1.1 Binomial Distribution

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

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

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.

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

Write down the probability distribution that describes .

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

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:

.

The random variable .  Find:

the largest value of  such that 

the largest value of  such that 

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

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

Abner asks random residents, who know the rules, whether they would like to join his baseball team. There’s an 80% chance that the resident will join his team.

Given that Abner needs at least 5 more players, find the smallest number of people that Abner should ask 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

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 .

The random variable . 

Find:

.

Find:

Find .

The random variable . 

Find the value of when .

Find the value of when 

Find the possible values of when .

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

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

For cans of a particular brand of soft drink labelled as containing , 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  of soft drink.

Write down the probability distribution that describes .

Find the probability that

contain less than 320 ml of soft drink.

The random variable . Find:

.

The random variable .  Find:

the largest value of  such that 

the smallest value of  such that

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

of the specimens testing positive for the characteristic.

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

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 .

On any given day during a normal five-day working week, there is a 60% chance that Yussuf catches a taxi to work. 

Find , the expected number of times Yussuf will catch a taxi to work during a normal five-day working week.

Find the probability that, during a normal five-day working week, Yussuf never catches a taxi.

Find the probability that, during a normal five-day working week, Yussuf catches a taxi once at the most.

The random variable . 

Given that , find the value of .

Find:

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

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

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

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

Zara is a gymnast. It is known that she has a 20% chance of making a mistake in any given routine.

Zara performs ten routines in a competition.

Find the probability that Zara makes a mistake in:

Given that Zara makes a mistake in at least 2 of her routines, find the probability that she makes a mistake in exactly 3 of her routines.

Find the probability that the number of routines in which Zara makes a mistake is less than one standard deviation away from the mean.

The random variable . Find:

.

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

	5	6	7	8	9	10	11	12
	0.0011	0.0040	0.0121	0.0312	0.0694	0.1350	0.2327	0.3592

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

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

the smallest value of such that 

the largest value of such that 

the smallest value of  such that  .

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

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.

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

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.