3.2 Binomial & Geometric 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.

The random variable .

Find:

Find:

Find .

Before Bucky can join the superhero club, he has to pass the superhero examination. Bucky can take the examination as many times as he likes but he only needs to pass once. It is modelled with the probability of him passing the examination on any one attempt being 0.16 and Bucky passing the examination is assumed to be independent of all previous attempts. 

Find the probability that:

Find the expected number of attempts Bucky will need to pass.

Give a criticism of the model.

The random variable  .

Find the value of when 

Find the possible values of  when 

The random variable 

Find the value of   when 

Find the possible values of  when .

Guglielma is rolling a biased dice, for which the probability of landing on a 5 is  .  

Find the probability that Guglielma rolls a 5 for the first time on the 21^st roll.

Guglielma 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.

 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

of the attendees.

Abner, an American baseball fanatic, has just moved to a town in which it is known that 85% 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.

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.

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.

Derek is watching cars pass his window. He knows there’s a 25% chance that a passing car will be blue. 

Find the probability that the first blue car Derek sees is the 8^th car to pass his window.

Find the probability that Derek will see a blue car within the first 8 cars that pass his window.

Find the expected number of cars to pass Derek’s window until he sees his first blue car.

The random variable 

Given that , find the value of .

Find:

 

The random variable 

Given that , find the value of .

Find:

Giovanni is rolling a biased dice, for which the probability of landing on a two is 0.25.  

Find the probability that Giovanni does not roll a two until his 11^th roll.

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.

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

contain less than 320 ml of soft drink.

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

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.

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.

David is responsible for market research at a company which specialises in revision materials for students. One day, David stands outside a library and attempts to ask market research questions to people who walk past him. It is modelled with being a 5% chance that a person stops, and it is assumed that a person’s decision to stop is independent of all previous people.

Find the expected number of people to walk past without stopping before someone stops.

Find the probability that:

Given that the first 20 people walk past without stopping, find the probability that at least one person stops out of the first 100 people who walk past.

Find the probability that the 10^th person and the 20^th person to walk past are the first two people to stop.

Find the minimum number of people needed so that there is at least a 99.9% chance that at least one person stops.

and  are independent geometric random variables with parameter .

The random variable 

Find the largest value for  such that .

Maifreda is rolling a biased dice which is numbered 1 to 6. The probability of landing on a prime number is  and the probability of landing on a square number is . 

Find the expected number of dice rolls it takes before the dice first lands on a 6.

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.

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.

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 q such that  

the largest value of  such that  

the smallest value of  such that  .