Binomial Distribution (Edexcel A Level Maths: Statistics): Exam Questions

Yuki selects 10 letters at random, one at a time with replacement, from the word

D E V I A T I O N

Find the probability that he selects the letter E at least 4 times.

A manufacturer of sweets knows that 8% of the bags of sugar delivered from supplier will be damp.

A random sample of 35 bags of sugar is taken from supplier .

Using a suitable model, find the probability that the number of bags of sugar that are damp is

(i) exactly 2

(ii) more than 3

A manufacturer makes metal hinges in large batches.

The hinges each have a probability of 0.015 of having a fault.

A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty.

The manufacturer's aim is for 95% of batches to be accepted.

Explain whether the manufacturer is likely to achieve its aim.

In the town of Wooster, Ohio, it is known that 90% of the residents prefer the locally produced Woostershire brand sauce when preparing a Caesar salad.  The other 10% of residents prefer another well-known brand.

 30 residents are chosen at random by a pollster.  Let the random variable  represent the number of those 30 residents that prefer Woostershire brand sauce.

 Suggest a suitable distribution for  and comment on any necessary assumptions.

Find the probability that

(i) 90% or more of the residents chosen prefer Woostershire brand sauce

(ii) none of the residents chosen prefer the other well-known brand.

The pollster knows that there is a greater than 97% chance of at least  of the 30 residents preferring Woostershire brand sauce, where  is the largest possible value that makes that statement true.

Find the value of .

A fair coin is tossed 20 times and the number of times it lands heads up is recorded.

Define a suitable distribution to model the number of times the coin lands heads up, and justify your choice.

Find the probability that the coin lands heads up 15 times.

For a jellyfish population in a certain area of the ocean, there is a 95% chance that any given jellyfish contains microplastic particles in its body.

State any assumptions that are required to model the number of jellyfish containing microplastic particles in their bodies in a sample of size  as a binomial distribution.

Using this model, for a sample size of 40, find the probability of

(i) exactly 38 jellyfish

(ii) all the jellyfish

having microplastic particles in their bodies.

Giovanni is rolling a biased dice, for which the probability of landing on a two is 0.25.  He rolls the dice 10 times and records the number of times that it lands on a two.  Find the probability that

(i) the dice lands on a two 4 times

(ii) the dice lands on a two 4 times, with the fourth two occurring on the final roll.

For cans of a particular brand of soft drink labelled as containing 330 ml, the actual volume of soft drink in a can varies.  Although the company’s quality control assures that the mean volume of soft drink in the cans remains at 330 ml, it is known from experience that the probability of any particular can of the soft drink containing less than 320 ml is 0.0296.

Tilly buys a pack of 24 cans of this soft drink.  It may be assumed that those 24 cans represent a random sample. Let represent the number of cans in the pack that contain less than 320 ml of soft drink.

 Write down the probability distribution that describes .

Find the probability that

(i) none of the cans

(ii) exactly two of the cans

(iii) at least two of the cans

contain less than 320 ml of soft drink.

The random variable .  Find:

(i)

(ii)

(iii)

(iv)

The random variable Find:

(i) the largest value of  such that  

(ii) the smallest value of  such that  

(iii) the largest value of such that  

In an experiment, the number of specimens testing positive for a certain characteristic is modelled by the random variable .  Find the probability of

(i) fewer than 20

(ii) no more than 20

(iii) at least 20

(iv) at most 20

(v) more than 20

of the specimens testing positive for the characteristic.

In a university 8% of students are members of the university dance club.

A random sample of 36 students is taken from the university.

The random variable represents the number of these students who are members of the dance club.

Using a suitable model for , find

(i)

(ii)

Only 40% of the university dance club members can dance the tango.

Find the probability that a student is a member of the university dance club and can dance the tango.

A random sample of 50 students is taken from the university.

Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango.

A machine fills packets with sweets and of the packets also contain a prize.

The packets of sweets are placed in boxes before being delivered to shops.

There are 40 packets of sweets in each box.

