belief. State your hypotheses clearly. (3 marks)
(b) State the actual significance level of the test, and explain why it is not exactly

5\%

. (2 marks)
(c) In the sample,

8

households put out a recycling bin. State the conclusion of the test in context. (2 marks)

The campaigner then observes a second random sample of

150

households in the town, of which

77

put out a recycling bin.
(d) Assuming that the council's claim is correct, use a suitable approximation to estimate the probability that

77

or fewer households in a random sample of

150

put out a recycling bin. Justify the use of your approximation, and comment on what your answer suggests about the council's claim. (5 marks)

●●●●●Level 512 marksStart

A craft kit contains wooden sticks, each of length

30

cm. A child snaps a stick into two pieces at a random point. The child then tries to make a triangle, using the two pieces as two of the sides and a rod of length

9

cm as the third side.
(a) Show that the probability that the child can make a triangle is

0.3

(4 marks)
(b) Find the expected length of the longer piece.

(2 marks)

The child snaps

10

sticks, one at a time, and tries to make a triangle with the two pieces of each stick and the rod.
(c) Find the probability that the child can make a triangle on at least

4

of the

10

attempts.

(3 marks)

In a school, the kit is used for

200

such attempts.
(d) Using a suitable approximation, find the probability that more than

70

of the

200

attempts produce a triangle.

(5 marks)

●●●●●Level 514 marksStart

Orla is practising archery.

Each time Orla shoots an arrow at a target, the probability that the arrow lands in the gold ring is

0.24

, independently of all other arrows.

In a round, Orla shoots

12

arrows at the target.

The random variable

X

represents the number of arrows that land in the gold ring in a round.
(a) Using a suitable distribution for

X

, find

(i)

\mathrm{P}(X = 3)

(ii)

\mathrm{P}(X \geqslant 5)

(3 marks)

Orla shoots one round each day for

25

days and records the number of arrows that land in the gold ring.
(b) Find the probability that there were at least

8

days when exactly

3

arrows landed in the gold ring.

(3 marks)
(c) Find an estimate for the total number of arrows that landed in the gold ring during these

25

days.

(1 mark)

Orla now shoots

150

arrows at the target.
(d) Use a normal approximation to estimate the probability that more than

42

of these arrows land in the gold ring.

(4 marks)

●●●●●Level 511 marksStart

A manufacturer uses a machine to make ceramic tiles.

The width of a tile,

W

mm, is normally distributed with

a mean of $46$ mm
a standard deviation of $x$ mm

Given that the proportion of tiles less than

44.5

mm in width is

4\%

(a) show that

x = 0.86

to 2 decimal places.

(2 marks)
(b) Calculate the proportion of tiles that are between

45

mm and

47

mm in width.

(1 mark)

The cost of producing a single tile is

18

p

A tile

where $W < 45$ is sold as a second for $25$ p
where $45 \leqslant W \leqslant 47$ is sold for $40$ p
where $W > 47$ is re-fired for an extra cost of $9$ p and then sold for $40$ p

(c) Calculate the expected profit per

400

of the tiles. Give your answer to the nearest pound.

(5 marks)

The same manufacturer makes plastic tile spacers in large batches.

The spacers each have a probability of

0.02

of being faulty.

A random sample of

150

spacers is taken from each batch and the batch is accepted if fewer than

7

of them are faulty.

The manufacturer's aim is for

95\%

of batches to be accepted.
(d) Explain whether the manufacturer is likely to achieve its aim.

(4 marks)

●●●●●Level 512 marksStart

In this question you must show detailed reasoning.

The probability that a seed of a particular variety of sunflower fails to germinate is 0.24, independently of other seeds.
(a) A grower plants 400 of these seeds. The number of seeds that fail to germinate is denoted by the random variable

Y

. Determine the smallest value of

a

such that

\mathrm{P}(Y > a) < 0.05

.

(3 marks)

In the expansion of

(0.24 + 0.76)^{40}

, the terms involving

0.24^r

and

0.24^{r+1}

are denoted by

T_r

and

T_{r+1}

respectively.
(b) Show that

\dfrac{T_{r+1}}{T_r} = \dfrac{6(40 - r)}{19(r + 1)}

.

(3 marks)

The number of seeds that fail to germinate in a packet of 40 of these seeds is modelled by the random variable

X

.
(c) (i) Determine the values of

r

for which

\mathrm{P}(X = r) \leqslant \mathrm{P}(X = r + 1)

.

(ii) Hence determine the most likely number of seeds in a packet of 40 that fail to germinate.

(6 marks)

●●●●●Level 512 marksStart

A city bike-hire scheme charges

£2

for a short ride,

£4

for a medium ride and

£7

for a long ride. The operator models the rides taken as short, medium and long in the ratio

2 : 1 : 1

respectively.

