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Discrete random variables and probability distributions

39 questions

LessonNot started
A courier collects parcels from two warehouses, AA and BB.
  • At warehouse AA, parcels of mass 11 kg and parcels of mass 33 kg are in the ratio 1:41 : 4 respectively
  • At warehouse BB, parcels of mass 44 kg and parcels of mass 66 kg are in the ratio 2:32 : 3 respectively
One parcel is selected at random from the very large number of parcels at warehouse AA, and two parcels are selected at random from the very large number of parcels at warehouse BB. The random variable TT represents the total mass, in kg, of the 33 parcels.
(a) Show that P(T=13)=0.456P(T = 13) = 0.456

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

(5 marks)

The random variable MM represents the median mass, in kg, of the 33 parcels.
(c) Find the sampling distribution of MM

(4 marks)
●●●●●Level 411 marksStart
Priya has two fair spinners.
  • Spinner AA has five sectors numbered 1, 2, 3, 4 and 5
  • Spinner BB has three sectors numbered 0, 1 and 2
The random variable XX represents the number that spinner AA lands on when it is spun once. The random variable YY represents the number that spinner BB lands on when it is spun once.
(a) Write down the name of the distribution of XX

(1 mark)
(b) Showing your working, find Var(X)\mathrm{Var}(X)

(3 marks)

Priya spins each spinner once.
(c) Find P(X+Y⩽3)\mathrm{P}(X + Y \leqslant 3)

(2 marks)

The random variable TT is the product of the number that spinner AA lands on and the number that spinner BB lands on.
(d) Find the probability distribution of TT

(4 marks)
●●●●●Level 410 marksStart
Hana is playing a game. She uses a bag of numbered tokens, a stopwatch and a puzzle.

The random variable XX represents the number on the token Hana takes from the bag.

The probability distribution of XX is given in the following table
xx681224
P(X=x)\mathrm{P}(X = x)aabbccdd
where aa, bb, cc and dd are probabilities.

To play the game
  • a token is taken at random from the bag to obtain a value of xx
  • Hana then has xx seconds to complete the puzzle
The event SS represents the event that Hana completes the puzzle within the time allowed.

To model this game Hana assumes that
  • P(S∣{X=x})=kx\mathrm{P}(S \mid \{X = x\}) = kx where kk is a constant
  • P(S∩{X=x})\mathrm{P}(S \cap \{X = x\}) should be the same whatever value of xx is obtained from the bag
Using Hana's model,
(a) show that a=43ba = \dfrac{4}{3}b

(2 marks)
(b) find the probability distribution of XX

(5 marks)

Ravi attempts the puzzle, a large number of times, with 6 seconds allowed.

He completes the puzzle within the time allowed 30% of the time.
(c) State, giving a reason, why Hana's model of this game is not suitable to describe Ravi playing the game for all values of XX

(1 mark)
●●●●●Level 48 marksStart
A pack contains a large number of cards. Each card is numbered 1, 2, 3, 4 or 5

A card is taken at random from the pack.

The random variable XX represents the number on the card, where

P(X=x)=k(6−x)x=1, 2, 3, 4, 5\mathrm{P}(X = x) = k(6 - x) \qquad x = 1,\ 2,\ 3,\ 4,\ 5

and kk is a constant.
(a) Find the value of kk

(2 marks)
(b) Find P(X⩾3)\mathrm{P}(X \geqslant 3)

(1 mark)
(c) Find P(X>2∣X<5)\mathrm{P}(X > 2 \mid X < 5)

(2 marks)

The card is replaced in the pack and a second card is taken at random.
(d) Find the probability that the total of the numbers on the two cards is 9

(2 marks)
●●●●●Level 47 marksStart
A game at a school fair uses a bag containing a large number of tokens. Each token is marked with one of the numbers 1, 3, 4, 6 or 8

The random variable XX represents the number on a token taken at random from the bag. The probability distribution of XX is given in the table below.

x13468P(X=x)0.150.250.30.20.1\begin{array}{c|ccccc} x & 1 & 3 & 4 & 6 & 8 \\ \hline \mathrm{P}(X = x) & 0.15 & 0.25 & 0.3 & 0.2 & 0.1 \end{array}
(a) Find P(3X+2<16)\mathrm{P}(3X + 2 < 16)

(1 mark)

Given that E(X)=4.1\mathrm{E}(X) = 4.1
(b) show that Var(X)=3.99\mathrm{Var}(X) = 3.99

(3 marks)

The random variable Y=aX−bY = aX - b, where aa and bb are positive constants.

Given that

E(Y)=14.5andVar(Y)=99.75\mathrm{E}(Y) = 14.5 \qquad \text{and} \qquad \mathrm{Var}(Y) = 99.75
(c) find the value of aa and the value of bb

(4 marks)

In a game, Priya and Tomasz each take one token at random from the bag, independently of each other. The number on Priya's token is X1X_1 and the number on Tomasz's token is X2X_2

Priya's score is given by the random variable S=X1S = X_1

Tomasz's score is given by the random variable T=10−X2T = 10 - X_2

The player with the greater score wins; if the two scores are equal, the game is drawn.
(d) Find the probability that Priya wins the game.

(4 marks)
●●●●●Level 412 marksStart
For a certain value of the constant pp, the random variable XX has the probability distribution given in the table.

x1236P(X=x)3pp0.16p2\begin{array}{c|cccc} x & 1 & 2 & 3 & 6 \\ \hline \mathrm{P}(X = x) & 3p & p & 0.16 & p^2 \end{array}

Two independent values, X1X_1 and X2X_2, of XX are found.

Determine P(X2=3X1∣X2>X1)\mathrm{P}(X_2 = 3X_1 \mid X_2 > X_1).

(8 marks)
●●●●●Level 48 marksStart
Each week a quiz awards a prize of £120, which is shared equally between the winners of that week. The number of winners in a randomly chosen week is modelled by the discrete random variable XX, which has the following probability distribution.

x1234P(X=x)0.050.30.450.2\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(X = x) & 0.05 & 0.3 & 0.45 & 0.2 \end{array}
(a) Show that E(1X)=0.4\mathrm{E}\left(\dfrac{1}{X}\right) = 0.4

(1 mark)
(b) Find Var(1X)\mathrm{Var}\left(\dfrac{1}{X}\right)

(3 marks)

The amount, in pounds, received by each winner in a randomly chosen week is Y=120XY = \dfrac{120}{X}
(c) Find

(i) E(Y)\mathrm{E}(Y)

(ii) Var(Y)\mathrm{Var}(Y)

(3 marks)
(d) Find P(Y⩾40∣X⩾2)\mathrm{P}(Y \geqslant 40 \mid X \geqslant 2)

(5 marks)
●●●●●Level 412 marksStart
A workshop makes decorative glass tiles. Small air bubbles occur at random in the glass at a mean rate of 3.53.5 per 1010 tiles.

The tiles are sold in boxes of 1010
(a) Suggest a suitable distribution for the number of air bubbles in a box of tiles.

(1 mark)
(b) State one assumption, in context, necessary for this distribution to be suitable.

(1 mark)
(c) Find the probability that a randomly selected box of tiles contains

(i) exactly 22 air bubbles,

(ii) at least 55 air bubbles.

(3 marks)

Tiles are also sold individually.
  • The cost to make each tile is £1.30
  • Each tile that contains no air bubbles is sold for £3.20
  • Each tile that contains at least one air bubble is sold for £0.90
(d) Find the expected profit made on each tile that is sold individually.

(4 marks)
●●●●●Level 49 marksStart
In each round of a game on a phone app, a star is hidden in one of 44 identical boxes. The player chooses one box.
  • If the box contains the star, the player gains 55 points.
  • If the box does not contain the star, the player loses 11 point.
Ines plays 2020 rounds of the game, choosing one box at random in each round.

Let XX be the number of rounds in which Ines finds the star.
(a) State the distribution of XX

(1 mark)

Let SS be the total number of points that Ines gains in the 2020 rounds.
(b) (i) Write down SS in terms of XX

(ii) Hence, show clearly that the expected value of SS is 1010

(4 marks)

A player who gains at least 2525 points in 2020 rounds receives a bonus.
(c) Find the probability that Ines receives a bonus.

(3 marks)

Records show that, for players who have completed the training level of the game, the probability of finding the star in the first round is 0.350.35

A random sample of 4040 players who have completed the training level is taken.

Given that the probability that more than nn but at most 1818 of these players find the star in the first round is 0.72480.7248 to 44 decimal places,
(d) find the value of nn

(4 marks)
●●●●●Level 412 marksStart
A bus company records the delay, in minutes, of each bus arriving at a busy stop. It models the delay of a randomly chosen bus as follows.
  • 45%45\% of buses have a delay of 00 minutes
  • 40%40\% of buses have a delay of 44 minutes
  • 15%15\% of buses have a delay of 1010 minutes
(a) Find the mean delay of the buses.

(2 marks)

A random sample of 33 buses is taken.
(b) List all the possible combinations of the delays of these 33 buses.

(2 marks)

The random variable MM represents the median delay, in minutes, of the 33 buses.
(c) Show that P(M=10)=0.06075P(M = 10) = 0.06075

(2 marks)
(d) Find the sampling distribution of MM

(4 marks)
●●●●●Level 410 marksStart
A vehicle recovery service records the number of call-outs it receives between midnight and 6 am each night. The results for 8080 randomly selected nights are shown in the table below.
Number of call-outs01234567
Number of nights716201711621
(a) Show that the mean number of call-outs per night is 2.52.5 and find the variance.

(3 marks)
(b) Explain why your answers to part (a) suggest that a Poisson distribution may be a suitable model for the number of call-outs per night.

(1 mark)

The number of call-outs per night is modelled by a Poisson distribution with mean 2.52.5
(c) Find the probability that, on a randomly selected night, the recovery service receives

(i) exactly 33 call-outs,

(ii) more than 44 call-outs.

(4 marks)

On two particular consecutive nights the recovery service received a total of 55 call-outs.
(d) Find the probability that at least 44 of these call-outs were received on the same night.

(5 marks)

The manager selects 1212 non-overlapping periods, each of two consecutive nights, at random.
(e) Find the probability that there are fewer than 33 call-outs in exactly 22 of these periods.

(3 marks)
●●●●●Level 416 marksStart
Bikes are returned to a docking station of a city cycle-hire scheme at random, at a mean rate of 88 per hour.
(a) Find the probability that, in a randomly selected 1515-minute period,

(i) exactly 33 bikes are returned,

(ii) at least 55 bikes are returned.

(3 marks)

Of the bikes returned to this docking station, 25%25\% are electric bikes.
(b) Show that the probability that at least one electric bike is returned in a randomly selected 1010-minute period is 0.28350.2835 to 44 decimal places.

(2 marks)

The station manager randomly selects 1212 non-overlapping 3030-minute periods.
(c) Find the probability that between 33 and 66 (inclusive) bikes are returned in exactly 88 of these 3030-minute periods.

(4 marks)

Each electric bike returned is plugged into a charging bank.
  • In each 1010-minute period in which no electric bike is returned, the charging bank costs £0.02 to run.
  • In each 1010-minute period in which at least one electric bike is returned, the charging bank costs £0.15 to run.
(d) Find the expected cost of running the charging bank for a period of 1212 hours.

(4 marks)
(e) Explain why a Poisson distribution with a mean rate of 88 per hour is unlikely to be a suitable model for the number of bikes returned to this docking station over a whole 2424-hour day.

(1 mark)
●●●●●Level 414 marksStart
A spinner has five sectors numbered 1, 2, 3, 4 and 5. The spinner is biased.

