1.5 Working with Distributions

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A cake recipe calls for a certain amount of flour to be used. The random variable  is the number of cakes that can be made, following the recipe exactly each time, from a bag containing a random amount of flour.

A student cuts a one-metre length of rope into two pieces at a random point. The random variable  is the difference in length between the two pieces of rope that result.

People are chosen at random from the UK population. The random variable  is the age of a randomly selected person, measured from their date and time of birth.

Each of the following histograms shows the distribution of results when a large number of measurements of the random variables or  are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.  Where a normal model is appropriate, suggest a real-world variable that might show such a distribution.

The random variable .

Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the normal approximation, find the largest value of  (where  is an integer) such that  .

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution .

On a European-style casino roulette wheel, the probability of the ball landing on a red number is .

The wheel is spun 36 times, and the ball lands on a red number  times.

Find .

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands on a red number, is recorded.

Write down the normal distribution that could be used to approximate the distribution of .

Use the distribution from (c)(ii) to approximate the probability that in 1000 spins the ball lands on a red number either less than 482 times or more than 491 times.

Due to a manufacturing irregularity, 41% of Adventure Dude action figures were produced with two left hands.  Although not especially rare, and therefore not especially collectible, these so-called ‘double left’ figures are nonetheless considered to be collector’s items by hard-core Adventure Dude fanatics.

A vintage toy shop has obtained 100 Adventure Dude action figures.  These may be assumed to represent a random sample.

Find the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer to 6 decimal places.

Use an appropriate normal approximation to approximate the probability that exactly 45 of the 100 figures are ‘double left’ figures.

Find the percentage error when using your normal approximation from part (b) to estimate the probability that exactly 45 of the 100 figures are ‘double left’ figures. Give your answer correct to two decimal places.

Fleur is a biology student researching daisies on a particular field. She randomly selects a square of field with length one metre and records the number of daisies in that area, denoted by the random variable . Fleur models  using a Poisson distribution with mean 5.

State the assumptions needed so that  follows a Poisson distribution.

Find the probability that there are more than 2 daisies in a randomly selected square with length one metre.

Using an approximating distribution, find the probability that the number of daisies, in a randomly selected square with length 6 metres, is between 175 and 190 inclusive.

Remy has a biased coin with the probability of it landing on tails being 0.05.

Using a Poisson approximation, find the probability that the coin lands on tails more than 4 times when Remy flips it 60 times.

Remy flips the coin 2000 times.

Using a normal approximation, find the probability that the coin lands on tails fewer than 120 times when Remy flips it 2000 times.

The random variable 

Write down the normal distribution which can be used to approximate . Explain why a normal distribution is suitable to approximate .

Using the normal approximation, find:

Using the normal approximation, find the largest integer value of  such that .

The random variable  Given that, use an approximating distribution to find the value of .

The random variable .

Given that and correct to 3 decimal places, use a suitable approximation to find the value of .

Given instead that and correct to 3 decimal places, use a suitable approximation to find the value of .

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A student cuts a one-metre length of rope into two pieces at a random point. The random variable  is the length of the shorter of these two pieces.

You ask a sample of students in your school about their preferences for after-school activities. The random variable  is the number of students who say they prefer participating in lawn bowling.

People are chosen at random from the UK population. The random variable  is the age of a randomly selected person, defined as the age which they turned on their most recent birthday.

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables – or – are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable.

For each of the following binomial random variables, :

• state, with reasons, whether  can be approximated by a normal distribution
• if appropriate, write down the normal approximation to  in the form  , giving the values of  and .

The random variable .

Explain why a normal distribution can be used to approximate .

Find, using the normal approximation:

.

Using the normal approximation, find the largest value of  such that .

Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution .

On a casino roulette wheel, the probability of the ball landing on a black number is  .

The wheel is spun 30 times, and the ball lands on a black number  times.

Find .

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands on a black number, is recorded.