A random sample of

2

rides is taken from the very large number of rides taken. The random variable

R

represents the range, in pounds, of the charges for the

2

rides.
(a) Show that, according to the model,

P(R = 5) = \frac14

(2 marks)
(b) Find the sampling distribution of

R

(4 marks)
(c) Find

E(R)

(2 marks)

The operator suspects that samples of

2

rides with a range of

£5

occur more often than the model suggests. She takes

20

random samples of

2

rides and finds that

9

of these samples have a range of

£5

(d) Test, at the

5\%

level of significance, whether or not there is evidence to support the operator's suspicion. State your hypotheses clearly.

(5 marks)

●●●●●Level 513 marksStart

The time,

T

minutes, that a pharmacy takes to prepare a prescription is modelled by a continuous uniform distribution over the interval

[a, b]

Given that

$\mathrm{E}(T) = 16$
$\mathrm{P}(T < 11) = 0.25$

(a) find the value of

a

and the value of

b

(3 marks)
(b) Show that the probability that a prescription takes more than

22

minutes to prepare is

0.2

(1 mark)
(c) Find the exact value of

\mathrm{E}(T^2)

(2 marks)

The pharmacy introduces a new system. The manager believes that, with the new system, the proportion of prescriptions that take more than

22

minutes to prepare is less than

0.2

A random sample of

40

prescriptions prepared with the new system is to be taken.
(d) Using a

5\%

level of significance, find the critical region for a test of the manager's belief. You should state your hypotheses clearly.

(4 marks)
(e) Find the actual significance level of this test.

(1 mark)

In the sample,

3

of the prescriptions took more than

22

minutes to prepare.
(f) Comment on the manager's belief in the light of this sample. Give a reason for your answer.

(2 marks)

●●●●●Level 513 marksStart

The normal distribution

38 questions

LessonNot started

In a particular year, the daily volume of milk produced by a cow on a dairy farm can be modelled by a normal distribution with mean

28

litres and standard deviation

3.5

litres.
(a) This normal distribution is represented in the diagram below.Label the values

28

and

31.5

on the horizontal axis.

(2 marks)
(b) State the probability that the daily volume of milk produced by a cow on the farm is less than

28

litres.

(1 mark)
(c) Find the probability that the daily volume of milk produced by a cow on the farm is more than

33

litres.

(1 mark)
(d) Find the probability that the daily volume of milk produced by a cow on the farm is more than

24

litres but less than

30

litres.

(1 mark)
(e) Determine the daily volume of milk exceeded by

95\%

of all cows on the farm.

(2 marks)
(f) In the following year, the farm changed the feed given to its cows.

The daily volumes of milk produced by

20

cows on the farm were selected at random. The total of these

20

daily volumes was

583

litres.

Carry out a hypothesis test at the

10\%

significance level to investigate whether the mean daily volume of milk produced by a cow on the farm in the following year has increased compared to the previous year.

You may assume that the daily volume of milk produced by a cow on the farm is still normally distributed with standard deviation

3.5

litres.

(7 marks)

●●●●●Level 514 marksStart

In a particular year, the wingspan of an adult barn owl ringed at a wildlife reserve, Reserve A, has a mean of

88.0

centimetres and a standard deviation of

2.4

centimetres.

The wingspans of

95\%

of these owls are between

83.3

centimetres and

92.7

centimetres.
(a) Comment on whether a normal distribution may be suitable to model the wingspan of an adult barn owl ringed at Reserve A in this particular year.

(3 marks)
(b) You may assume that the wingspan of an adult barn owl ringed at Reserve A may be modelled by a normal distribution with mean

88.0

centimetres and standard deviation

2.4

centimetres.

(i) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is

90

centimetres.

(ii) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is between

85

centimetres and

91

centimetres.

(iii) Two adult barn owls ringed at Reserve A are chosen at random.

Calculate the probability that both of their wingspans are between

85

centimetres and

91

centimetres.

(3 marks)
(c) The summarised data for the wingspans,

w

centimetres, of a random sample of

25

adult barn owls ringed at a second reserve, Reserve B, is given below.

\sum w = 2127.5 \qquad \sum (w - \bar{w})^2 = 230.64

Use this data to calculate estimates of the mean and standard deviation of the wingspans of adult barn owls ringed at Reserve B.

(3 marks)
(d) Using your answers from part (c), compare the wingspans of adult barn owls ringed at Reserve A and adult barn owls ringed at Reserve B.

(2 marks)

●●●●●Level 511 marksStart

The mass of flour used each day by a bakery may be modelled by a normal distribution with mean

86

kg and standard deviation

7

kg.
(a) State the probability that the mass of flour used on any given day is exactly

86

kg.