The discrete random variable XX represents the number on the sector on which the spinner lands when it is spun once. The cumulative distribution function of XX is shown in the table below.

x12345F(x)k40k+4403k404k−240k+3040\begin{array}{c|ccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline \mathrm{F}(x) & \dfrac{k}{40} & \dfrac{k + 4}{40} & \dfrac{3k}{40} & \dfrac{4k - 2}{40} & \dfrac{k + 30}{40} \end{array}

where kk is a positive constant.
(a) Show that k=10k = 10

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

(3 marks)
(c) Using your answer to part (b), write down the mode of XX

(1 mark)
(d) Calculate Var(3X−5)\mathrm{Var}(3X - 5)

(5 marks)
●●●●●Level 410 marksStart
A hardware shop has a very large number of packs of nails. Each pack contains either 5050 nails or 8080 nails. The proportion of packs that contain 5050 nails is pp

A random sample of 33 packs is taken. The random variable Xˉ\bar{X} represents the mean number of nails in a pack for these 33 packs. The sampling distribution of Xˉ\bar{X} is shown in the table below.
xˉ\bar{x}5050606070708080
P(Xˉ=xˉ)P(\bar{X} = \bar{x})aa225512\frac{225}{512}bb27512\frac{27}{512}
(a) Find the value of pp

(2 marks)
(b) Find the value of aa and the value of bb

(3 marks)

The random variable RR represents the range of the numbers of nails in the 33 packs.
(c) Find the sampling distribution of RR

(3 marks)
(d) Find E(Xˉ)E(\bar{X})

(2 marks)
●●●●●Level 410 marksStart

The binomial distribution as a model

32 questions

LessonNot started
2%2\% of items produced by a machine are faulty. A sample of 3030 items is taken. Suggest a model for the number of faulty items, and find the probability that none are faulty, to 4 d.p.
●●●●●Level 43 marksStart
A shop records the number of customers entering in a one-hour period. Give one reason why this number cannot be modelled by a binomial distribution.
●●●●●Level 41 markStart
A bag contains a large number of beads, 30%30\% of which are red. A handful of 1212 is taken. Explain why the number of red beads can be modelled approximately by a binomial distribution, despite the sampling being without replacement.
●●●●●Level 41 markStart
A machine produces components, of which 8%8\% are faulty. A random sample of 2525 components is taken, and XX is the number of faulty components in the sample.
(a) State two conditions needed for XX to be modelled by a binomial distribution. (2 marks)
(b) Find P(X⩽3)\mathrm{P}(X \leqslant 3), giving your answer to 44 decimal places. (2 marks)
(c) Find P(X⩾2)\mathrm{P}(X \geqslant 2), giving your answer to 44 decimal places. (2 marks)

The machine is serviced. A new random sample of 2525 components is taken, and none of them is faulty.
(d) Test, at the 5%5\% level of significance, whether the proportion of faulty components has decreased. (4 marks)
●●●●●Level 410 marksStart
A library records every book that is returned to it.

It is found that 8%8\% of all books returned to this library are returned late.

A sample of 5050 returned books is selected. The number of books in the sample that were returned late is denoted by the random variable XX
(a) State two assumptions necessary for XX to be modelled by a binomial distribution.

(2 marks)
(b) Assume that XX can be modelled by a binomial distribution.

(i) Find P(X=2)P(X = 2)

(ii) Find P(X<6)P(X < 6)

(iii) Find P(X≥7)P(X \ge 7)

(5 marks)
(c) In a random sample of 6060 books returned to a different library, the number of books returned late, YY, may be modelled by Y∼B(60,p)Y \sim \text{B}(60, p)

The standard deviation of YY is 33

Calculate the possible values of pp

(3 marks)
●●●●●Level 410 marksStart
The random variable X∼B(45,0.2)X \sim \text{B}(45, 0.2)
(a) Find the mean of XX

(1 mark)
(b) Show that the variance of XX is 7.27.2

(1 mark)
(c) Find P(X<6)P(X < 6)

(1 mark)
(d) Find P(X≥12)P(X \ge 12)

(2 marks)
(e) Find P(7≤X≤11)P(7 \le X \le 11)

(2 marks)
(f) The coaches used for a company's sightseeing tours each have 4545 seats.

The proportion of passengers on these tours who are aged over 7070 is known to be 20%20\%

Make two comments on whether it would be appropriate to use XX to model the number of passengers on a coach who are aged over 7070 on one of these tours.

(2 marks)
●●●●●Level 49 marksStart
It is known that 41% of households in a large city have at least one pet. A researcher selects a random sample of 1200 households in the city. The random variable XX is defined as the number of households in the sample that have at least one pet.

Given that P(X⩽n)<0.05\mathrm{P}(X \leqslant n) < 0.05, calculate the largest possible value of nn.

(5 marks)
●●●●●Level 45 marksStart
A manufacturer produces components, each of which is defective with probability 0.010.01, independently of all others. A batch contains 300300 components.
(a) Explain why the number of defective components in a batch may be modelled approximately by a Poisson distribution, and state its parameter. (2 marks)
(b) Use this approximation to find the probability that a batch contains exactly 22 defective components. (2 marks)
(c) Use this approximation to find the probability that a batch contains at most 11 defective component. (2 marks)
●●●●●Level 46 marksStart
On average, 64%64\% of the tomato plants grown from seed at a nursery produce ripe fruit within ten weeks of planting.

A gardener buys a tray of 2020 of these tomato plants.

The number of plants in the tray that produce ripe fruit within ten weeks can be modelled by the distribution B(20,0.64)\text{B}(20, 0.64)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 1313 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(c) Find the probability that at most 1010 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(d) Find the probability that more than 1515 of the plants in the tray produce ripe fruit within ten weeks.

(2 marks)
(e) Find the mean number of plants in the tray that produce ripe fruit within ten weeks.

(1 mark)
(f) Find the standard deviation of the number of plants in the tray that produce ripe fruit within ten weeks.

(2 marks)
●●●●●Level 48 marksStart
On his way to school each morning, Yusuf passes through 99 sets of traffic lights.

The probability that any particular set of lights is green when Yusuf reaches it is 0.450.45, independently of all the other sets of lights.

The random variable XX represents the number of sets of lights that are green on Yusuf's journey to school on a randomly chosen morning.
(a) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

Yusuf makes this journey on each of 2020 school mornings.
(b) Find the probability that there were at least 44 mornings when fewer than 33 sets of lights were green.

(3 marks)
(c) Find an estimate for the total number of sets of lights that were green on Yusuf's journey to school during these 2020 mornings.

(1 mark)
●●●●●Level 47 marksStart
A workshop makes wooden beads and 15\tfrac{1}{5} of the beads are also painted gold.

The beads are packed into bags before being sent to craft shops.

There are 1414 beads in each bag.

The random variable TT represents the number of beads that are painted gold in each bag.
(a) State a condition needed for TT to be modelled by B ⁣(14,15)\mathrm{B}\!\left(14, \tfrac{1}{5}\right)

(1 mark)

A bag is selected at random.
(b) Using T∼B ⁣(14,15)T \sim \mathrm{B}\!\left(14, \tfrac{1}{5}\right) find

(i) the probability that the bag has exactly 33 beads that are painted gold,

(ii) the probability that the bag has fewer than 22 beads that are painted gold.

(2 marks)

Iris owns a craft shop and buys 2525 bags of these beads.
(c) Find the probability that exactly 44 of these 2525 bags have fewer than 22 beads that are painted gold.

(2 marks)

Iris claims that the proportion of these beads that are painted gold is less than 15\tfrac{1}{5}

A random sample of 6060 of these beads is taken and 66 of them are painted gold.
(d) Use a suitable test to assess Iris's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 49 marksStart
A running club has 169 members who run mainly on roads and 91 members who run mainly on trails. A random sample of 18 members is selected to attend a meeting about the club's new training programme. The number of members in the sample who run mainly on trails is denoted by XX.
(a) State a suitable binomial model for XX.

(1 mark)

Use your model to answer the following.
(b) (i) Write down an expression for P(X=x)\mathrm{P}(X = x).

(ii) State, in set notation, the values of xx for which your expression is valid.

(iii) Explain the connection between the expression in part (b)(i) and the binomial expansion of (0.65+0.35)n(0.65 + 0.35)^n, for a specific value of nn which should be stated.

(4 marks)
(c) Find P(4⩽X⩽8)\mathrm{P}(4 \leqslant X \leqslant 8).

(2 marks)
(d) State one disadvantage of using a random sample in this context.

(1 mark)
●●●●●Level 48 marksStart
In each round of a game on a phone app, a star is hidden in one of 44 identical boxes. The player chooses one box.
  • If the box contains the star, the player gains 55 points.
  • If the box does not contain the star, the player loses 11 point.
Ines plays 2020 rounds of the game, choosing one box at random in each round.

Let XX be the number of rounds in which Ines finds the star.
(a) State the distribution of XX

(1 mark)

Let SS be the total number of points that Ines gains in the 2020 rounds.
(b) (i) Write down SS in terms of XX

(ii) Hence, show clearly that the expected value of SS is 1010

(4 marks)

A player who gains at least 2525 points in 2020 rounds receives a bonus.
(c) Find the probability that Ines receives a bonus.

(3 marks)

Records show that, for players who have completed the training level of the game, the probability of finding the star in the first round is 0.350.35

A random sample of 4040 players who have completed the training level is taken.

Given that the probability that more than nn but at most 1818 of these players find the star in the first round is 0.72480.7248 to 44 decimal places,
(d) find the value of nn

(4 marks)
●●●●●Level 412 marksStart
A garden centre sells packets of poppy seeds. Each seed grows into a plant with red flowers with probability pp, independently of the other seeds.

Each small packet contains nn seeds.

The mean number of seeds in a small packet that grow into plants with red flowers is 4.54.5 and the variance is 3.153.15
(a) Find the value of nn and the value of pp

(3 marks)

Each large packet contains 2525 seeds. The random variable RR represents the number of seeds in a large packet that grow into plants with red flowers.
(b) Find the probability that RR exceeds the expected number of seeds in a large packet that grow into plants with red flowers.

(3 marks)
(c) Find the largest value of rr such that P(R<r)<0.05\mathrm{P}(R < r) < 0.05

(2 marks)

Jamal buys 66 large packets.
(d) Find the probability that, in at least 55 of these 66 packets, the number of seeds that grow into plants with red flowers exceeds the expected number.

(3 marks)
●●●●●Level 411 marksStart

Calculating binomial probabilities P(X = r)

40 questions

LessonNot started
A moth trap is set in a garden every night in July. Moths of a particular species are caught in the trap at random, at a mean rate of 1.41.4 per night.
(a) Find the probability that, on a randomly chosen night in July, at least 33 moths of this species are caught.

(2 marks)
(b) Find the probability that, in a randomly chosen week of 77 nights in July, at least 33 moths of this species are caught on exactly 22 of the nights.

(3 marks)

A moth expert believes that more moths of this species are caught when the trap uses a new type of bulb. The new bulb is used for 55 nights in July, and the total number of moths of this species caught in the 55 nights is recorded.
(c) Write down suitable null and alternative hypotheses for a test of the expert's belief.

(1 mark)
(d) Using a 5%5\% level of significance, find the critical region for this test and state its associated probability.

(3 marks)

In the 55 nights, 1313 moths of this species are caught.
(e) State the conclusion of the test, giving a reason for your answer.