The random variable represents the number of packets of sweets that contain a prize in each box.

State a condition needed for to be modelled by

A box is selected at random.

Using find

(i) the probability that the box has exactly 6 packets containing a prize,

(ii) the probability that the box has fewer than 3 packets containing a prize.

Kamil’s sweet shop buys 5 boxes of these sweets.

Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize.

A nursery has a sack containing a large number of coloured beads of which 14% are coloured red.

Aliya takes a random sample of 18 beads from the sack to make a bracelet.

State a suitable binomial distribution to model the number of red beads in Aliya’s bracelet.

Use this binomial distribution to find the probability that

(i) Aliya has just 1 red bead in her bracelet,

(ii) there are at least 4 red beads in Aliya’s bracelet.

Comment on the suitability of a binomial distribution to model this situation.

A fair dice is rolled 24 times and the number of times it lands on a 4 is recorded.

Define a suitable distribution to model the number of times the dice lands on a 4, and justify your choice.

Find the probability that the dice lands on a ‘4’ four times.

For a population of squirrels in a certain area of woodland, there is a 92% chance that any given squirrel was born in that area of woodland.  Squirrels born in that area of woodland are referred to by researchers as being ‘local’.

State any assumptions that are required to model the number of local squirrels in a sample of size  as a binomial distribution.

Using this model, for a sample size of 50, find the probability of

(i) exactly 45 squirrels

(ii) all but one of the squirrels

being local.

Guglielma is rolling a biased dice, for which the probability of landing on a 5 is  She rolls the dice twenty times and records the number of times that it lands on a 5.  Find the probability that

(i) the dice lands on a ‘5’ four times

(ii) the dice lands on a ‘5’ four times, but the final ‘5’ does not occur on the final roll.

For bars of a particular brand of chocolate labelled as weighing , the actual weight of the bars varies.  Although the company’s quality control assures that the mean weight of the bars remains at 300 g, it is known from experience that the probability of any particular bar of the chocolate weighing between 297 g and 303 g is 0.9596.  For bars outside that range, the proportion of underweight bars is equal to the proportion of overweight bars.

 Millie buys 25 bars of this chocolate to hand out as snacks at her weekly Chocophiles club meeting.  It may be assumed that those 25 bars represent a random sample.  Let represent the number of bars out of those 25 that weigh less than 297 g.

Write down the probability distribution that describes .

The chocolate fanaticism of the club members means that no bars weighing less than 297 g can be handed out as snacks at their meetings.

Given that 24 people (including Millie) will be attending the meeting, find the probability that there will be enough bars to hand out to

(i) all

(ii) all but one, but not all

of the attendees.

After an incident where there were not enough chocolate bars weighing 297 g or more to hand out to all of a meeting’s attendees, Millie decides to reorganise the way she runs the meetings.  She will still only buy 25 of the chocolate bars each week, but she wants to reduce the number of attendees to make sure that she will have a certainty of at least 99.9% of being able to hand out a chocolate bar to every single attendee (including herself).

Work out the greatest number of attendees that a meeting will be able to have under this new system.

The random variable .  Find:

(i)

(ii)

(iii)

(iv)

The random variable Find:

(i) the largest value of such that  

(ii) the largest value of  such that 

(iii) the smallest value of  such that  

Abner, an American baseball fanatic, has just moved to a town in which it is known that 25% of the residents are familiar with the rules of the game.

 Abner takes a random sample of 40 residents of the town. Find the probability that

(i) fewer than 13

(ii) no more than 13

(iii) more than 13

(iv) at most 13 but at least 5

of the residents in Abner’s sample are familiar with the rules of baseball.

Abner would like to field a town team for an upcoming regional baseball tournament.  Abner is intending to play himself, but he needs to find enough other players to fill up the team.  As he does not yet know anyone in the town, he decides to take another random sample of residents in hopes of finding enough other players.  Only people familiar with the rules of baseball are able to be included in the team, but it may be assumed that anyone in the town familiar with the rules would also be willing to join Abner’s team.