(1 mark)
(b) To attract more customers, the bakery decides to change the range of bread that it bakes.

Following the decision, it was found that over a

15

-day period a total mass of

1353

kg of flour was used.

It can be assumed that the distribution of the mass of flour used each day is still normal with standard deviation

7

kg, and that the

15

-day period can be regarded as a random sample.

Investigate, at the

5\%

level of significance, whether the mean daily mass of flour used has changed.

(7 marks)
(c) The bakery's manager claims that if a different sample of

15

days had been used, the hypothesis test in part (b) would have given the same result.

Comment on the validity of this claim.

(2 marks)

●●●●●Level 510 marksStart

A manufacturer uses a machine to make ceramic tiles.

The width of a tile,

W

mm, is normally distributed with

a mean of $46$ mm
a standard deviation of $x$ mm

Given that the proportion of tiles less than

44.5

mm in width is

4\%

(a) show that

x = 0.86

to 2 decimal places.

(2 marks)
(b) Calculate the proportion of tiles that are between

45

mm and

47

mm in width.

(1 mark)

The cost of producing a single tile is

18

p

A tile

where $W < 45$ is sold as a second for $25$ p
where $45 \leqslant W \leqslant 47$ is sold for $40$ p
where $W > 47$ is re-fired for an extra cost of $9$ p and then sold for $40$ p

(c) Calculate the expected profit per

400

of the tiles. Give your answer to the nearest pound.

(5 marks)

The same manufacturer makes plastic tile spacers in large batches.

The spacers each have a probability of

0.02

of being faulty.

A random sample of

150

spacers is taken from each batch and the batch is accepted if fewer than

7

of them are faulty.

The manufacturer's aim is for

95\%

of batches to be accepted.
(d) Explain whether the manufacturer is likely to achieve its aim.

(4 marks)

●●●●●Level 512 marksStart

The masses, in grams, of a random sample of 240 potatoes of a certain variety were measured. The results are summarised in the histogram.One of the 240 potatoes is chosen at random, and its mass,

X

grams, is noted.
(a) Show that

\mathrm{P}(160 < X < 180) = 0.183

, correct to 3 significant figures.

(2 marks)

Priya suggests that the distribution of

X

can be well modelled by the distribution

\mathrm{N}(190, 900)

.
(b) (i) Give a brief justification for the use of the normal distribution in this context.

(ii) Give a brief justification for the choice of the parameter values 190 and 900.

(3 marks)
(c) Use Priya's model to find

\mathrm{P}(160 < X < 180)

.

(1 mark)

Omar suggests a different model. He uses the midpoints of the classes to calculate estimates,

m

and

s

, for the mean and standard deviation respectively, in grams, of the 240 masses. He then uses the distribution

\mathrm{N}(m, s^2)

as his model.
(d) Use Omar's model to find

\mathrm{P}(160 < X < 180)

.

(4 marks)

The table shows the probabilities of

X

lying in three further ranges, obtained from the histogram and from the two models. Two of the entries,

p

and

q

, are missing.

	$X < 160$	$180 < X < 220$	$X > 220$
Histogram	0.175	0.458	0.183
Priya's model	$p$	0.472	0.159
Omar's model	0.184	$q$	0.197

(e) (i) Find the values of

p

and

q

.

(ii) By considering the different ranges of values of

X

given in the table, together with your answers to parts (a),
(c) and (d), discuss how well the two models fit the original distribution.

(4 marks)

●●●●●Level 514 marksStart

A warehouse uses machines to wrap parcels.

The time,

T

seconds, taken by machine A to wrap a parcel can be modelled by

T \sim \mathrm{N}(24, 2^2)

(a) (i) Using standardisation, show that the probability that machine A takes more than 27.4 seconds to wrap a randomly selected parcel is 0.0446

(ii) Using standardisation and showing your working, find the probability that the time machine A takes to wrap a randomly selected parcel is 25 seconds, correct to the nearest 5 seconds.

(7 marks)

The time,

V

seconds, taken by machine B to wrap a parcel can be modelled by

V \sim \mathrm{N}(\mu, \sigma^2)

(b) Given that

\mathrm{P}(V > 30) = 0.0228

and

\mathrm{P}(V < 18) = 0.2

(i) find two equations in terms of

\mu

and

\sigma

(ii) Hence, showing your working, find the value of

\mu

and the value of

\sigma

(6 marks)

●●●●●Level 513 marksStart

A train company runs services from a city centre to an airport and from the city centre to a port.

The journey time,

X

minutes, from the city centre to the airport is normally distributed with mean 46 and standard deviation 4

One of these journeys to the airport is selected at random.
(a) (i) Using standardisation, find

\mathrm{P}(X > 51)

(ii) Hence, find

\mathrm{P}(41 < X < 51)

(4 marks)

A random sample of 5 journeys to the airport is taken.
(b) Find the probability that exactly 2 of these journeys each take more than 51 minutes.