(2 marks)
●●●●●Level 411 marksStart
The times taken by a large group of athletes to run 400400 m are modelled by a normal distribution with mean 52.452.4 seconds and standard deviation 2.82.8 seconds.
(a) Find the probability that a randomly chosen athlete takes less than 5050 seconds, giving your answer to 33 significant figures. (2 marks)
(b) Find the probability that a randomly chosen athlete takes between 5050 and 5555 seconds, giving your answer to 33 significant figures. (3 marks)
(c) The fastest 15%15\% of athletes qualify for a final. Find the qualifying time, giving your answer to 11 decimal place. (3 marks)
(d) Three athletes are chosen at random. Find the probability that exactly one of them takes less than 5050 seconds, giving your answer to 33 significant figures. (3 marks)
●●●●●Level 411 marksStart
A library records every book that is returned to it.

It is found that 8%8\% of all books returned to this library are returned late.

A sample of 5050 returned books is selected. The number of books in the sample that were returned late is denoted by the random variable XX
(a) State two assumptions necessary for XX to be modelled by a binomial distribution.

(2 marks)
(b) Assume that XX can be modelled by a binomial distribution.

(i) Find P(X=2)P(X = 2)

(ii) Find P(X<6)P(X < 6)

(iii) Find P(X≥7)P(X \ge 7)

(5 marks)
(c) In a random sample of 6060 books returned to a different library, the number of books returned late, YY, may be modelled by Y∼B(60,p)Y \sim \text{B}(60, p)

The standard deviation of YY is 33

Calculate the possible values of pp

(3 marks)
●●●●●Level 410 marksStart
The random variable X∼B(45,0.2)X \sim \text{B}(45, 0.2)
(a) Find the mean of XX

(1 mark)
(b) Show that the variance of XX is 7.27.2

(1 mark)
(c) Find P(X<6)P(X < 6)

(1 mark)
(d) Find P(X≥12)P(X \ge 12)

(2 marks)
(e) Find P(7≤X≤11)P(7 \le X \le 11)

(2 marks)
(f) The coaches used for a company's sightseeing tours each have 4545 seats.

The proportion of passengers on these tours who are aged over 7070 is known to be 20%20\%

Make two comments on whether it would be appropriate to use XX to model the number of passengers on a coach who are aged over 7070 on one of these tours.

(2 marks)
●●●●●Level 49 marksStart
A seed company claims that 85%85\% of its parsnip seeds germinate. A gardener plants the seeds in trays of 2020, and the number of seeds in a tray that germinate is modelled by the distribution B(20, 0.85)\mathrm{B}(20,\ 0.85).
(a) Find the probability that, in a randomly chosen tray, exactly 1818 seeds germinate. (2 marks)
(b) Find the probability that, in a randomly chosen tray, at least 1717 seeds germinate. (2 marks)

The gardener suspects that the company's claim overstates the germination rate. In the next tray planted, only 1313 of the 2020 seeds germinate.
(c) Test, at the 5%5\% significance level, whether there is evidence to support the gardener's suspicion. State your hypotheses clearly. (5 marks)
●●●●●Level 49 marksStart
An online bookshop finds that 2.5%2.5\% of the orders it sends out are returned.
(a) State one assumption, in context, that is needed for a binomial distribution to model the number of orders returned out of a batch of orders.

(1 mark)

On a particular day the bookshop sends out 160160 orders.
(b) Using a suitable approximation, find the probability that more than 66 of these orders are returned.

(3 marks)

The bookshop sends out 160160 orders on each of 55 days.
(c) Find the probability that more than 66 orders are returned on exactly 11 of these days.

(3 marks)

The bookshop changes the way it packs its orders. After the change, a random sample of 400400 orders is taken, and 55 of them are returned.
(d) Using a suitable approximation, test, at the 5%5\% level of significance, whether or not there is evidence that the proportion of orders returned has decreased. State your hypotheses clearly.

(5 marks)
●●●●●Level 412 marksStart
A speed camera is positioned beside a busy road. Vehicles pass the camera at random, at a mean rate of 88 per minute. Of these vehicles, 15%15\% are travelling above the speed limit, independently of each other.
(a) State two assumptions, in context, that are needed for the number of vehicles passing the camera in one minute to be modelled by a Poisson distribution.

(2 marks)
(b) Find the probability that at least 33 vehicles travelling above the speed limit pass the camera in a 22-minute period.

(3 marks)
(c) Using a suitable approximation, find the probability that more than 8080 vehicles travelling above the speed limit pass the camera in one hour.

(4 marks)

Given that exactly 44 vehicles pass the camera in a particular 3030-second period,
(d) find the probability that at least one of these 44 vehicles is travelling above the speed limit.

(2 marks)
●●●●●Level 411 marksStart
During the winter, a mountain rescue team is called out at random, at a mean rate of 0.60.6 per day.
(a) Find the probability that the team is called out at least twice in a 33-day period.

(2 marks)
(b) Show, by using a suitable approximation, that the probability of more than 2424 call-outs in a 3030-day period is 0.060.06, correct to 22 decimal places.

Show your working clearly.

(Solutions relying on calculator technology are not acceptable.)

(5 marks)

A winter season is made up of 55 separate 3030-day periods.
(c) Find the probability that the team is called out more than 2424 times in at least 22 of these 55 periods.

(3 marks)

The team also records the number of people it rescues each day.
(d) Explain why a Poisson distribution is unlikely to be a suitable model for the number of people rescued in a day.

(1 mark)
●●●●●Level 411 marksStart
On average, 64%64\% of the tomato plants grown from seed at a nursery produce ripe fruit within ten weeks of planting.

A gardener buys a tray of 2020 of these tomato plants.

The number of plants in the tray that produce ripe fruit within ten weeks can be modelled by the distribution B(20,0.64)\text{B}(20, 0.64)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 1313 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(c) Find the probability that at most 1010 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(d) Find the probability that more than 1515 of the plants in the tray produce ripe fruit within ten weeks.

(2 marks)
(e) Find the mean number of plants in the tray that produce ripe fruit within ten weeks.

(1 mark)
(f) Find the standard deviation of the number of plants in the tray that produce ripe fruit within ten weeks.

(2 marks)
●●●●●Level 48 marksStart
At a fairground stall, each player pays to have one go at a game.

Each time a player has a go, the probability that the player wins a prize is 0.180.18, independently of all other goes.

Maya has 1616 goes at this game.

The random variable XX represents the number of prizes that Maya wins.
(a) Using a suitable distribution for XX, find

(i) P(X=4)\mathrm{P}(X = 4)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

During one weekend there are 300300 goes at this stall.
(b) Use a normal approximation to estimate the probability that more than 6262 prizes are won during this weekend.

(3 marks)
●●●●●Level 46 marksStart
A theatre has found that, over a long period, 30%30\% of the people who attend a performance buy a programme.

A random sample of 2525 people who attend a performance is taken.

The random variable XX represents the number of these people who buy a programme.
(a) Using a suitable distribution for XX, find

(i) P(X=9)\mathrm{P}(X = 9)

(ii) P(X⩽5)\mathrm{P}(X \leqslant 5)

(3 marks)

The price of a programme is then increased.

The manager of the theatre claims that the proportion of people attending a performance who buy a programme has fallen.

After the increase, a random sample of 5050 people who attend a performance is taken and 99 of them buy a programme.
(b) Use a suitable test to assess the manager's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 47 marksStart
On his way to school each morning, Yusuf passes through 99 sets of traffic lights.

The probability that any particular set of lights is green when Yusuf reaches it is 0.450.45, independently of all the other sets of lights.

The random variable XX represents the number of sets of lights that are green on Yusuf's journey to school on a randomly chosen morning.
(a) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

Yusuf makes this journey on each of 2020 school mornings.
(b) Find the probability that there were at least 44 mornings when fewer than 33 sets of lights were green.

(3 marks)
(c) Find an estimate for the total number of sets of lights that were green on Yusuf's journey to school during these 2020 mornings.

(1 mark)
●●●●●Level 47 marksStart
A workshop makes wooden beads and 15\tfrac{1}{5} of the beads are also painted gold.

The beads are packed into bags before being sent to craft shops.

There are 1414 beads in each bag.

The random variable TT represents the number of beads that are painted gold in each bag.
(a) State a condition needed for TT to be modelled by B ⁣(14,15)\mathrm{B}\!\left(14, \tfrac{1}{5}\right)

(1 mark)

A bag is selected at random.
(b) Using T∼B ⁣(14,15)T \sim \mathrm{B}\!\left(14, \tfrac{1}{5}\right) find

(i) the probability that the bag has exactly 33 beads that are painted gold,

(ii) the probability that the bag has fewer than 22 beads that are painted gold.

(2 marks)

Iris owns a craft shop and buys 2525 bags of these beads.
(c) Find the probability that exactly 44 of these 2525 bags have fewer than 22 beads that are painted gold.

(2 marks)

Iris claims that the proportion of these beads that are painted gold is less than 15\tfrac{1}{5}

A random sample of 6060 of these beads is taken and 66 of them are painted gold.
(d) Use a suitable test to assess Iris's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 49 marksStart
A running club has 169 members who run mainly on roads and 91 members who run mainly on trails. A random sample of 18 members is selected to attend a meeting about the club's new training programme. The number of members in the sample who run mainly on trails is denoted by XX.
(a) State a suitable binomial model for XX.

(1 mark)

Use your model to answer the following.
(b) (i) Write down an expression for P(X=x)\mathrm{P}(X = x).

(ii) State, in set notation, the values of xx for which your expression is valid.

(iii) Explain the connection between the expression in part (b)(i) and the binomial expansion of (0.65+0.35)n(0.65 + 0.35)^n, for a specific value of nn which should be stated.

(4 marks)
(c) Find P(4⩽X⩽8)\mathrm{P}(4 \leqslant X \leqslant 8).

(2 marks)
(d) State one disadvantage of using a random sample in this context.

(1 mark)
●●●●●Level 48 marksStart
A cereal company puts a sticker in every box of cereal it makes. Each sticker is either gold or silver, and gold stickers and silver stickers are in the ratio 1:31 : 3 respectively.

A child collects stars for her stickers:
  • 33 stars for each gold sticker
  • 11 star for each silver sticker
A random sample of 44 boxes is taken from the very large number of boxes made. The random variable XX represents the total number of stars for the 44 stickers.
(a) Show that P(X=6)=2764P(X = 6) = \frac{27}{64}

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

(5 marks)

A random sample of nn sets of 44 boxes is taken. The random variable YY represents the number of these nn sets that give a total of exactly 1010 stars.
(c) Calculate the minimum value of nn such that P(Y⩾1)>0.9P(Y \geqslant 1) > 0.9

(3 marks)
●●●●●Level 410 marksStart
Nadia and Owen are playing a game.
  • The game consists of four rounds.
  • In each round, P(Nadia wins the round)=0.75\mathrm{P}(\text{Nadia wins the round}) = 0.75 and P(Owen wins the round)=0.25\mathrm{P}(\text{Owen wins the round}) = 0.25.
  • The outcome of each round is independent of the outcome of every other round.
  • If one player wins at least three of the four rounds, then they win the game, otherwise neither player wins.
(a) Determine the probability of the following events.

(i) Nadia wins the game.

(ii) Neither player wins the game.

(5 marks)

Nadia and Owen now decide to play a match which consists of a series of games.
  • In each game, if a player wins the game then they win the match.
  • If neither player wins the game then the players play another game.
(b) Determine the probability that Owen wins the match.