Write down the normal distribution that could be used to approximate the distribution of .

Use the distribution from (c)(ii) to approximate the probability that in at least one half of the 1000 spins the ball lands on a black number.

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a zombie toy as a prize.  A random sample of 100 packets is taken.

Find the probability that exactly 49 of the packets contain a prize. Give your answer to 6 decimal places

Write down the normal distribution that could be used to approximate the distribution for the number of the 100 packets that contain a prize.

Dominique owns a grocery store, and she models the number of customers entering the store during a 10-minute period using a Poisson distribution with mean 7.

State the assumptions that Dominique has made by using a Poisson distribution to model the number of customers entering her store during a 10-minute period.

Find the probability that no more than 4 customers enter the store during a 10-minute period.

Explain why a normal distribution can be used to approximate the distribution for the number of customers entering the store in a one-hour period. State the appropriate normal distribution.

Using the normal approximation, find the probability that more than 50 customers enter the store within a one-hour period.

In England, it is known that 5% of the population have ginger hair. Kenneth, the owner of a hairdressing salon, has 40 appointments available each day. He models the number of clients with ginger hair that attend his salon in a day using a binomial distribution.

State the assumptions that Kenneth has made by using a binomial distribution.

Assuming all 40 appointments are filled, find the probability that at least one person has ginger hair.

Assuming all 40 appointments are filled each day, the number of clients with ginger hair attending an appointment over a two-day period is denoted . Explain why a Poisson distribution can be used to approximate . State the appropriate Poisson distribution.

Using the Poisson distribution, find the probability at most 5 clients with ginger hair will have an appointment at the salon over the two-day period.

The random variable is approximated by .

Explain why this approximation is valid and state the values of .

Hence find the percentage error when a Poisson distribution is used to approximate

The random variable is approximated by .

Explain why this approximation is valid and state the values of  and .

Hence find the percentage error when a normal distribution is used to approximate 

Charlotte, an online tutor, receives messages from students at an average rate of 4 per half an hour.

State the assumptions that are needed in order to use a Poisson distribution to model the number of messages Charlotte receives from students during a fixed period of time.

Find the probability that Charlotte receives no fewer than 2 messages in a half-hour period.

Charlotte works as an online tutor for 4 hours per day and works 5 days a week.

Using an approximating distribution, find the probability that Charlotte receives more than 30 messages during a 4-hour working day.

Hence find the probability that Charlotte receives more than 30 messages in a day for exactly 2 days in a 5-day working week.

The random variable

Write down the normal distribution which can be used to approximate .

Using the normal approximation, find:

Using the normal approximation, find the largest integer value of  such that .

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

A string collector decides to measure all the pieces of string in his collection. The random variable  is the length of a randomly chosen piece of string from the collection, rounded to the nearest centimetre.

A random sample of 1000 people is chosen from the US population, and the number of siblings each has is recorded. The random variable  is the mean number of siblings for the 1000 people in the sample.

The masses of individual eggs in the nests of a particular species of bird are measured and recorded. The random variable  is the total mass of the eggs in a randomly chosen nest.

Each of the following histograms shows the distribution of results when a large number of measurements of the specified random variables – or  – are made.  In each case, state with a reason whether a normal distribution would be appropriate for modelling the random variable, and suggest a real-world variable that might show such a distribution.

The random variable .

Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

Using the appropriate normal approximation, find the smallest value of (where )  such that  .

On the roulette wheel at the Dunes Oddstacker casino, the probability of the ball landing on a red number is  , and the probability of the ball landing on a black number is the same.  In general, the majority of bettors will lose their bets if the ball lands neither on a red number nor on a black number.

The wheel is spun 50 times, and the ball lands on a number that is either red or black  times.

Find .

In a separate experiment, the wheel is spun 1000 times and , the number of times the ball lands neither on a red number nor on a black number, is recorded.

Explain why a normal approximation would be appropriate in this case.