(3 marks)

The journey time,

Y

minutes, from the city centre to the port is normally distributed with mean 102 and standard deviation

\sigma

The 90th percentile for the journey time to the port is 117.38 minutes to 2 decimal places.
(c) Show that

\sigma = 12

to the nearest minute.

(2 marks)

Given that

\mathrm{P}(X > k) = \mathrm{P}(Y < 2k) = p

, where

k

and

p

are constants,
(d) (i) find the value of

k

(ii) find the value of

p

(5 marks)

●●●●●Level 514 marksStart

The thickness,

T

mm, of a paperback book printed by a publisher is normally distributed with

T \sim \mathrm{N}(24, 3^2)

(a) Find the probability that a randomly selected paperback has a thickness of less than 20.4 mm.

(3 marks)
(b) Find

(i) the upper quartile (

Q_3

) of

T

(ii) the lower quartile (

Q_1

) of

T

(4 marks)

A value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

is defined as an outlier.
(c) Find these two outlier limits.

(3 marks)

A paperback is selected at random.
(d) Using standardisation, show that the probability that this paperback is an outlier is 0.007 to 3 decimal places.

(3 marks)

Given that this paperback is not an outlier,
(e) showing your working, find the probability that its thickness is greater than 27 mm.

(4 marks)

●●●●●Level 517 marksStart

The random variable

W

has the distribution

\mathrm{N}(\mu, \sigma^2)

.
(a) Find

\mathrm{P}(W > \mu + 0.5\sigma)

.

(1 mark)
(b) Given that

\mathrm{P}(W > 63) = 0.12

and

\mathrm{P}(W < 48) = 0.35

, determine the values of

\mu

and

\sigma

.

(6 marks)

The random variables

A

and

B

have the distributions

\mathrm{N}(15, 16)

and

\mathrm{N}(20, 25)

respectively.
(c) It is given that

\mathrm{P}(A < b) = \mathrm{P}(B > c)

, where

b < 15

and

c > 20

.

Determine

b

in terms of

c

.

(2 marks)

●●●●●Level 59 marksStart

Finding normal probabilities

51 questions

LessonNot started

In a particular year, the daily volume of milk produced by a cow on a dairy farm can be modelled by a normal distribution with mean

28

litres and standard deviation

3.5

litres.
(a) This normal distribution is represented in the diagram below.Label the values

28

and

31.5

on the horizontal axis.

(2 marks)
(b) State the probability that the daily volume of milk produced by a cow on the farm is less than

28

litres.

(1 mark)
(c) Find the probability that the daily volume of milk produced by a cow on the farm is more than

33

litres.

(1 mark)
(d) Find the probability that the daily volume of milk produced by a cow on the farm is more than

24

litres but less than

30

litres.

(1 mark)
(e) Determine the daily volume of milk exceeded by

95\%

of all cows on the farm.

(2 marks)
(f) In the following year, the farm changed the feed given to its cows.

The daily volumes of milk produced by

20

cows on the farm were selected at random. The total of these

20

daily volumes was

583

litres.

Carry out a hypothesis test at the

10\%

significance level to investigate whether the mean daily volume of milk produced by a cow on the farm in the following year has increased compared to the previous year.

You may assume that the daily volume of milk produced by a cow on the farm is still normally distributed with standard deviation

3.5

litres.

(7 marks)

●●●●●Level 514 marksStart

Heights are modelled by

X \sim N(170, 64)

cm.
(a) Find

P(X > 180)

.
(b) Find

P(160 < X < 175)

.
(c) Find the height exceeded by

10\%

of people. Give probabilities to 4 d.p. and the height to 3 s.f.

●●●●●Level 57 marksStart

In a particular year, the wingspan of an adult barn owl ringed at a wildlife reserve, Reserve A, has a mean of

88.0

centimetres and a standard deviation of

2.4

centimetres.

The wingspans of

95\%

of these owls are between

83.3

centimetres and

92.7

centimetres.
(a) Comment on whether a normal distribution may be suitable to model the wingspan of an adult barn owl ringed at Reserve A in this particular year.

(3 marks)
(b) You may assume that the wingspan of an adult barn owl ringed at Reserve A may be modelled by a normal distribution with mean

88.0

centimetres and standard deviation

2.4

centimetres.

(i) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is

90

centimetres.

(ii) Find the probability that the wingspan of a randomly selected adult barn owl ringed at Reserve A is between

85

centimetres and

91

centimetres.

(iii) Two adult barn owls ringed at Reserve A are chosen at random.

Calculate the probability that both of their wingspans are between

85

centimetres and

91

centimetres.