(3 marks)
●●●●●Level 48 marksStart
A garden centre sells a very large number of bamboo canes. Each cane has a length of 1.21.2 m, 1.51.5 m or 2.42.4 m, and canes of these lengths are in the ratio 2:a:12 : a : 1 respectively, where aa is a positive integer.

A random sample of 22 canes is taken. The random variable RR represents the range, in metres, of the lengths of the 22 canes.

Given that P(R=0)=1532P(R = 0) = \frac{15}{32}
(a) by forming and solving an equation in aa, show that a=5a = 5

(4 marks)
(b) find the sampling distribution of RR

(4 marks)

Five random samples of 22 canes are taken.
(c) Find the probability that exactly 22 of these 55 samples have a range of 00

(2 marks)
●●●●●Level 410 marksStart
A florist has a bucket of 3030 roses. 1515 of the roses cost £2£2 each, 1010 of the roses cost £3£3 each and 55 of the roses cost £5£5 each.

A random sample of 22 roses is taken from the bucket. The random variable MM represents the mean price, in pounds, of the 22 roses.
(a) Show that P(M=5)=287P(M = 5) = \frac{2}{87}

(1 mark)
(b) Find the sampling distribution of MM

(6 marks)

On each of nn days, a random sample of 22 roses is taken from a full bucket of 3030 roses made up in the same way. The random variable YY represents the number of these nn days on which both roses cost £5£5
(c) Calculate the minimum value of nn such that P(Y=0)<0.25P(Y = 0) < 0.25

(3 marks)
●●●●●Level 410 marksStart
Orders are received by a pizza takeaway at random, at a mean rate of 66 per hour during the evening.
(a) Find the probability that, in a randomly selected 1010-minute period during the evening, the takeaway receives

(i) no more than 11 order,

(ii) at least 44 orders.

(3 marks)

The manager randomly selects 1515 non-overlapping 3030-minute periods during the evening.
(b) Find the probability that there are between 22 and 55 (inclusive) orders received in exactly 1212 of these 3030-minute periods.

(4 marks)

Given that exactly 55 orders were received in a particular 3030-minute period during the evening,
(c) find the probability that more orders were received in the first 1010 minutes of this period than in the last 2020 minutes.

(4 marks)
(d) Explain why the Poisson distribution is unlikely to be valid for the number of pizzas ordered in a 1010-minute period.

(1 mark)
●●●●●Level 412 marksStart
A garden centre sells packets of poppy seeds. Each seed grows into a plant with red flowers with probability pp, independently of the other seeds.

Each small packet contains nn seeds.

The mean number of seeds in a small packet that grow into plants with red flowers is 4.54.5 and the variance is 3.153.15
(a) Find the value of nn and the value of pp

(3 marks)

Each large packet contains 2525 seeds. The random variable RR represents the number of seeds in a large packet that grow into plants with red flowers.
(b) Find the probability that RR exceeds the expected number of seeds in a large packet that grow into plants with red flowers.

(3 marks)
(c) Find the largest value of rr such that P(R<r)<0.05\mathrm{P}(R < r) < 0.05

(2 marks)

Jamal buys 66 large packets.
(d) Find the probability that, in at least 55 of these 66 packets, the number of seeds that grow into plants with red flowers exceeds the expected number.

(3 marks)
●●●●●Level 411 marksStart
A vehicle recovery service records the number of call-outs it receives between midnight and 6 am each night. The results for 8080 randomly selected nights are shown in the table below.
Number of call-outs01234567
Number of nights716201711621
(a) Show that the mean number of call-outs per night is 2.52.5 and find the variance.

(3 marks)
(b) Explain why your answers to part (a) suggest that a Poisson distribution may be a suitable model for the number of call-outs per night.

(1 mark)

The number of call-outs per night is modelled by a Poisson distribution with mean 2.52.5
(c) Find the probability that, on a randomly selected night, the recovery service receives

(i) exactly 33 call-outs,

(ii) more than 44 call-outs.

(4 marks)

On two particular consecutive nights the recovery service received a total of 55 call-outs.
(d) Find the probability that at least 44 of these call-outs were received on the same night.

(5 marks)

The manager selects 1212 non-overlapping periods, each of two consecutive nights, at random.
(e) Find the probability that there are fewer than 33 call-outs in exactly 22 of these periods.

(3 marks)
●●●●●Level 416 marksStart
Bikes are returned to a docking station of a city cycle-hire scheme at random, at a mean rate of 88 per hour.
(a) Find the probability that, in a randomly selected 1515-minute period,

(i) exactly 33 bikes are returned,

(ii) at least 55 bikes are returned.

(3 marks)

Of the bikes returned to this docking station, 25%25\% are electric bikes.
(b) Show that the probability that at least one electric bike is returned in a randomly selected 1010-minute period is 0.28350.2835 to 44 decimal places.

(2 marks)

The station manager randomly selects 1212 non-overlapping 3030-minute periods.
(c) Find the probability that between 33 and 66 (inclusive) bikes are returned in exactly 88 of these 3030-minute periods.

(4 marks)

Each electric bike returned is plugged into a charging bank.
  • In each 1010-minute period in which no electric bike is returned, the charging bank costs £0.02 to run.
  • In each 1010-minute period in which at least one electric bike is returned, the charging bank costs £0.15 to run.
(d) Find the expected cost of running the charging bank for a period of 1212 hours.

(4 marks)
(e) Explain why a Poisson distribution with a mean rate of 88 per hour is unlikely to be a suitable model for the number of bikes returned to this docking station over a whole 2424-hour day.

(1 mark)
●●●●●Level 414 marksStart

Cumulative binomial probabilities

39 questions

LessonNot started
A machine produces components, of which 8%8\% are faulty. A random sample of 2525 components is taken, and XX is the number of faulty components in the sample.
(a) State two conditions needed for XX to be modelled by a binomial distribution. (2 marks)
(b) Find P(X⩽3)\mathrm{P}(X \leqslant 3), giving your answer to 44 decimal places. (2 marks)
(c) Find P(X⩾2)\mathrm{P}(X \geqslant 2), giving your answer to 44 decimal places. (2 marks)

The machine is serviced. A new random sample of 2525 components is taken, and none of them is faulty.
(d) Test, at the 5%5\% level of significance, whether the proportion of faulty components has decreased. (4 marks)
●●●●●Level 410 marksStart
A library records every book that is returned to it.

It is found that 8%8\% of all books returned to this library are returned late.

A sample of 5050 returned books is selected. The number of books in the sample that were returned late is denoted by the random variable XX
(a) State two assumptions necessary for XX to be modelled by a binomial distribution.

(2 marks)
(b) Assume that XX can be modelled by a binomial distribution.

(i) Find P(X=2)P(X = 2)

(ii) Find P(X<6)P(X < 6)

(iii) Find P(X≥7)P(X \ge 7)

(5 marks)
(c) In a random sample of 6060 books returned to a different library, the number of books returned late, YY, may be modelled by Y∼B(60,p)Y \sim \text{B}(60, p)

The standard deviation of YY is 33

Calculate the possible values of pp

(3 marks)
●●●●●Level 410 marksStart
The random variable X∼B(45,0.2)X \sim \text{B}(45, 0.2)
(a) Find the mean of XX

(1 mark)
(b) Show that the variance of XX is 7.27.2

(1 mark)
(c) Find P(X<6)P(X < 6)

(1 mark)
(d) Find P(X≥12)P(X \ge 12)

(2 marks)
(e) Find P(7≤X≤11)P(7 \le X \le 11)

(2 marks)
(f) The coaches used for a company's sightseeing tours each have 4545 seats.

The proportion of passengers on these tours who are aged over 7070 is known to be 20%20\%

Make two comments on whether it would be appropriate to use XX to model the number of passengers on a coach who are aged over 7070 on one of these tours.

(2 marks)
●●●●●Level 49 marksStart
A survey carried out in a town in 2020 found that 30%30\% of adults regularly cycled to work. Since then a network of cycle lanes has opened in the town. A council officer believes that the proportion of adults in the town who regularly cycle to work has changed. To investigate, she asks 2525 adults leaving the town's leisure centre one Saturday morning whether they regularly cycle to work, and uses XX, the number who say yes, to carry out a hypothesis test.
(a) Explain why the officer should use a two-tailed test. (1 mark)
(b) Stating your hypotheses clearly, determine the critical region for the test at the 5%5\% significance level, showing the probabilities you use. Find the actual significance level of the test. (5 marks)
(c) In fact 1212 of the 2525 adults say that they regularly cycle to work. State, with a reason, the conclusion of the test in context. (2 marks)
(d) Explain why the way the officer chose her sample makes the conclusion of the test unreliable. (1 mark)
●●●●●Level 49 marksStart
It is known that 41% of households in a large city have at least one pet. A researcher selects a random sample of 1200 households in the city. The random variable XX is defined as the number of households in the sample that have at least one pet.

Given that P(X⩽n)<0.05\mathrm{P}(X \leqslant n) < 0.05, calculate the largest possible value of nn.

(5 marks)
●●●●●Level 45 marksStart
Seeds of a certain plant germinate with probability 0.30.3. A gardener plants 2020 seeds in a new compost, and claims the compost increases the germination rate. Let XX be the number of seeds that germinate, and assume X∼B(20,p)X \sim B(20, p).
(a) State two conditions needed for a binomial model to be suitable here. (2 marks)
(b) Assuming p=0.3p = 0.3, find (i) P(X=4)P(X = 4) and (ii) P(X≥7)P(X \ge 7). (4 marks)
(c) Using a 5%5\% level of significance, find the critical region for a test of the gardener's claim, and state the actual significance level of the test. (5 marks)
(d) In fact 1111 of the 2020 seeds germinate. State the conclusion of the test in context. (2 marks)
●●●●●Level 413 marksStart
It is known that 25%25\% of all visitors to a museum buy something from the museum's gift shop.

The museum moves its gift shop to the entrance hall.

Priya decides to investigate whether the proportion of visitors to the museum who buy something from the gift shop has changed.

Priya will carry out a hypothesis test at the 10%10\% significance level on a random sample of 3030 visitors to the museum.
(a) State suitable null and alternative hypotheses for Priya's test.

(1 mark)
(b) Using a 10%10\% level of significance, find the critical region for Priya's test.

(5 marks)
(c) In her random sample, Priya finds 1212 visitors who buy something from the gift shop.

State Priya's conclusion in context.

(1 mark)
●●●●●Level 47 marksStart
On average, 64%64\% of the tomato plants grown from seed at a nursery produce ripe fruit within ten weeks of planting.

A gardener buys a tray of 2020 of these tomato plants.

The number of plants in the tray that produce ripe fruit within ten weeks can be modelled by the distribution B(20,0.64)\text{B}(20, 0.64)
(a) State one assumption, in context, required for this distribution to be used.

(1 mark)
(b) Find the probability that exactly 1313 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(c) Find the probability that at most 1010 of the plants in the tray produce ripe fruit within ten weeks.

(1 mark)
(d) Find the probability that more than 1515 of the plants in the tray produce ripe fruit within ten weeks.

(2 marks)
(e) Find the mean number of plants in the tray that produce ripe fruit within ten weeks.

(1 mark)
(f) Find the standard deviation of the number of plants in the tray that produce ripe fruit within ten weeks.

(2 marks)
●●●●●Level 48 marksStart
A bus company operates the bus services in a town.

Two years ago, 35%35\% of the company's passengers paid for their journey with cash.

The company has since installed contactless card readers on all of its buses.

One month after the card readers were installed, 4040 passengers were selected at random and 88 of them were found to have paid for their journey with cash.