Use an appropriate normal distribution to approximate the probability that in 1000 spins the number of times the ball lands neither on a red number nor on a black number is neither 80 or more nor less than 75.

Due to a production error, 58% of Bobbie Sue dolls were manufactured with proportions that might be seen on a real human being.  Plastic surgeons have been buying up these so-called ‘realie’ dolls, out of a concern that if too many of them are seen by the general public then numbers of people seeking plastic surgery will decrease.

A toy shop has received an order of 100 Bobbie Sue dolls.  These may be assumed to represent a random sample.

Calculate the percentage error when an appropriate normal approximation is used to calculate probability that exactly 58 of the 100 dolls are ‘realie’ dolls.

Jacob, the owner of a café, knows that the probability of a customer ordering a latte is 30% and the probability of a customer ordering a green macchiato is 3%. A random sample of 100 customers is observed over a week for research purposes.

Use suitable approximations, justifying your choices, to estimate the probability that in this sample

at least one quarter of customers order a latte,

no more than 2 customers order a green macchiato.

By considering the mean and variance of the distribution , explain why a Poisson distribution should only be used as an approximation when the value of  is small.

From data regarding previous cohorts at a prestigious college, it is known that there is a 99% chance that a student, who attends the college, completes their assignments on time.

In total, there are 3000 students studying at the college. Using a normal approximation, find the probability that fewer than 2980 students complete their assignments on time. State any assumptions that are needed.

There are 250 people studying mathematics at the college. By first defining a relevant binomial distribution with a small value of , use a Poisson approximation to find the probability that no fewer than 249 of the students complete their assignments on time.

The random variable

Explain why a normal distribution can be used to approximate .

Find, using the appropriate normal approximation:

.

Find the smallest integer value of  such that

Alzena works as an IT technician for a large computer company. Problems arise at an average rate of 8.4 per hour. If more than 90 problems arise during an 8-hour shift then she goes home and searches for a new job, otherwise she goes home happy.

Use an approximating distribution to show that the probability that Alzena looks for a new job after an 8-hour shift is roughly 0.2%.

Justify the use of your approximating distribution in part (a).

In a year, Alzena works 250 8-hour shifts. You may assume that these shifts are independent of each other and that the probability of Alzena looking for a new job after an 8-hour shift is 0.2%. Use a suitable approximation to find the probability that Alzena searches for a new job, after an 8-hour shift, more than 3 times in a year.

Justify the use of your approximating distribution in part (c).

It is known that 0.75% of the US population have the blood type of AB negative. During one week, a hospital receives 200 blood donations from different people.

Using a suitable approximation, find the probability that at least 5 out of the 200 donations will contain AB negative blood.

Justify the use of the approximating distribution used in part (a).

Explain why a binomial distribution may not be appropriate to model the number of the 200 blood donations that contain AB negative blood.

Mick runs a business selling cages for chinchillas. He receives orders from customers at an average rate of 6 per week.

At the beginning of a particular week, Mick has three cages in stock and is unable to obtain any more that week. Find the probability that he will be able to fulfil all orders from customers that week.

Mick visits his supplier to buy more cages. His supplier tells Mick that she is going away on holiday so this will be Mick’s last restock for 40 days. He wants to buy enough stock so that there is at least a 90% chance that he will be able to fulfil all orders over the next 40 days. It is known that Mick receives orders seven days a week.

Use a suitable approximation to find the smallest number of cages that Mick should buy from his supplier.

In the magical land of Attiland, it is known that the number of raindrops that land in a specific area of ground in one second follows a Poisson distribution. Karnac, the researcher of Attiland, draws a circle on the ground with a radius of one metre. Karnac notices that rain falls in that area at an average rate of 12 raindrops per second. The mean number of raindrops landing within a circle on the ground is proportional to the area of the circle.

Using an approximating distribution, find the radius of the circle that should be drawn on the ground so that there is a 99% chance of more than 1000 raindrops landing in the circle within one second.