Working with Distributions (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

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 $G$. It can be assumed that clients on each day are independent of each other. Explain why a Poisson distribution can be used to approximate $G$. 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 $X~B(60,0.08)$ is approximated by $Y~Po\left(\lambda \right)$.

Explain why this approximation is valid and state the value of $\lambda$.

(i) Find $P(X=4)$, giving your answer to 6 decimal places.

(ii) Find $P(Y=4)$, giving your answer to 6 decimal places.

Hence find the percentage error when a Poisson distribution is used to approximate $P(X=4)$.

The random variable $X~Po\left(40\right)$ is approximated by $Y~N(\mu ,{\sigma }^2)$.

Explain why this approximation is valid and state the values of $\mu \mathrm{and}\sigma$.

(i) Find $P(37<X\le 39)$, giving your answer to 6 decimal places.

(ii) Use $Y$ to find an approximate value for $P(37<X\le 39)$, giving your answer to 3 significant figures.

Hence find the percentage error when a normal distribution is used to approximate $P(37<X\le 39)$.

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 $S~Po\left(75\right)$.

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

Using the normal approximation, find:

(i) $P(60\le S<80)$

(ii) $P(S\le 85)$

Using the normal approximation, find the largest integer value of $s$ such that $P(S\le s)<0.1$

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 $D$. Fleur models $D$ using a Poisson distribution with mean 5.

State the assumptions needed so that $D$ 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 $T~Po\left(200\right)$.

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

Using the normal approximation, find:

(i) $P(180<T\le 230)$

(ii) $P(T>175)$

Using the normal approximation, find the largest integer value of $t$ such that $P(200-t<T<200+t)<0.5$

A shop manager buys a box of 100 small chocolate bars. The mass, in grams, of a chocolate bar can be modelled using $N(85,1.5^2)$.

Find the probability that a randomly selected chocolate bar is less than 81 g.

Explain why a Poisson distribution can be used as an approximation for the number of chocolate bars in the box of 100 that weigh less than 81 g.

Using a suitable approximation, find the probability that more than one chocolate bar in the box weighs less than 81 g.

The random variable $X~Po\left(\lambda \right)$. Given that $P(X\ge 100)=0.999$, use an approximating distribution to find the value of $\lambda$.

The random variable $Y~B(n,p)$.

(i) Given that $p=0.00025$ and $P(Y>0)=0.847$ correct to 3 decimal places, use a suitable approximation to find the value of $n$.

(ii) Given instead that $p=0.6$ and $P(Y>450)=0.486$ correct to 3 decimal places, use a suitable approximation to find the value of $n$.

In the town of Ozbourne, there is a 2.5% chance that a resident is secretly a goblin. Harry is hunting goblins and captures a group of 200 residents. It can be assumed that these 200 residents form a random sample.

Using a binomial distribution, find the probability that exactly 5 of the residents that Harry captures are goblins. Give your answer to 6 decimal places.

By calculating percentage errors, determine whether a Poisson or normal approximation is more accurate when used to approximate the probability in part (a).

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

(i) At least one quarter of customers order a latte,

(ii) No more than 2 customers order a green macchiato.

By considering the mean and variance of the distribution $B(n,p)$, explain why a Poisson distribution should only be used as an approximation when the value of $p$ 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 $p$, use a Poisson approximation to find the probability that no fewer than 249 of the students complete their assignments on time.

The random variable $W~Po\left(105\right)$.

Explain why a normal distribution can be used to approximate $W$.

Find, using the appropriate normal approximation:

(i) $P(80<W<100)$

(ii) $P\left(\right|W-105|\ge 15)$

Find the smallest integer value of $w$ such that $P(10+w<W<200-w)<0.9$

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.