(3 marks)
(c) The summarised data for the wingspans,

w

centimetres, of a random sample of

25

adult barn owls ringed at a second reserve, Reserve B, is given below.

\sum w = 2127.5 \qquad \sum (w - \bar{w})^2 = 230.64

Use this data to calculate estimates of the mean and standard deviation of the wingspans of adult barn owls ringed at Reserve B.

(3 marks)
(d) Using your answers from part (c), compare the wingspans of adult barn owls ringed at Reserve A and adult barn owls ringed at Reserve B.

(2 marks)

●●●●●Level 511 marksStart

A manufacturer uses a machine to make ceramic tiles.

The width of a tile,

W

mm, is normally distributed with

a mean of $46$ mm
a standard deviation of $x$ mm

Given that the proportion of tiles less than

44.5

mm in width is

4\%

(a) show that

x = 0.86

to 2 decimal places.

(2 marks)
(b) Calculate the proportion of tiles that are between

45

mm and

47

mm in width.

(1 mark)

The cost of producing a single tile is

18

p

A tile

where $W < 45$ is sold as a second for $25$ p
where $45 \leqslant W \leqslant 47$ is sold for $40$ p
where $W > 47$ is re-fired for an extra cost of $9$ p and then sold for $40$ p

(c) Calculate the expected profit per

400

of the tiles. Give your answer to the nearest pound.

(5 marks)

The same manufacturer makes plastic tile spacers in large batches.

The spacers each have a probability of

0.02

of being faulty.

A random sample of

150

spacers is taken from each batch and the batch is accepted if fewer than

7

of them are faulty.

The manufacturer's aim is for

95\%

of batches to be accepted.
(d) Explain whether the manufacturer is likely to achieve its aim.

(4 marks)

●●●●●Level 512 marksStart

The masses, in grams, of a random sample of 240 potatoes of a certain variety were measured. The results are summarised in the histogram.One of the 240 potatoes is chosen at random, and its mass,

X

grams, is noted.
(a) Show that

\mathrm{P}(160 < X < 180) = 0.183

, correct to 3 significant figures.

(2 marks)

Priya suggests that the distribution of

X

can be well modelled by the distribution

\mathrm{N}(190, 900)

.
(b) (i) Give a brief justification for the use of the normal distribution in this context.

(ii) Give a brief justification for the choice of the parameter values 190 and 900.

(3 marks)
(c) Use Priya's model to find

\mathrm{P}(160 < X < 180)

.

(1 mark)

Omar suggests a different model. He uses the midpoints of the classes to calculate estimates,

m

and

s

, for the mean and standard deviation respectively, in grams, of the 240 masses. He then uses the distribution

\mathrm{N}(m, s^2)

as his model.
(d) Use Omar's model to find

\mathrm{P}(160 < X < 180)

.

(4 marks)

The table shows the probabilities of

X

lying in three further ranges, obtained from the histogram and from the two models. Two of the entries,

p

and

q

, are missing.

	$X < 160$	$180 < X < 220$	$X > 220$
Histogram	0.175	0.458	0.183
Priya's model	$p$	0.472	0.159
Omar's model	0.184	$q$	0.197

(e) (i) Find the values of

p

and

q

.

(ii) By considering the different ranges of values of

X

given in the table, together with your answers to parts (a),
(c) and (d), discuss how well the two models fit the original distribution.

(4 marks)

●●●●●Level 514 marksStart

A warehouse uses machines to wrap parcels.

The time,

T

seconds, taken by machine A to wrap a parcel can be modelled by

T \sim \mathrm{N}(24, 2^2)

(a) (i) Using standardisation, show that the probability that machine A takes more than 27.4 seconds to wrap a randomly selected parcel is 0.0446

(ii) Using standardisation and showing your working, find the probability that the time machine A takes to wrap a randomly selected parcel is 25 seconds, correct to the nearest 5 seconds.

(7 marks)

The time,

V

seconds, taken by machine B to wrap a parcel can be modelled by

V \sim \mathrm{N}(\mu, \sigma^2)

(b) Given that

\mathrm{P}(V > 30) = 0.0228

and

\mathrm{P}(V < 18) = 0.2

(i) find two equations in terms of

\mu

and

\sigma

(ii) Hence, showing your working, find the value of

\mu

and the value of

\sigma

(6 marks)

●●●●●Level 513 marksStart

A train company runs services from a city centre to an airport and from the city centre to a port.

The journey time,

X

minutes, from the city centre to the airport is normally distributed with mean 46 and standard deviation 4

One of these journeys to the airport is selected at random.
(a) (i) Using standardisation, find

\mathrm{P}(X > 51)

(ii) Hence, find

\mathrm{P}(41 < X < 51)

(4 marks)

A random sample of 5 journeys to the airport is taken.
(b) Find the probability that exactly 2 of these journeys each take more than 51 minutes.