Using a binomial distribution, carry out a hypothesis test at the 5%5\% significance level to investigate the claim that, since the card readers were installed, there has been a decrease in the proportion of passengers who pay for their journey with cash.

(6 marks)
●●●●●Level 46 marksStart
At a fairground stall, each player pays to have one go at a game.

Each time a player has a go, the probability that the player wins a prize is 0.180.18, independently of all other goes.

Maya has 1616 goes at this game.

The random variable XX represents the number of prizes that Maya wins.
(a) Using a suitable distribution for XX, find

(i) P(X=4)\mathrm{P}(X = 4)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

During one weekend there are 300300 goes at this stall.
(b) Use a normal approximation to estimate the probability that more than 6262 prizes are won during this weekend.

(3 marks)
●●●●●Level 46 marksStart
A theatre has found that, over a long period, 30%30\% of the people who attend a performance buy a programme.

A random sample of 2525 people who attend a performance is taken.

The random variable XX represents the number of these people who buy a programme.
(a) Using a suitable distribution for XX, find

(i) P(X=9)\mathrm{P}(X = 9)

(ii) P(X⩽5)\mathrm{P}(X \leqslant 5)

(3 marks)

The price of a programme is then increased.

The manager of the theatre claims that the proportion of people attending a performance who buy a programme has fallen.

After the increase, a random sample of 5050 people who attend a performance is taken and 99 of them buy a programme.
(b) Use a suitable test to assess the manager's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 47 marksStart
On his way to school each morning, Yusuf passes through 99 sets of traffic lights.

The probability that any particular set of lights is green when Yusuf reaches it is 0.450.45, independently of all the other sets of lights.

The random variable XX represents the number of sets of lights that are green on Yusuf's journey to school on a randomly chosen morning.
(a) Using a suitable distribution for XX, find

(i) P(X=5)\mathrm{P}(X = 5)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

Yusuf makes this journey on each of 2020 school mornings.
(b) Find the probability that there were at least 44 mornings when fewer than 33 sets of lights were green.

(3 marks)
(c) Find an estimate for the total number of sets of lights that were green on Yusuf's journey to school during these 2020 mornings.

(1 mark)
●●●●●Level 47 marksStart
A workshop makes wooden beads and 15\tfrac{1}{5} of the beads are also painted gold.

The beads are packed into bags before being sent to craft shops.

There are 1414 beads in each bag.

The random variable TT represents the number of beads that are painted gold in each bag.
(a) State a condition needed for TT to be modelled by B ⁣(14,15)\mathrm{B}\!\left(14, \tfrac{1}{5}\right)

(1 mark)

A bag is selected at random.
(b) Using T∼B ⁣(14,15)T \sim \mathrm{B}\!\left(14, \tfrac{1}{5}\right) find

(i) the probability that the bag has exactly 33 beads that are painted gold,

(ii) the probability that the bag has fewer than 22 beads that are painted gold.

(2 marks)

Iris owns a craft shop and buys 2525 bags of these beads.
(c) Find the probability that exactly 44 of these 2525 bags have fewer than 22 beads that are painted gold.

(2 marks)

Iris claims that the proportion of these beads that are painted gold is less than 15\tfrac{1}{5}

A random sample of 6060 of these beads is taken and 66 of them are painted gold.
(d) Use a suitable test to assess Iris's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
●●●●●Level 49 marksStart
A ferry company claims that no more than 8% of its sailings leave port late. A travel journalist believes that the actual proportion is greater than 8%. In order to test the company's claim, the journalist records, for a random sample of 150 sailings, whether each one leaves port late, and carries out a hypothesis test, at the 5% significance level, using the null hypothesis p=0.08p = 0.08.
(a) Given that the journalist's null hypothesis is correct, determine the probability that the journalist will conclude that the company's claim is incorrect.

(5 marks)
(b) The journalist finds that 17 of the 150 sailings leave port late. A colleague says

“17 out of 150 is more than 11%, so there is evidence that the actual proportion of sailings that leave port late is greater than 8%.”

Criticise this statement.

(1 mark)
●●●●●Level 46 marksStart
Last year, 35% of the customers at a café paid in cash. The owner plans to test whether the proportion of customers at the café who pay in cash this year is less than 0.35. The hypotheses for the test will be H0:p=0.35\mathrm{H}_0: p = 0.35 and H1:p<0.35\mathrm{H}_1: p < 0.35.

The owner chooses a random sample of 300 customers this year and notes the number XX who pay in cash.

The random variable YY has a normal distribution which is a good approximation to the distribution of XX when the null hypothesis is correct.

The significance level of the test is 2.5%.
(a) Find the value of yy such that P(Y<y)=0.025\mathrm{P}(Y < y) = 0.025.

(2 marks)
(b) Use your answer to part (a) to determine the largest value of XX that would result in rejecting the null hypothesis.

(3 marks)
●●●●●Level 45 marksStart
A running club has 169 members who run mainly on roads and 91 members who run mainly on trails. A random sample of 18 members is selected to attend a meeting about the club's new training programme. The number of members in the sample who run mainly on trails is denoted by XX.
(a) State a suitable binomial model for XX.

(1 mark)

Use your model to answer the following.
(b) (i) Write down an expression for P(X=x)\mathrm{P}(X = x).

(ii) State, in set notation, the values of xx for which your expression is valid.

(iii) Explain the connection between the expression in part (b)(i) and the binomial expansion of (0.65+0.35)n(0.65 + 0.35)^n, for a specific value of nn which should be stated.

(4 marks)
(c) Find P(4⩽X⩽8)\mathrm{P}(4 \leqslant X \leqslant 8).

(2 marks)
(d) State one disadvantage of using a random sample in this context.

(1 mark)
●●●●●Level 48 marksStart
Gwen is a driving instructor. She claims that 75% of her pupils pass their practical driving test at their first attempt. Tomasz suspects that Gwen is over-confident about the first-attempt pass rate amongst her pupils. Tomasz contacts a random sample of 24 of Gwen's former pupils, and finds that 13 of them passed at their first attempt. He then performs a hypothesis test.

Carry out the test at the 5% significance level.

(7 marks)
●●●●●Level 47 marksStart
In each round of a game on a phone app, a star is hidden in one of 44 identical boxes. The player chooses one box.
  • If the box contains the star, the player gains 55 points.
  • If the box does not contain the star, the player loses 11 point.
Ines plays 2020 rounds of the game, choosing one box at random in each round.

Let XX be the number of rounds in which Ines finds the star.
(a) State the distribution of XX

(1 mark)

Let SS be the total number of points that Ines gains in the 2020 rounds.
(b) (i) Write down SS in terms of XX

(ii) Hence, show clearly that the expected value of SS is 1010

(4 marks)

A player who gains at least 2525 points in 2020 rounds receives a bonus.
(c) Find the probability that Ines receives a bonus.

(3 marks)

Records show that, for players who have completed the training level of the game, the probability of finding the star in the first round is 0.350.35

A random sample of 4040 players who have completed the training level is taken.

Given that the probability that more than nn but at most 1818 of these players find the star in the first round is 0.72480.7248 to 44 decimal places,
(d) find the value of nn

(4 marks)
●●●●●Level 412 marksStart
A garden centre sells packets of poppy seeds. Each seed grows into a plant with red flowers with probability pp, independently of the other seeds.

Each small packet contains nn seeds.

The mean number of seeds in a small packet that grow into plants with red flowers is 4.54.5 and the variance is 3.153.15
(a) Find the value of nn and the value of pp

(3 marks)

Each large packet contains 2525 seeds. The random variable RR represents the number of seeds in a large packet that grow into plants with red flowers.
(b) Find the probability that RR exceeds the expected number of seeds in a large packet that grow into plants with red flowers.

(3 marks)
(c) Find the largest value of rr such that P(R<r)<0.05\mathrm{P}(R < r) < 0.05

(2 marks)

Jamal buys 66 large packets.
(d) Find the probability that, in at least 55 of these 66 packets, the number of seeds that grow into plants with red flowers exceeds the expected number.

(3 marks)
●●●●●Level 411 marksStart

The normal distribution

38 questions

LessonNot started
The times taken by a large group of athletes to run 400400 m are modelled by a normal distribution with mean 52.452.4 seconds and standard deviation 2.82.8 seconds.
(a) Find the probability that a randomly chosen athlete takes less than 5050 seconds, giving your answer to 33 significant figures. (2 marks)
(b) Find the probability that a randomly chosen athlete takes between 5050 and 5555 seconds, giving your answer to 33 significant figures. (3 marks)
(c) The fastest 15%15\% of athletes qualify for a final. Find the qualifying time, giving your answer to 11 decimal place. (3 marks)
(d) Three athletes are chosen at random. Find the probability that exactly one of them takes less than 5050 seconds, giving your answer to 33 significant figures. (3 marks)
●●●●●Level 411 marksStart
The masses, in grams, of apples from an orchard are normally distributed with mean 165165 g and standard deviation 1212 g.
(a) Find the probability that a randomly chosen apple has a mass greater than 180180 g. (2 marks)
(b) Find the probability that a randomly chosen apple has a mass between 150150 g and 175175 g. (3 marks)
(c) The heaviest 10%10\% of apples are graded as "large". Find the least mass of a large apple. (3 marks)
●●●●●Level 48 marksStart
The time taken by a courier company to deliver a parcel, YY minutes, is modelled by a normal distribution with mean μ\mu and standard deviation σ\sigma.

Given that P(Y<22)=0.15P(Y < 22) = 0.15 and P(Y>35)=0.2P(Y > 35) = 0.2,
(a) find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(b) A parcel is known to have taken more than 2222 minutes to deliver. Find the probability that it took less than 3030 minutes. (3 marks)
(c) The company's records show that a small number of parcels take several hours to deliver. Explain why this suggests that the normal distribution may not be a suitable model for delivery times. (1 mark)
●●●●●Level 410 marksStart
The weights of apples from an orchard, WW grams, are modelled by a normal distribution.

It is known that 10%10\% of the apples weigh less than 120120 g, and 5%5\% of the apples weigh more than 190190 g.
(a) Find the value of the mean μ\mu and the value of the standard deviation σ\sigma, giving each answer to 33 significant figures. (6 marks)
(b) An apple is classed as large if it weighs more than 175175 g.

Find the probability that a randomly chosen apple is large, giving your answer to 33 significant figures. (2 marks)
(c) Find the weight that is exceeded by 25%25\% of the apples, giving your answer to the nearest gram. (3 marks)
●●●●●Level 411 marksStart
The time taken for a courier to deliver a parcel is modelled as X∼N(30,42)X \sim N(30, 4^2) minutes.
(a) Find P(X>35)P(X > 35). (2 marks)
(b) Find the time dd such that 10%10\% of deliveries take less than dd minutes. (3 marks)

A rival company's delivery times are modelled as Y∼N(μ,σ2)Y \sim N(\mu, \sigma^2). It is known that P(Y<20)=0.25P(Y < 20) = 0.25 and P(Y>32)=0.10P(Y > 32) = 0.10.
(c) Find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(d) Give one reason why a normal distribution might be a suitable model for delivery times, and one potential limitation. (2 marks)
●●●●●Level 413 marksStart
As part of a study, the masses, mm grams, of a random sample of 600 eggs from one farm were recorded. The results are summarised in the table.
Mass
(g)		45⩽m<5545 \leqslant m < 5555⩽m<6055 \leqslant m < 6060⩽m<6560 \leqslant m < 6565⩽m<7565 \leqslant m < 75
Frequency8821423662
(a) Calculate estimates of the mean and standard deviation of these masses.

