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 $A$ will be damp.

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

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.

For a jellyfish population in a certain area of the ocean, 95% of the jellyfish contain microplastic particles in its body.

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

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

(i) exactly 38 jellyfish have microplastic particles in their bodies,

(ii) at least 36 jellyfish have microplastic particles in their bodies.

A snowboarder is trying to perform the Poptart trick. The snowboarder has a success rate of 25% of completing the trick.

The snowboarder will model the number of times they can expect to successfully complete the Poptart trick, out of their next 12 attempts, using the random variable $X~\mathrm{B}(12,0.25).$

Suggest a reason why the binomial model may not be suitable in this case.

Using the model, find the probability that the snowboarder

(i) successfully completes the Poptart trick more than 3 times in their next 12 attempts

(ii) fails to successfully complete the trick on any of their next 12 attempts.

Give your answers to three significant figures.

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 $L$ represent the number of cans in the pack that contain less than 320 ml of soft drink.

 Write down the probability distribution that describes $L$.

Find the probability that exactly two of the cans contain less than 320 ml of soft drink.

Find the probability that at least two of the cans contain less than 320 ml of soft drink.

For the random variable $X~\mathrm{B}(40,0.15)$,  find:

(i) $\mathrm{P}(X<10)$

(ii) $\mathrm{P}(X\ge 6)$

(iii) $\mathrm{P}(2<X\le 13)$

In an experiment, the number of specimens testing positive for a certain characteristic is modelled by the random variable $X~\mathrm{B}(50,0.35)$.  Find the probability of

(i) fewer than 20

(ii) no more than 20

(iii) at least 20

(iv) more than 20

of the specimens testing positive for the characteristic.

A fair six-sided spinner has sides numbered 1, 2, 3, 4, 5 and 6.

The spinner is spun once and the score of the side it lands on is recorded.

Write the name of the distribution that can be used to model the score of the side it lands on.

The spinner is spun 30 times.

The random variable $X$ represents the number of times the spinner lands on 5. Find the probability that the spinner lands on 5 at least 6 times.

Find $\mathrm{P}\left(3\le X\le 8\right)$.

The random variable $X~\mathrm{B}(50,0.3)$.  Find:

(i) $\mathrm{P}(X>20)$

(ii) $\mathrm{P}(7\le X<16)$

(iv) $\mathrm{P}(X<8\mathrm{or}X>16)$

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 $X$ represents the number of these students who are members of the dance club.

Using a suitable model for $X$, find

(i) $\mathrm{P}\left(X=4\right)$

(ii) $\mathrm{P}\left(X\ge 7\right)$

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 $\frac{1}{7}$ 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 $T$ represents the number of packets of sweets that contain a prize in each box.

State a condition needed for $T$ to be modelled by $\mathrm{B}\left(40,\frac{1}{7}\right)$

A box is selected at random.

Using $T~\mathrm{B}\left(40,\frac{1}{7}\right)$ 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 manufacturer produces light bulbs. It is known that 5% of the bulbs are defective. A quality control officer takes a random sample of 50 bulbs.

State two assumptions required to model the number of defective bulbs, $D$, using a binomial distribution $\mathrm{B}(50,0.05)$.

Find the probability that

(i) exactly 2 bulbs are defective,

(ii) at least 3 bulbs are defective.

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.

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 the dice lands on a two 4 times.

Find the probability that the dice lands on a two 4 times, with the fourth two occurring on the final roll.

Bars of a particular brand of chocolate are labelled as weighing 300 g. The actual weight of the bars varies.  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 leads weekly Chocophiles club meetings. She 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 $U$ represent the number of bars out of those 25 that weigh less than 297 g.

Write down the probability distribution that describes $U$.

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.

There are 24 club members in total at a weekly meeting. Find the probability that Millie has enough chocolate bars that can be handed out to all 24 members.

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 member.

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

In the town of Edinboro, Pennsylvania, a festival of 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.

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

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

Find the probability that there are more residents in the sample that oppose the festival than residents in the sample that are in favour of the festival.

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

Find the value of $k$.

In each round of a game, two fair six-sided dice numbered 1 to 6 are rolled and the numbers showing on the dice are added together.  The player wins a point in a round if the sum of the two numbers is greater than 7.

A player plays 10 rounds of the game. Find the probability that the player wins no more than 5 points.