(3 marks)

The journey time,

Y

minutes, from the city centre to the port is normally distributed with mean 102 and standard deviation

\sigma

The 90th percentile for the journey time to the port is 117.38 minutes to 2 decimal places.
(c) Show that

\sigma = 12

to the nearest minute.

(2 marks)

Given that

\mathrm{P}(X > k) = \mathrm{P}(Y < 2k) = p

, where

k

and

p

are constants,
(d) (i) find the value of

k

(ii) find the value of

p

(5 marks)

●●●●●Level 514 marksStart

The thickness,

T

mm, of a paperback book printed by a publisher is normally distributed with

T \sim \mathrm{N}(24, 3^2)

(a) Find the probability that a randomly selected paperback has a thickness of less than 20.4 mm.

(3 marks)
(b) Find

(i) the upper quartile (

Q_3

) of

T

(ii) the lower quartile (

Q_1

) of

T

(4 marks)

A value that is greater than

Q_3 + 1.5 \times (Q_3 - Q_1)

or smaller than

Q_1 - 1.5 \times (Q_3 - Q_1)

is defined as an outlier.
(c) Find these two outlier limits.

(3 marks)

A paperback is selected at random.
(d) Using standardisation, show that the probability that this paperback is an outlier is 0.007 to 3 decimal places.

(3 marks)

Given that this paperback is not an outlier,
(e) showing your working, find the probability that its thickness is greater than 27 mm.

(4 marks)

●●●●●Level 517 marksStart

The random variable

W

has the distribution

\mathrm{N}(\mu, \sigma^2)

.
(a) Find

\mathrm{P}(W > \mu + 0.5\sigma)

.

(1 mark)
(b) Given that

\mathrm{P}(W > 63) = 0.12

and

\mathrm{P}(W < 48) = 0.35

, determine the values of

\mu

and

\sigma

.

(6 marks)

The random variables

A

and

B

have the distributions

\mathrm{N}(15, 16)

and

\mathrm{N}(20, 25)

respectively.
(c) It is given that

\mathrm{P}(A < b) = \mathrm{P}(B > c)

, where

b < 15

and

c > 20

.

Determine

b

in terms of

c

.

(2 marks)

●●●●●Level 59 marksStart

Finding μ and σ

33 questions

LessonNot started

The mass,

X

grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

. Records show that

2.28\%

of loaves have mass less than

775

g and that

91.04\%

of loaves have mass between

775

g and

831

g.
(a) Find the values of

\mu

and

\sigma

. (5 marks)

The bakery installs a new oven. The baker believes that loaves baked in the new oven have a greater mean mass. It may be assumed that the standard deviation is unchanged. A random sample of

25

loaves baked in the new oven has mean mass

812.5

g.
(b) Test the baker's belief at the

5\%

level of significance. State your hypotheses clearly. (5 marks)
(c) Explain how, if at all, your conclusion would change if the baker had believed only that the mean mass had changed, with no view as to the direction of the change. (2 marks)
(d) A second random sample of

n

loaves from the new oven also has mean mass

812.5

g. Find the smallest value of

n

for which this sample would provide significant evidence, at the

5\%

level, that the mean mass has changed. (2 marks)

●●●●●Level 514 marksStart

The weights of apples from an orchard,

W

grams, are modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

.

It is known that

10\%

of the apples weigh less than

120

g and that

5\%

of the apples weigh more than

190

g.
(a) Find the value of

\mu

and the value of

\sigma

, giving each answer to

3

significant figures. (6 marks)

An apple is classed as large if it weighs more than

160

g.
(b) Show that the probability that a randomly chosen apple is large is approximately

0.35

. (2 marks)

A wholesaler buys a random sample of

200

apples from the orchard.
(c) Using a suitable approximation, which you should justify, estimate the probability that at least

80

of these apples are large. Use

\mathrm{P}(\text{large}) = 0.35

. (5 marks)

●●●●●Level 513 marksStart

Component lifetimes are

N(\mu, \sigma^2)

hours.

10\%

last under

800

and

5\%

last over

1200

.
(a) Find

\mu

and

\sigma

to 3 s.f.
(b) Hence find

P(X > 1000)

, to 3 d.p.

●●●●●Level 56 marksStart

The lifetimes of a type of battery are modelled as

N(\mu, \sigma^{2})

hours.

20\%

of batteries last less than

1000

hours and

10\%

last more than

1500

hours.

Find

\mu

and

\sigma

, giving your answers to 3 significant figures. (6 marks)

●●●●●Level 56 marksStart

A warehouse uses machines to wrap parcels.