Given that an American baseball team must have a minimum of nine players, find the smallest number of people that Abner should include in his sample in order to have at least a 90% chance of filling up his team.

In the town of Edinboro, Pennsylvania, a festival of trimmed below the forehead hairstyles is held every year, known as the Edinboro Fringe Festival.  It is known that 70% of the residents of the town are in favour of the festival because of the tourism revenue it brings in.  The other 30% of residents oppose the festival because of the sometimes hostile reactions of the large number of tourists who arrive every year thinking they had actually made bookings to attend another well-known fringe festival.

25 residents are chosen at random by a local newspaper reporter.  Let the random variable  represent the number of those 25 residents that are in favour of the festival.

Suggest a suitable distribution for and comment on any necessary assumptions.

Find the probability that

(i) 76% or more of the residents chosen are in favour of the festival

(ii) more of the residents chosen oppose the festival than are in favour of it.

The reporter knows that the chance of  or more of the 25 residents being opposed to the festival is less than 0.5%, where  is the smallest possible value that makes that statement true.

Find the value of .

In an experiment a group of children each repeatedly throw a dart at a target.

For each child, the random variable represents the number of times the dart hits the target in the first 10 throws.

Peta models as

State two assumptions Peta needs to make to use her model.

Using Peta’s model, find .

For each child the random variable represents the number of the throw on which the dart first hits the target.

Using Peta’s assumptions about this experiment, find .

Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models as

where is a constant.

Find the value of .

Using Thomas’ model, find .

Explain how Peta’s and Thomas’ models differ in describing the probability that a dart hits the target in this experiment.

Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.

One of the 184 days is selected at random.

Find the probability that it has a daily mean total cloud cover of 6 or greater.

Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution.

She uses the random variable to denote the daily mean total cloud cover and believes that .

Using Magali’s model,

(i)  find

(ii)  find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7.

Explain whether or not your answers to part (b) support the use of Magali’s model.

There were 28 days that had a daily mean total cloud cover of 8.

For these 28 days, the daily mean total cloud cover for the following day is shown in the table below.

Find the proportion of these days when the daily mean total cloud cover was 6 or greater.

Comment on Magali’s model in light of your answer to part (d).

Two fair dice are rolled and the numbers showing on the dice are added together.  This is done 18 times and the number of times the sum is not equal to 7 or 11 is recorded.

Define a suitable distribution to model the number of times the sum is not equal to 7 or 11, and justify your choice.

Find the probability that the sum of the two dice is not equal to 7 or 11 exactly fourteen times.

Researchers studying malaria in a certain geographical region know that there is an 80% chance of any given female mosquito in the region carrying the malaria parasite. 

State any assumptions that are required to model the number of female mosquitoes that carry the malaria parasite in a sample of  female mosquitoes as a binomial distribution.

Male mosquitoes do not bite humans and therefore are unable to transmit the malaria parasite to a human.  A female mosquito is only able to transmit the malaria parasite to a human if it is carrying the malaria parasite itself.

Given that 50% of the mosquitoes in the region are male, find the probability that in a random sample of six mosquitoes none of them are able to transmit the malaria parasite to a human. Give your answer as an exact value.

Maifreda is rolling a biased dice, for which the probability of landing on a prime number is  and the probability of landing on a square number is .  She rolls the dice twenty times and records the number of times that it lands on a 6.  Find the probability that

(i) the dice lands on a ‘6’ four times

(ii) the dice lands on a ‘6’ four times, but all of those sixes occur within the first rolls (where 420).

Your answer for (ii) should be given in terms of , in the form

where    is a binomial coefficient, and  and are constants to be found.

Although a particular manufacturer of academic gowns advertises the material of their gowns as being 93% silk, the actual silk content of the gowns varies.  Although the manufacturer’s quality control protocols assure that the mean percentage of silk in the gowns remains at 93%, it is known from experience that the probability of the silk content of any particular gown being between 90% and 95% is 0.9805.  For gowns falling outside that range, the probability that a gown contains less than 90% silk is exactly half the probability that a gown contains more than 95% silk.