If a player earns a star if they win more than 5 points in 10 rounds of the game. Four friends each play 10 rounds of the game. Find the probability that at least one of them earns a star.

For the random variable $X~\mathrm{B}(40,0.25)$, find:

(i) the largest value of $k$ such that  $\text{P}(X<k)<0.10$

(ii) the smallest value of $r$ such that  $\mathrm{P}(X\ge r)<0.05$

(iii) the largest value of $s$ such that  $\mathrm{P}(X>s)>0.95.$

For the random variable $X~\mathrm{B}(50,0.75)$, find:

(i) $\mathrm{P}(30\le X<40)$

(ii) $\mathrm{P}(X\le 29\mathrm{or}X>39)$

Find $\mathrm{P}\left(X\ge 30|X<40\right)$.

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

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

Peta models $H$ as $\mathrm{B}(10,0.1)$

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

Using Peta’s model, find $\mathrm{P}\left(H\ge 4\right)$.

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

Using Peta’s assumptions about this experiment, find $\mathrm{P}\left(F=5\right)$.

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 $\mathrm{P}(F=n)$ as

$\mathrm{P}(F=n)=0.01+(n-1)\times \alpha$

where $\alpha$ is a constant.

Find the value of $\alpha$.

Using Thomas’ model, find $\mathrm{P}(F=5)$.

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 $X$ to denote the daily mean total cloud cover and believes that $X~\mathrm{B}(8,0.76)$.

Using Magali’s model,

(i)  find $\mathrm{P}\left(X\ge 6\right)$

(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).

Guglielma is rolling a biased dice, for which the probability of landing on a 5 is $\frac{2}{11}.$ She rolls the dice twenty times and records the number of times that it lands on a 5.  Find the probability that the dice lands on a 5 at least four times.

Given that the dice lands on a 5 at least four times, find the probability that the dice does not land on a 5 in the first three rolls.

The table below contains part of the cumulative distribution function for the random variable $R~\mathrm{B}(30,p)$, where $p$ is an unknown constant.

Using the table above, find $\mathrm{P}\left(X=13\right)$.

You do not need to find the value of $p$.

$S$ and $T$ are random variables defined as:

• $S=30-R$

• $T~\mathrm{B}(30,1-p)$

Using the table above in part (a), find:

(i) the smallest value of $a$ such that  $\mathrm{P}(R>a)<0.21$

(ii) the largest value of $b$ such that  $\mathrm{P}(S>b)>0.93$

(iii) the smallest value of $c$ such that  $\mathrm{P}(T<c)>0.988$.

You do not need to find the value of $p$.

92% of squirrels in a population were born in that area of woodland. Squirrels born in that area of woodland are referred to by researchers as being local.

A sample of 50 squirrels from that area is taken. State two assumptions that are required to model the number of local squirrels in the sample as a binomial distribution.

Using a binomial model, find the probability that

(i) exactly 45 squirrels are local,

(ii) at least 45 squirrels are local.

Given that at least 45 squirrels in the sample are local, find the probability that fewer than 46 squirrels in the sample are local.

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.

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  $X$ represent the number of those 25 residents that are opposed to the Slurry Fest.

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

Find the probability that fewer than five residents are in favour of the festival.

Given that fewer than five residents are in favour of the festival, show that there is more than a 99% chance that there is at least one resident in favour of the festival.

A manufacturer produces gowns for university students. It is known from experience that 1.3% of the gowns from this manufacturer contain 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 $W$ represent the number of gowns out of those 100 that contain more than 95% silk.

(i) Write down the probability distribution that describes $W$.

(ii) The probabilities given by this model are the terms of the binomial expansion of an expression of the form ${\left(a+b\right)}^n$. Write down this expression, using appropriate values of $a$, $b$ and $n$.

(iii) Find the expected value for the number of gowns in the order that contain more than 95% silk.

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.

Find the probability that all four of the government statisticians can be provided with gowns containing more than 95% silk from the order of 100 gowns.

Due to a mix-up at the ceremony, the four government statisticians receiving honorary degrees are handed gowns at random from the order of 100 gowns.  It is revealed that exactly four of the 100 gowns in the order contain less than 90% silk.

Let $S$ be the number of the four government statisticians receiving honorary degrees that receive a gown containing less than 90% silk.

Explain why $S$ should not be modelled by a binomial distribution.

Find $\mathrm{P}\left(S=1\right)$.