The time,

T

seconds, taken by machine A to wrap a parcel can be modelled by

T \sim \mathrm{N}(24, 2^2)

(a) (i) Using standardisation, show that the probability that machine A takes more than 27.4 seconds to wrap a randomly selected parcel is 0.0446

(ii) Using standardisation and showing your working, find the probability that the time machine A takes to wrap a randomly selected parcel is 25 seconds, correct to the nearest 5 seconds.

(7 marks)

The time,

V

seconds, taken by machine B to wrap a parcel can be modelled by

V \sim \mathrm{N}(\mu, \sigma^2)

(b) Given that

\mathrm{P}(V > 30) = 0.0228

and

\mathrm{P}(V < 18) = 0.2

(i) find two equations in terms of

\mu

and

\sigma

(ii) Hence, showing your working, find the value of

\mu

and the value of

\sigma

(6 marks)

●●●●●Level 513 marksStart

(a) The heights of a species of plant are normally distributed with mean

\mu

cm and standard deviation

\sigma

cm. It is found that

15\%

of the plants are shorter than

20

cm and

20\%

are taller than

35

cm.

Find the values of

\mu

and

\sigma

. (5 marks)
(b) A fair six-sided die is thrown

180

times. Using a suitable approximation, find the probability that a six is obtained at least

40

times. (3 marks)

●●●●●Level 58 marksStart

A train company runs services from a city centre to an airport and from the city centre to a port.

The journey time,

X

minutes, from the city centre to the airport is normally distributed with mean 46 and standard deviation 4

One of these journeys to the airport is selected at random.
(a) (i) Using standardisation, find

\mathrm{P}(X > 51)

(ii) Hence, find

\mathrm{P}(41 < X < 51)

(4 marks)

A random sample of 5 journeys to the airport is taken.
(b) Find the probability that exactly 2 of these journeys each take more than 51 minutes.

(3 marks)

The journey time,

Y

minutes, from the city centre to the port is normally distributed with mean 102 and standard deviation

\sigma

The 90th percentile for the journey time to the port is 117.38 minutes to 2 decimal places.
(c) Show that

\sigma = 12

to the nearest minute.

(2 marks)

Given that

\mathrm{P}(X > k) = \mathrm{P}(Y < 2k) = p

, where

k

and

p

are constants,
(d) (i) find the value of

k

(ii) find the value of

p

(5 marks)

●●●●●Level 514 marksStart

The random variable

W

has the distribution

\mathrm{N}(\mu, \sigma^2)

.
(a) Find

\mathrm{P}(W > \mu + 0.5\sigma)

.

(1 mark)
(b) Given that

\mathrm{P}(W > 63) = 0.12

and

\mathrm{P}(W < 48) = 0.35

, determine the values of

\mu

and

\sigma

.

(6 marks)

The random variables

A

and

B

have the distributions

\mathrm{N}(15, 16)

and

\mathrm{N}(20, 25)

respectively.
(c) It is given that

\mathrm{P}(A < b) = \mathrm{P}(B > c)

, where

b < 15

and

c > 20

.

Determine

b

in terms of

c

.

(2 marks)

●●●●●Level 59 marksStart

Approximating a binomial with a normal

24 questions

LessonNot started

The weights of apples from an orchard,

W

grams, are modelled by a normal distribution with mean

\mu

and standard deviation

\sigma

.

It is known that

10\%

of the apples weigh less than

120

g and that

5\%

of the apples weigh more than

190

g.
(a) Find the value of

\mu

and the value of

\sigma

, giving each answer to

3

significant figures. (6 marks)

An apple is classed as large if it weighs more than

160

g.
(b) Show that the probability that a randomly chosen apple is large is approximately

0.35

. (2 marks)

A wholesaler buys a random sample of

200

apples from the orchard.
(c) Using a suitable approximation, which you should justify, estimate the probability that at least

80

of these apples are large. Use

\mathrm{P}(\text{large}) = 0.35

. (5 marks)

●●●●●Level 513 marksStart

A council claims that

60\%

of the households in a town put out a recycling bin each week. A campaigner believes that the true proportion is lower than this.

The campaigner observes a random sample of

20

households in the town and records the number,

X

, that put out a recycling bin in a particular week.
(a) Using a

5\%

level of significance, find the critical region for a test of the campaigner's belief. State your hypotheses clearly. (3 marks)
(b) State the actual significance level of the test, and explain why it is not exactly

5\%

. (2 marks)
(c) In the sample,

8

households put out a recycling bin. State the conclusion of the test in context. (2 marks)

The campaigner then observes a second random sample of

150

households in the town, of which

77

put out a recycling bin.
(d) Assuming that the council's claim is correct, use a suitable approximation to estimate the probability that

77

or fewer households in a random sample of

150

put out a recycling bin. Justify the use of your approximation, and comment on what your answer suggests about the council's claim. (5 marks)