(2 marks)

The masses, xx grams, of a random sample of 600 eggs from a second farm were also found. It was proposed that the distribution of the masses of eggs from this farm should be modelled by the random variable X∼N(61,36)X \sim \mathrm{N}(61, 36).
(b) Use the model to find P(55<X<64)\mathrm{P}(55 < X < 64).

(1 mark)
(c) Use the model to determine x1x_1 such that P(58<X<x1)=0.5\mathrm{P}(58 < X < x_1) = 0.5, giving your answer correct to five significant figures.

(3 marks)

In the sample from the second farm, the lightest egg had a mass of 52 g and the heaviest egg had a mass of 70 g.
(d) Use these results to show that the model may not be appropriate.

(1 mark)
(e) In the light of these results, suggest a different value for one of the parameters of the model.

(2 marks)
●●●●●Level 49 marksStart
(a) The lengths, LL centimetres, of runner beans of a certain variety are modelled by the distribution N(18.5,6.25)\mathrm{N}(18.5, 6.25).

Find P(15<L<20)\mathrm{P}(15 < L < 20).

(1 mark)
(b) The masses, in grams, of strawberries of a certain variety are modelled by the distribution N(μ,σ2)\mathrm{N}(\mu, \sigma^2).

You are given that 15% of these strawberries have masses greater than 27 g and 40% have masses less than 21 g.

Determine the values of μ\mu and σ\sigma. Give your answers correct to 2 decimal places.

(5 marks)
(c) The diagram shows the probability distribution of a normal variable, XX.246810121416182022x(i) Write down estimates of the xx-coordinates of the points of inflection on the graph.

(ii) Hence write down an estimate of the standard deviation of XX, explaining your method.

(2 marks)
●●●●●Level 48 marksStart
A factory makes rubber seals for washing machines.

The diameter of a seal, DD mm, is modelled as being normally distributed with mean 24.624.6 mm and standard deviation 0.350.35 mm

A seal is selected at random.
(a) Find the probability that this seal has a diameter of more than 25.225.2 mm

(1 mark)
(b) Find the probability that this seal has a diameter between 24.224.2 mm and 25.025.0 mm

(2 marks)

The same factory also makes a larger seal.

The random variable E∼N(μ,σ2)E \sim \mathrm{N}(\mu, \sigma^2) represents the diameter, in mm, of a larger seal.

Given that P(E<39.5)=0.02\mathrm{P}(E < 39.5) = 0.02 and P(E>42)=0.30\mathrm{P}(E > 42) = 0.30
(c) calculate the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 48 marksStart
The records of a rowing club show that the time, TT minutes, taken by a member of the club to complete a 55 km test on a rowing machine is normally distributed with mean 19.419.4 minutes and standard deviation 1.61.6 minutes.
(a) Find the proportion of these members who complete the 55 km test in less than 1818 minutes.

(1 mark)

The records also show that the distance, SS metres, rowed by a member of the club in a 33 minute test is normally distributed with mean 820820 m and standard deviation 4545 m

The coach would like to use these distributions to estimate the proportion of members who can complete the 55 km test in less than 1818 minutes and can row more than 0.90.9 km in the 33 minute test.
(b) State a necessary assumption about TT and SS for the coach to calculate an estimate of this proportion.

(1 mark)
(c) Find the coach's estimate of this proportion.

(3 marks)
(d) Find the distance, to the nearest metre, that is exceeded in the 33 minute test by only the best 10%10\% of these members.

(2 marks)
●●●●●Level 47 marksStart
A fish farm grades its trout by length. The length, LL cm, of a trout on the farm is modelled by L∼N(38,42)L \sim \mathrm{N}(38, 4^2)

Each trout is placed in one of five grades, as shown in the table below, where kk is a constant.
GradeUndersizedSmallStandardLargeSpecimen
Length (LL cm)L<30L < 3030≤L<3530 \le L < 3535≤L<k35 \le L < kk≤L<46k \le L < 46L≥46L \ge 46
One trout is selected at random from the farm.
(a) Find the probability that this trout is graded Small.

(3 marks)

15% of the trout are graded Large.
(b) Find the value of kk, giving your answer to one decimal place.

(4 marks)

The farm sells the longest 20% of its Specimen trout to a restaurant.
(c) Find the length of the shortest trout that is sold to the restaurant, giving your answer to one decimal place.

(5 marks)
●●●●●Level 412 marksStart
The time, XX minutes, that a customer spends in a bookshop is normally distributed with mean mm minutes and variance 25

Given that

P(m−3k<X<m+3k)=0.8\mathrm{P}(m - 3k < X < m + 3k) = 0.8
(a) find the value of kk

(4 marks)

The time, YY minutes, that a customer spends in the café next to the bookshop is normally distributed with mean μ\mu minutes and standard deviation σ\sigma minutes.

Given that

5μ=12σandP(Y>42)=0.03595\mu = 12\sigma \qquad \text{and} \qquad \mathrm{P}(Y > 42) = 0.0359
(b) find the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 49 marksStart
The masses, in grams, of chocolate truffles made by a confectioner have the distribution N(14.2,0.64)\mathrm{N}(14.2, 0.64).
(a) Find the probability that a truffle chosen at random has a mass between 13.5 and 15.0 grams.

(1 mark)
(b) The heaviest 10% of truffles are sold as 'large'.

Find the minimum mass of a large truffle.

(1 mark)
(c) The truffles are packed in boxes, each containing 12 randomly selected truffles.

Find the probability that a box chosen at random has a total mass of more than 172 g.

(4 marks)
●●●●●Level 46 marksStart
A typing test measures a person's typing speed, WW words per minute. For one trainee, WW can be modelled by a normal distribution with mean 62 and standard deviation 8
(a) Using standardisation, show that the probability that the trainee types at a speed of less than 58 words per minute on a randomly chosen attempt is 0.3085

(2 marks)

Attempts with the highest 10% of the trainee's speeds are rated as "excellent".
(b) Use standardisation to find the lowest speed, in words per minute, that would be rated as excellent.

(3 marks)

To pass the typing test the trainee needs a speed of at least 58 words per minute. The trainee has up to 3 attempts at the test. Once the trainee passes, they do not use any of their remaining attempts. The speeds on different attempts are independent.

Given that the trainee passed the test,
(c) find the probability that the trainee passed on their second attempt with a speed of more than 72 words per minute.

(5 marks)
●●●●●Level 410 marksStart
A music streaming service takes a random sample of 160 songs from one of its playlists and records the length, in seconds, of each song.

The lengths of the songs in the sample are summarised in the following box plot.150200216248310Length (seconds)(a) Use linear interpolation to estimate the probability that a randomly chosen song from this sample is shorter than 232 seconds.

(2 marks)

The service takes the quartiles from this sample and decides to label any song whose length is at least Q3+1.5×(Q3−Q1)Q_3 + 1.5 \times (Q_3 - Q_1) as "extended".
(b) Find the shortest length of a song that the service would label as extended.

(1 mark)

An analyst suggests that the lengths of the songs on the playlist may be modelled by a normal distribution.
(c) Explain whether or not the box plot supports this suggestion.

(1 mark)

The analyst records the length of every song on the playlist and classifies any song whose length is more than 2.5 standard deviations from the mean as unusual.

Assuming that the lengths of the songs on the playlist may be modelled by a normal distribution,
(d) find the probability that a randomly selected song from the playlist would be classified as unusual.

(2 marks)

The mean length of the songs on the playlist is 226 seconds.

Given that a song of length 350 seconds is classified as unusual,
(e) find the maximum possible value of the standard deviation of the lengths of the songs on the playlist.

(2 marks)
●●●●●Level 48 marksStart

Finding normal probabilities

51 questions

LessonNot started
A machine fills bottles with water. The volume of water in a bottle, XX ml, is modelled by X∼N(500,82)X \sim N(500, 8^2).
(a) Find P(X<490)P(X < 490). (2 marks)
(b) Find the probability that a randomly chosen bottle contains within 1010 ml of the mean. (3 marks)
(c) A crate contains 55 bottles, filled independently. Find the probability that every bottle in the crate contains at least 490490 ml. (3 marks)
(d) The machine is recalibrated: the mean stays at 500500 ml but the standard deviation, σ\sigma ml, changes. Find the largest value of σ\sigma for which P(X<490)≤0.02P(X < 490) \le 0.02. (4 marks)
●●●●●Level 412 marksStart
The times taken by a large group of athletes to run 400400 m are modelled by a normal distribution with mean 52.452.4 seconds and standard deviation 2.82.8 seconds.
(a) Find the probability that a randomly chosen athlete takes less than 5050 seconds, giving your answer to 33 significant figures. (2 marks)
(b) Find the probability that a randomly chosen athlete takes between 5050 and 5555 seconds, giving your answer to 33 significant figures. (3 marks)
(c) The fastest 15%15\% of athletes qualify for a final. Find the qualifying time, giving your answer to 11 decimal place. (3 marks)
(d) Three athletes are chosen at random. Find the probability that exactly one of them takes less than 5050 seconds, giving your answer to 33 significant figures. (3 marks)
●●●●●Level 411 marksStart
The masses, in grams, of apples from an orchard are normally distributed with mean 165165 g and standard deviation 1212 g.
(a) Find the probability that a randomly chosen apple has a mass greater than 180180 g. (2 marks)
(b) Find the probability that a randomly chosen apple has a mass between 150150 g and 175175 g. (3 marks)
(c) The heaviest 10%10\% of apples are graded as "large". Find the least mass of a large apple. (3 marks)
●●●●●Level 48 marksStart
The time taken by a courier company to deliver a parcel, YY minutes, is modelled by a normal distribution with mean μ\mu and standard deviation σ\sigma.

Given that P(Y<22)=0.15P(Y < 22) = 0.15 and P(Y>35)=0.2P(Y > 35) = 0.2,
(a) find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(b) A parcel is known to have taken more than 2222 minutes to deliver. Find the probability that it took less than 3030 minutes. (3 marks)
(c) The company's records show that a small number of parcels take several hours to deliver. Explain why this suggests that the normal distribution may not be a suitable model for delivery times. (1 mark)
●●●●●Level 410 marksStart
The weights of apples from an orchard, WW grams, are modelled by a normal distribution.

It is known that 10%10\% of the apples weigh less than 120120 g, and 5%5\% of the apples weigh more than 190190 g.
(a) Find the value of the mean μ\mu and the value of the standard deviation σ\sigma, giving each answer to 33 significant figures. (6 marks)
(b) An apple is classed as large if it weighs more than 175175 g.