Camford University has received an order of 100 gowns from the manufacturer.  It may be assumed that those  gowns represent a random sample.  Let  represent the number of gowns out of those 100 that have a silk content greater than 95%.

Write down the probability distribution that describes .

At an upcoming ceremony the university’s Department of Obfuscation is going to be awarding honorary degrees to four government statisticians.  The university prefers whenever possible to provide the recipients of such degrees with gowns containing more than 95% silk.

Out of the order of 100 gowns, find the probability that there will be enough gowns containing more than 95% silk to provide

(i) all

(ii) all but one (but not all)

(iii) less than half

of the honorary degree recipients with such a gown.

Due to a mix-up at the ceremony, the four honorary degree recipients are simply handed gowns at random from the order of 100 gowns.  It had previously been determined that exactly one of the 100 gowns in the order contained less than 90% silk, and the university is worried that if one of the honorary degree recipients received that gown then the university’s government grant funding will be cut.

Work out the probability that one of the honorary degree recipients received the gown containing less than 90% silk.

The random variable .  Find:

(i)

(ii)

(iii)

The table below contains part of the cumulative distribution function for the random variable  :

The random variable  is defined in terms of  as ,  while the random variable  .                                            

Using the table above, and showing your working, find:

(i) the smallest value of q such that  

(ii) the largest value of  such that  

(iii) the smallest value of  such that  .

After falling asleep while reading his A Level mathematics textbook, Gwion awakens to find that he has been transported in his sleep to the magical kingdom of Statistica.  As every statistics student knows, the kingdom of Statistica has a very large population (it has been reputed to be nearly infinite), and the chance that any given resident of the kingdom will welcome a newcomer with tea and cakes is 97%.

Gwion takes a random sample of 50 residents of the kingdom. Find the probability that of those 50 residents

(i) all 50

(ii) no more than 46

(iii) more than 25 but at most 49

(iv) at least 40 but fewer than 46

will welcome Gwion (who is a newcomer) with tea and cakes.

Luckily for Gwion, all 50 residents in his first sample welcome him with tea and cakes.  From one of them, however, Gwion learns an unsettling fact: those residents of the kingdom who will not welcome a newcomer with tea and cakes, will instead insist on making the newcomer sit a five-hour statistics mock exam paper.  Even worse, each such resident will insist on making the newcomer sit a different five-hour statistics mock exam paper

Gwion wants to take another random sample of residents of the kingdom, in hopes that one of them will be able to advise him on how to get home.  However he does not want to have to sit more than one five-hour statistics mock exam paper.

Work out the size of the largest random sample that Gwion can take such that he will have a greater than 95% chance of not having to sit more than one five-hour statistics mock exam paper.

In Surry County, North Carolina, local farmers and agricultural equipment suppliers gather each year to celebrate at the Surry Slurry Fest.  It is known that 80% of the residents of the county are opposed to the Slurry Fest because of the mess it leaves behind on local roads, fields and government buildings.  The other 20% of residents are in favour of the Slurry Fest because it is (according to them) “one heck of a good ol’ time”.

An organiser of the rival Surry ♥ Curry Not Slurry food festival is attempting to gather evidence to support his campaign to have the Surry Slurry Fest banned.  He selects 25 county residents at random in order to poll them about their opinions on the Slurry Fest.  Let the random variable  represent the number of those 25 residents that are opposed to the Slurry Fest.

Suggest a suitable distribution for  and comment on any necessary assumptions.

Find the probability that

(i) 90% or more of the residents chosen are opposed to the Slurry Fest

(ii) a majority of the residents chosen are in favour of the Slurry Fest.

Before he is able to conduct his poll of the selected residents, the food festival organiser is interviewed by a local newspaper.  He would like to be able to predict with at least 90% certainty that not more than a given percentage of the 25 residents selected for the poll will be in favour of the Slurry Fest.

Given that the organiser would like his prediction to support his anti-Slurry Fest campaign in the strongest manner possible, determine the ‘given percentage’ that he should quote to the newspaper.