●●●●●Level 512 marksStart

The continuous random variable

X

is uniformly distributed over the interval

[-2k, 3k]

, where

k

is a positive constant.
(a) Write down

\mathrm{E}(X)

in terms of

k

and show that

\mathrm{Var}(X) = \dfrac{25k^2}{12}

(3 marks)
(b) Find

\mathrm{E}(X^2)

in terms of

k

(2 marks)

The random variable

Y

is defined by

Y = |X|

(c) Show that

\mathrm{P}(Y \leqslant y) = \dfrac{2y}{5k}

for

0 \leqslant y \leqslant 2k

(3 marks)
(d) Find

\mathrm{P}(Y \leqslant y)

for

2k < y \leqslant 3k

, giving your answer in terms of

y

and

k

(2 marks)
(e) Find the median of

Y

in terms of

k

(2 marks)

Given that

k = 2

, a random sample of

50

independent observations of

X

is taken.
(f) Using a suitable approximation, find the probability that more than

25

of these observations satisfy

|X| < 3

(4 marks)

●●●●●Level 516 marksStart

A craft kit contains wooden sticks, each of length

30

cm. A child snaps a stick into two pieces at a random point. The child then tries to make a triangle, using the two pieces as two of the sides and a rod of length

9

cm as the third side.
(a) Show that the probability that the child can make a triangle is

0.3

(4 marks)
(b) Find the expected length of the longer piece.

(2 marks)

The child snaps

10

sticks, one at a time, and tries to make a triangle with the two pieces of each stick and the rod.
(c) Find the probability that the child can make a triangle on at least

4

of the

10

attempts.

(3 marks)

In a school, the kit is used for

200

such attempts.
(d) Using a suitable approximation, find the probability that more than

70

of the

200

attempts produce a triangle.

(5 marks)

●●●●●Level 514 marksStart

40\%

of items from a production line are type A. In a sample of

150

, let

X

be the number of type A.
(a) State a suitable normal approximation for

X

.
(b) Estimate

P(X > 65)

, to 3 d.p.

●●●●●Level 55 marksStart

10\%

of items from a production line are faulty. A batch of

900

items is inspected.

Use a normal approximation to estimate the probability that at most

75

items are faulty, giving your answer to 3 significant figures. (5 marks)

●●●●●Level 55 marksStart

Orla is practising archery.

Each time Orla shoots an arrow at a target, the probability that the arrow lands in the gold ring is

0.24

, independently of all other arrows.

In a round, Orla shoots

12

arrows at the target.

The random variable

X

represents the number of arrows that land in the gold ring in a round.
(a) Using a suitable distribution for

X

, find

(i)

\mathrm{P}(X = 3)

(ii)

\mathrm{P}(X \geqslant 5)

(3 marks)

Orla shoots one round each day for

25

days and records the number of arrows that land in the gold ring.
(b) Find the probability that there were at least

8

days when exactly

3

arrows landed in the gold ring.

(3 marks)
(c) Find an estimate for the total number of arrows that landed in the gold ring during these

25

days.

(1 mark)

Orla now shoots

150

arrows at the target.
(d) Use a normal approximation to estimate the probability that more than

42

of these arrows land in the gold ring.

(4 marks)

●●●●●Level 511 marksStart

(a) The heights of a species of plant are normally distributed with mean

\mu

cm and standard deviation

\sigma

cm. It is found that

15\%

of the plants are shorter than

20

cm and

20\%

are taller than

35

cm.

Find the values of

\mu

and

\sigma

. (5 marks)
(b) A fair six-sided die is thrown

180

times. Using a suitable approximation, find the probability that a six is obtained at least

40

times. (3 marks)

●●●●●Level 58 marksStart

A factory makes a very large number of floor tiles, and each tile is stamped with a serial number when it is made. Each tile is graded

A

,

B

or

C

and sold for

£3

,

£2

or

£1

respectively. Tiles of grades

A

,

B

and

C

are made in the ratio

1 : 6 : 3

respectively.

A quality inspector checks a random sample of the tiles made each day, rather than inspecting every tile.
(a) (i) Suggest a suitable sampling frame for the inspector to use.

(ii) Identify the sampling units.

(2 marks)

A random sample of

3

tiles is taken. The random variable

M

represents the median price, in pounds, of the

3

tiles.
(b) Show that

P(M = 1) = 0.216

(2 marks)
(c) Find the sampling distribution of

M

(4 marks)

On each of

150

days, a random sample of

3

tiles is taken. The random variable

Y

represents the number of these

150

samples that have a median price of

£1

(d) Using a suitable approximation, find the probability that fewer than

25

of these samples have a median price of

£1

(5 marks)

●●●●●Level 513 marksStart