Find the probability that a randomly chosen apple is large, giving your answer to 33 significant figures. (2 marks)
(c) Find the weight that is exceeded by 25%25\% of the apples, giving your answer to the nearest gram. (3 marks)
●●●●●Level 411 marksStart
A machine fills jars with honey. The mass of honey in a jar is modelled by a normal distribution with mean 410410 g and standard deviation 1212 g.
(a) Find the probability that a randomly chosen jar contains more than 428428 g of honey. (2 marks)
(b) Determine the mass mm such that 99%99\% of jars contain less than mm grams, giving your answer to 3 significant figures. (3 marks)

A consumer group believes the machine has drifted and is now underfilling jars. The masses of a random sample of 4040 jars have mean 405.4405.4 g. The standard deviation is assumed to remain 1212 g.
(c) Test, at the 5%5\% significance level, whether there is evidence that the mean mass of honey per jar has fallen below 410410 g. State your hypotheses clearly, and state whether your conclusion would change at the 1%1\% level. (6 marks)
●●●●●Level 411 marksStart
The time taken for a courier to deliver a parcel is modelled as X∼N(30,42)X \sim N(30, 4^2) minutes.
(a) Find P(X>35)P(X > 35). (2 marks)
(b) Find the time dd such that 10%10\% of deliveries take less than dd minutes. (3 marks)

A rival company's delivery times are modelled as Y∼N(μ,σ2)Y \sim N(\mu, \sigma^2). It is known that P(Y<20)=0.25P(Y < 20) = 0.25 and P(Y>32)=0.10P(Y > 32) = 0.10.
(c) Find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(d) Give one reason why a normal distribution might be a suitable model for delivery times, and one potential limitation. (2 marks)
●●●●●Level 413 marksStart
The mass, in grams, of an apple from a particular orchard is modelled by the normal distribution N(180,  25)\mathrm{N}\left(180,\; 25\right).
(a) Find the probability that a randomly chosen apple has a mass greater than 190190 g, giving your answer to 33 significant figures. (2 marks)
(b) Find the mass exceeded by 10%10\% of the apples, giving your answer to 11 decimal place. (3 marks)

A new fertiliser is used. A random sample of 1616 apples from the treated trees has a mean mass of 183.5183.5 g. The standard deviation is assumed to be unchanged.
(c) Test, at the 5%5\% level of significance, whether the mean mass of the apples has increased. (5 marks)
●●●●●Level 410 marksStart
As part of a study, the masses, mm grams, of a random sample of 600 eggs from one farm were recorded. The results are summarised in the table.
Mass
(g)		45⩽m<5545 \leqslant m < 5555⩽m<6055 \leqslant m < 6060⩽m<6560 \leqslant m < 6565⩽m<7565 \leqslant m < 75
Frequency8821423662
(a) Calculate estimates of the mean and standard deviation of these masses.

(2 marks)

The masses, xx grams, of a random sample of 600 eggs from a second farm were also found. It was proposed that the distribution of the masses of eggs from this farm should be modelled by the random variable X∼N(61,36)X \sim \mathrm{N}(61, 36).
(b) Use the model to find P(55<X<64)\mathrm{P}(55 < X < 64).

(1 mark)
(c) Use the model to determine x1x_1 such that P(58<X<x1)=0.5\mathrm{P}(58 < X < x_1) = 0.5, giving your answer correct to five significant figures.

(3 marks)

In the sample from the second farm, the lightest egg had a mass of 52 g and the heaviest egg had a mass of 70 g.
(d) Use these results to show that the model may not be appropriate.

(1 mark)
(e) In the light of these results, suggest a different value for one of the parameters of the model.

(2 marks)
●●●●●Level 49 marksStart
The masses of apples are distributed as N(180,152)\mathrm{N}(180, 15^2) g and the masses of oranges, independently, as N(210,202)\mathrm{N}(210, 20^2) g.

An apple and an orange are chosen at random.
(a) State the distribution of DD, the mass of the orange minus the mass of the apple. (3 marks)
(b) Find the probability that the orange is heavier than the apple. (3 marks)
●●●●●Level 46 marksStart
A machine fills bottles so that the mass of a filled bottle is distributed as N(750,52)\mathrm{N}(750, 5^2) g, independently for each bottle.

A crate contains 55 bottles.
(a) State the distribution of the total mass TT of the bottles in a crate. (3 marks)
(b) Find P(T<3740)\mathrm{P}(T < 3740). (3 marks)
(c) Explain why the variance of TT is not 52×52=6255^2 \times 5^2 = 625. (1 mark)
●●●●●Level 47 marksStart
A poultry farm supplies hens' eggs to a wholesaler.

The masses of the hens' eggs, MM grams, are normally distributed with mean 61.561.5 and standard deviation 4.24.2
(a) (i) Find P(M<55)P(M < 55)

(ii) Find P(M>70)P(M > 70)

(iii) The wholesaler only accepts hens' eggs with masses between 5353 grams and 7373 grams.

Find the proportion of hens' eggs that the wholesaler accepts.

(3 marks)
(b) The farm also supplies duck eggs to the wholesaler.

The masses of these duck eggs are normally distributed.

It is found that 12%12\% of these duck eggs have a mass less than 6464 grams.

It is found that 20%20\% of these duck eggs have a mass greater than 8080 grams.

Find the mean and standard deviation of the mass, in grams, of the duck eggs supplied by the farm.

(6 marks)
●●●●●Level 49 marksStart
(a) The lengths, LL centimetres, of runner beans of a certain variety are modelled by the distribution N(18.5,6.25)\mathrm{N}(18.5, 6.25).

Find P(15<L<20)\mathrm{P}(15 < L < 20).

(1 mark)
(b) The masses, in grams, of strawberries of a certain variety are modelled by the distribution N(μ,σ2)\mathrm{N}(\mu, \sigma^2).

You are given that 15% of these strawberries have masses greater than 27 g and 40% have masses less than 21 g.

Determine the values of μ\mu and σ\sigma. Give your answers correct to 2 decimal places.

(5 marks)
(c) The diagram shows the probability distribution of a normal variable, XX.246810121416182022x(i) Write down estimates of the xx-coordinates of the points of inflection on the graph.

(ii) Hence write down an estimate of the standard deviation of XX, explaining your method.

(2 marks)
●●●●●Level 48 marksStart
A factory makes rubber seals for washing machines.

The diameter of a seal, DD mm, is modelled as being normally distributed with mean 24.624.6 mm and standard deviation 0.350.35 mm

A seal is selected at random.
(a) Find the probability that this seal has a diameter of more than 25.225.2 mm

(1 mark)
(b) Find the probability that this seal has a diameter between 24.224.2 mm and 25.025.0 mm

(2 marks)

The same factory also makes a larger seal.

The random variable E∼N(μ,σ2)E \sim \mathrm{N}(\mu, \sigma^2) represents the diameter, in mm, of a larger seal.

Given that P(E<39.5)=0.02\mathrm{P}(E < 39.5) = 0.02 and P(E>42)=0.30\mathrm{P}(E > 42) = 0.30
(c) calculate the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 48 marksStart
The records of a rowing club show that the time, TT minutes, taken by a member of the club to complete a 55 km test on a rowing machine is normally distributed with mean 19.419.4 minutes and standard deviation 1.61.6 minutes.
(a) Find the proportion of these members who complete the 55 km test in less than 1818 minutes.

(1 mark)

The records also show that the distance, SS metres, rowed by a member of the club in a 33 minute test is normally distributed with mean 820820 m and standard deviation 4545 m

The coach would like to use these distributions to estimate the proportion of members who can complete the 55 km test in less than 1818 minutes and can row more than 0.90.9 km in the 33 minute test.
(b) State a necessary assumption about TT and SS for the coach to calculate an estimate of this proportion.

(1 mark)
(c) Find the coach's estimate of this proportion.

(3 marks)
(d) Find the distance, to the nearest metre, that is exceeded in the 33 minute test by only the best 10%10\% of these members.

(2 marks)
●●●●●Level 47 marksStart
A study was made of the times taken by the trainees at the Northgate depot to complete a standard assembly task. It was found that these times were normally distributed with a mean of 46.046.0 minutes and a standard deviation of 5.25.2 minutes.
(a) Find the proportion of these trainees who complete the task in less than 4040 minutes.

(1 mark)

A manager claimed that the mean time taken by the trainees at the Riverside depot was different from the mean time taken by the trainees at the Northgate depot.

A random sample of 2020 trainees at the Riverside depot had a mean time of 44.244.2 minutes

The manager assumed that the standard deviation of the times was 5.25.2 minutes for the trainees at both depots.
(b) Use a suitable test to assess the manager's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
(c) Find the p-value for the test in part (b).

(1 mark)
●●●●●Level 46 marksStart
A study was made of the loaves of bread baked at the Millgate bakery. It was found that the masses of these loaves were normally distributed with a mean of 802802 g and a standard deviation of 1212 g.
(a) Find the proportion of these loaves with a mass of more than 820820 g.

(1 mark)

A student claimed that the mean mass of the loaves baked at the Riverbank bakery was different from the mean mass of the loaves baked at the Millgate bakery.

A random sample of 3030 loaves baked at the Riverbank bakery had a mean mass of 796.5796.5 g

The student assumed that the standard deviation of the masses of the loaves was 1212 g both for the Millgate bakery and for the Riverbank bakery.
(b) Use a suitable test to assess the student's claim.

You should
  • state your hypotheses clearly
  • use a 5%5\% level of significance
(4 marks)
(c) Find the p-value for the test in part (b).

(1 mark)
●●●●●Level 46 marksStart
A fish farm grades its trout by length. The length, LL cm, of a trout on the farm is modelled by L∼N(38,42)L \sim \mathrm{N}(38, 4^2)

Each trout is placed in one of five grades, as shown in the table below, where kk is a constant.
GradeUndersizedSmallStandardLargeSpecimen
Length (LL cm)L<30L < 3030≤L<3530 \le L < 3535≤L<k35 \le L < kk≤L<46k \le L < 46L≥46L \ge 46
One trout is selected at random from the farm.
(a) Find the probability that this trout is graded Small.

(3 marks)

15% of the trout are graded Large.
(b) Find the value of kk, giving your answer to one decimal place.

(4 marks)

The farm sells the longest 20% of its Specimen trout to a restaurant.
(c) Find the length of the shortest trout that is sold to the restaurant, giving your answer to one decimal place.

(5 marks)
●●●●●Level 412 marksStart
The time, XX minutes, that a customer spends in a bookshop is normally distributed with mean mm minutes and variance 25

Given that

P(m−3k<X<m+3k)=0.8\mathrm{P}(m - 3k < X < m + 3k) = 0.8
(a) find the value of kk

(4 marks)

The time, YY minutes, that a customer spends in the café next to the bookshop is normally distributed with mean μ\mu minutes and standard deviation σ\sigma minutes.

Given that

5μ=12σandP(Y>42)=0.03595\mu = 12\sigma \qquad \text{and} \qquad \mathrm{P}(Y > 42) = 0.0359
(b) find the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 49 marksStart
The masses, in grams, of chocolate truffles made by a confectioner have the distribution N(14.2,0.64)\mathrm{N}(14.2, 0.64).
(a) Find the probability that a truffle chosen at random has a mass between 13.5 and 15.0 grams.

(1 mark)
(b) The heaviest 10% of truffles are sold as 'large'.

Find the minimum mass of a large truffle.

(1 mark)
(c) The truffles are packed in boxes, each containing 12 randomly selected truffles.

Find the probability that a box chosen at random has a total mass of more than 172 g.

(4 marks)
●●●●●Level 46 marksStart
A typing test measures a person's typing speed, WW words per minute. For one trainee, WW can be modelled by a normal distribution with mean 62 and standard deviation 8
(a) Using standardisation, show that the probability that the trainee types at a speed of less than 58 words per minute on a randomly chosen attempt is 0.3085

(2 marks)

Attempts with the highest 10% of the trainee's speeds are rated as "excellent".
(b) Use standardisation to find the lowest speed, in words per minute, that would be rated as excellent.

(3 marks)

To pass the typing test the trainee needs a speed of at least 58 words per minute. The trainee has up to 3 attempts at the test. Once the trainee passes, they do not use any of their remaining attempts. The speeds on different attempts are independent.

Given that the trainee passed the test,
(c) find the probability that the trainee passed on their second attempt with a speed of more than 72 words per minute.

(5 marks)
●●●●●Level 410 marksStart
A music streaming service takes a random sample of 160 songs from one of its playlists and records the length, in seconds, of each song.

The lengths of the songs in the sample are summarised in the following box plot.150200216248310Length (seconds)(a) Use linear interpolation to estimate the probability that a randomly chosen song from this sample is shorter than 232 seconds.

(2 marks)

The service takes the quartiles from this sample and decides to label any song whose length is at least Q3+1.5×(Q3−Q1)Q_3 + 1.5 \times (Q_3 - Q_1) as "extended".
(b) Find the shortest length of a song that the service would label as extended.

(1 mark)

An analyst suggests that the lengths of the songs on the playlist may be modelled by a normal distribution.
(c) Explain whether or not the box plot supports this suggestion.

(1 mark)

The analyst records the length of every song on the playlist and classifies any song whose length is more than 2.5 standard deviations from the mean as unusual.

Assuming that the lengths of the songs on the playlist may be modelled by a normal distribution,
(d) find the probability that a randomly selected song from the playlist would be classified as unusual.

(2 marks)

The mean length of the songs on the playlist is 226 seconds.

Given that a song of length 350 seconds is classified as unusual,
(e) find the maximum possible value of the standard deviation of the lengths of the songs on the playlist.

(2 marks)
●●●●●Level 48 marksStart

Finding μ and σ

33 questions

LessonNot started
X∼N(μ,σ2)X \sim N(\mu, \sigma^{2}) with P(X<10)=0.05P(X < 10) = 0.05 and P(X<70)=0.99P(X < 70) = 0.99.

Find μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
●●●●●Level 46 marksStart
A machine fills bottles with water. The volume of water in a bottle, XX ml, is modelled by X∼N(500,82)X \sim N(500, 8^2).
(a) Find P(X<490)P(X < 490). (2 marks)
(b) Find the probability that a randomly chosen bottle contains within 1010 ml of the mean. (3 marks)
(c) A crate contains 55 bottles, filled independently. Find the probability that every bottle in the crate contains at least 490490 ml. (3 marks)
(d) The machine is recalibrated: the mean stays at 500500 ml but the standard deviation, σ\sigma ml, changes. Find the largest value of σ\sigma for which P(X<490)≤0.02P(X < 490) \le 0.02. (4 marks)
●●●●●Level 412 marksStart
The time taken by a courier company to deliver a parcel, YY minutes, is modelled by a normal distribution with mean μ\mu and standard deviation σ\sigma.

Given that P(Y<22)=0.15P(Y < 22) = 0.15 and P(Y>35)=0.2P(Y > 35) = 0.2,
(a) find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(b) A parcel is known to have taken more than 2222 minutes to deliver. Find the probability that it took less than 3030 minutes. (3 marks)
(c) The company's records show that a small number of parcels take several hours to deliver. Explain why this suggests that the normal distribution may not be a suitable model for delivery times. (1 mark)
●●●●●Level 410 marksStart
The weights of apples from an orchard, WW grams, are modelled by a normal distribution.

It is known that 10%10\% of the apples weigh less than 120120 g, and 5%5\% of the apples weigh more than 190190 g.
(a) Find the value of the mean μ\mu and the value of the standard deviation σ\sigma, giving each answer to 33 significant figures. (6 marks)
(b) An apple is classed as large if it weighs more than 175175 g.

Find the probability that a randomly chosen apple is large, giving your answer to 33 significant figures. (2 marks)
(c) Find the weight that is exceeded by 25%25\% of the apples, giving your answer to the nearest gram. (3 marks)
●●●●●Level 411 marksStart
The time taken for a courier to deliver a parcel is modelled as X∼N(30,42)X \sim N(30, 4^2) minutes.
(a) Find P(X>35)P(X > 35). (2 marks)
(b) Find the time dd such that 10%10\% of deliveries take less than dd minutes. (3 marks)

A rival company's delivery times are modelled as Y∼N(μ,σ2)Y \sim N(\mu, \sigma^2). It is known that P(Y<20)=0.25P(Y < 20) = 0.25 and P(Y>32)=0.10P(Y > 32) = 0.10.
(c) Find the values of μ\mu and σ\sigma, giving your answers to 3 significant figures. (6 marks)
(d) Give one reason why a normal distribution might be a suitable model for delivery times, and one potential limitation. (2 marks)
●●●●●Level 413 marksStart
The mass, MM grams, of a loaf of bread produced by a bakery is modelled by a normal distribution with mean μ\mu and standard deviation σ\sigma. Records show that 10.56%10.56\% of loaves have a mass less than 780780 g and 3.04%3.04\% have a mass greater than 830830 g.
(a) Find the values of μ\mu and σ\sigma. (5 marks)

The bakery installs a new oven. A random sample of 4040 loaves baked in the new oven has a mean mass of 795.2795.2 g. It is assumed that the standard deviation of the masses is unchanged.
(b) Test, at the 5%5\% level of significance, whether the mean mass of loaves baked in the new oven is different from the value of μ\mu found in part (a). (4 marks)
(c) The manager says, \"The test proves that the new oven has not changed the mean mass.\" Explain why this statement is not justified. (1 mark)
●●●●●Level 410 marksStart
The time taken by a courier to complete a delivery round in a city, TT minutes, is modelled by a normal distribution with mean μ\mu and standard deviation σ\sigma. Records show that P(T<40)=0.0668\mathrm{P}(T < 40) = 0.0668 and P(T>58)=0.1587\mathrm{P}(T > 58) = 0.1587.
(a) Determine the values of μ\mu and σ\sigma. (4 marks)

The courier company introduces a route-planning app and claims that it reduces the time taken to complete a round. After the app is introduced, the times taken for a random sample of 3030 rounds are recorded. The standard deviation of the times is assumed to be unchanged.
(b) Stating your hypotheses clearly, find the critical region for a test of the company's claim at the 5%5\% significance level, giving your answer in terms of the sample mean Tˉ\bar{T}. (4 marks)
(c) The mean time for the sample of 3030 rounds is 48.248.2 minutes. Complete the test, stating your conclusion in context.

The company's analyst says that this result proves the app reduces the time taken to complete a round. Comment on the analyst's claim. (3 marks)
●●●●●Level 411 marksStart
A poultry farm supplies hens' eggs to a wholesaler.

The masses of the hens' eggs, MM grams, are normally distributed with mean 61.561.5 and standard deviation 4.24.2
(a) (i) Find P(M<55)P(M < 55)

(ii) Find P(M>70)P(M > 70)

(iii) The wholesaler only accepts hens' eggs with masses between 5353 grams and 7373 grams.

Find the proportion of hens' eggs that the wholesaler accepts.

(3 marks)
(b) The farm also supplies duck eggs to the wholesaler.

The masses of these duck eggs are normally distributed.

It is found that 12%12\% of these duck eggs have a mass less than 6464 grams.

It is found that 20%20\% of these duck eggs have a mass greater than 8080 grams.

Find the mean and standard deviation of the mass, in grams, of the duck eggs supplied by the farm.

(6 marks)
●●●●●Level 49 marksStart
(a) The lengths, LL centimetres, of runner beans of a certain variety are modelled by the distribution N(18.5,6.25)\mathrm{N}(18.5, 6.25).

Find P(15<L<20)\mathrm{P}(15 < L < 20).

(1 mark)
(b) The masses, in grams, of strawberries of a certain variety are modelled by the distribution N(μ,σ2)\mathrm{N}(\mu, \sigma^2).

You are given that 15% of these strawberries have masses greater than 27 g and 40% have masses less than 21 g.

Determine the values of μ\mu and σ\sigma. Give your answers correct to 2 decimal places.

(5 marks)
(c) The diagram shows the probability distribution of a normal variable, XX.246810121416182022x(i) Write down estimates of the xx-coordinates of the points of inflection on the graph.

(ii) Hence write down an estimate of the standard deviation of XX, explaining your method.

(2 marks)
●●●●●Level 48 marksStart
A factory makes rubber seals for washing machines.

The diameter of a seal, DD mm, is modelled as being normally distributed with mean 24.624.6 mm and standard deviation 0.350.35 mm

A seal is selected at random.
(a) Find the probability that this seal has a diameter of more than 25.225.2 mm

(1 mark)
(b) Find the probability that this seal has a diameter between 24.224.2 mm and 25.025.0 mm

(2 marks)

The same factory also makes a larger seal.

The random variable E∼N(μ,σ2)E \sim \mathrm{N}(\mu, \sigma^2) represents the diameter, in mm, of a larger seal.

Given that P(E<39.5)=0.02\mathrm{P}(E < 39.5) = 0.02 and P(E>42)=0.30\mathrm{P}(E > 42) = 0.30
(c) calculate the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 48 marksStart
The time, XX minutes, that a customer spends in a bookshop is normally distributed with mean mm minutes and variance 25

Given that

P(m−3k<X<m+3k)=0.8\mathrm{P}(m - 3k < X < m + 3k) = 0.8
(a) find the value of kk

(4 marks)

The time, YY minutes, that a customer spends in the café next to the bookshop is normally distributed with mean μ\mu minutes and standard deviation σ\sigma minutes.

Given that

5μ=12σandP(Y>42)=0.03595\mu = 12\sigma \qquad \text{and} \qquad \mathrm{P}(Y > 42) = 0.0359
(b) find the value of μ\mu and the value of σ\sigma

(5 marks)
●●●●●Level 49 marksStart
A bakery makes fruit bread. The number of raisins in the dough is modelled by a Poisson distribution with a mean of 99 raisins per 100100 g of dough.
(a) State one assumption, in context, that is needed for this model to be valid.

(1 mark)

A bread roll is made from 5050 g of the dough.
(b) Find the probability that the roll contains fewer than 22 raisins.

(2 marks)

A loaf is made from mm grams of the dough. When a normal approximation is used, the probability that the loaf contains at most 6767 raisins is 0.06680.0668
(c) Find the value of mm

(8 marks)
(d) Give a reason why the normal approximation used in part (c) is suitable.

(1 mark)
●●●●●Level 412 marksStart

Approximating a binomial with a normal

24 questions

LessonNot started
A manufacturer claims that 40%40\% of the customers who receive a free sample of its new product go on to buy the product. A marketing analyst believes that the true proportion is lower than this.

The analyst takes a random sample of 2525 customers who received a free sample, and records the number, XX, who went on to buy the product.
(a) Using a 5%5\% level of significance, find the critical region for a test of the analyst's belief. State your hypotheses clearly, and state the actual significance level of the test. (5 marks)
(b) In the sample, 66 of the 2525 customers bought the product. State the conclusion of the test in context. (2 marks)

The analyst then takes a second random sample of 300300 customers who received a free sample, and finds that 104104 of them bought the product.
(c) Assuming that the manufacturer's claim is true, use a suitable approximation to estimate the probability that 104104 or fewer customers in a random sample of 300300 buy the product. (4 marks)
(d) Comment on what your answer to part (c) suggests about the manufacturer's claim, and explain why the approximation used in part (c) is valid. (2 marks)
●●●●●Level 413 marksStart
At a fairground stall, each player pays to have one go at a game.

Each time a player has a go, the probability that the player wins a prize is 0.180.18, independently of all other goes.

Maya has 1616 goes at this game.

The random variable XX represents the number of prizes that Maya wins.
(a) Using a suitable distribution for XX, find

(i) P(X=4)\mathrm{P}(X = 4)

(ii) P(X<3)\mathrm{P}(X < 3)

(3 marks)

During one weekend there are 300300 goes at this stall.
(b) Use a normal approximation to estimate the probability that more than 6262 prizes are won during this weekend.

(3 marks)
●●●●●Level 46 marksStart