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

Dominique owns a grocery store. She models the number of customers entering the store during a 10-minute period by the distribution $\text{Po}\left(7\right)$.

State two assumptions that are required for the Poisson model to be valid in this context.

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

Explain why a normal distribution can be used to approximate the number of customers entering the store in a 1-hour period, and state the parameters of this normal distribution.

Use this approximating distribution to find the probability that more than 50 customers enter the store in a randomly chosen 1-hour period.

It is known that 5% of the population of a country have ginger hair. Kenneth, the owner of a hairdressing salon, has 40 appointments available each day. Assume that all 40 appointments are filled each day, by 40 different clients chosen at random from the population.

State the distribution of the number of clients with ginger hair on one day, giving the values of any parameters.

Find the probability that at least one client on a randomly chosen day has ginger hair.

The number of clients with ginger hair over a two-day period is denoted by $G$.

State, with a justification, a suitable approximating distribution for $G$, giving the values of any parameters.

Use your approximating distribution to find the probability that at most 5 clients with ginger hair attend the salon over the two-day period.

The random variable $X$ has the distribution $\text{Po}\left(40\right)$.

State a reason why $X$ can be approximated by a normal distribution $Y~\text{N}(\mu ,{\sigma }^2)$, and state the values of $\mu$ and $\sigma$.

(i) Find $\text{P}(37<X\le 39)$.

(ii) Use $Y$ to find an approximate value for $\text{P}(37<X\le 39)$.

A random variable $S$ has the distribution $\text{Po}\left(75\right)$.

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

Use the normal approximation to find

(i) $\text{P}(60\le S<80)$

(ii) $\text{P}(S\le 85)$.

Use the normal approximation to find the largest integer value of $s$ such that $\text{P}(S\le s)<0.1$.

Jacob, the owner of a café, knows that customers arrive at the café at random times, at a constant average rate of 7.5 per hour. He also knows that 3% of customers order a green macchiato.

Use suitable approximating distributions, justifying your choices, to estimate the probability that

(i) at least 25 customers arrive at the café during a randomly chosen 4-hour morning,

(ii) in a random sample of 100 customers, no more than 2 order a green macchiato.

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

Use a suitable approximating distribution to find the probability that at least 5 of the 200 donations contain AB negative blood.

Justify the use of your approximating distribution 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.

Charlotte, an online tutor, receives messages from students at an average rate of 4 per half an hour. She models the number of messages she receives in a fixed period of time using a Poisson distribution.

State two assumptions that are required for the Poisson model to be valid in this context.

Find the probability that Charlotte receives at least 2 messages in a randomly chosen half-hour period.

Charlotte works as an online tutor for 4 hours each day.

Use a suitable approximating distribution to find the probability that Charlotte receives more than 30 messages during a randomly chosen 4-hour working day.

Charlotte works 5 days a week.

Hence find the probability that Charlotte receives more than 30 messages in a day on exactly 2 of the 5 days in a randomly chosen working week.

Fleur is a biology student researching daisies in a large field. She randomly selects a square of side 1 m in the field and records the number of daisies in the square, $D$. Fleur models $D$ by a Poisson distribution with mean 5.

State two assumptions that are required for the Poisson model to be valid in this context.

Find the probability that a randomly selected square of side 1 m contains more than 2 daisies.

Use a suitable approximating distribution to find the probability that a randomly selected square of side 6 m contains between 175 and 190 daisies inclusive.

Remy has a biased coin. The probability that the coin lands on tails on any flip is 0.05.

Remy flips the coin 60 times. Use a suitable approximating distribution to find the probability that the coin lands on tails more than 4 times.

Justify your approximating distribution.

Remy flips the coin $n$ times, where $n>50$. The probability that the coin lands on tails at least once is less than 0.95.

Use an approximating distribution to find the largest possible value of $n$.

The random variable $T$ has the distribution $\text{Po}\left(200\right)$.

State a suitable approximating normal distribution for $T$, and explain why a normal distribution is suitable in this case.

Use the normal approximation to find

(i) $\text{P}(180<T\le 230)$

(ii) $\text{P}(T>175)$.

Use the normal approximation to find the largest integer value of $t$ such that $\text{P}(200-t<T<200+t)<0.5$.

A shop manager buys a box of 100 small chocolate bars. The mass, in grams, of a chocolate bar is modelled by the distribution $\text{N}(85,1.5^2)$.

Find the probability that a randomly chosen chocolate bar has a mass of less than 81 g.

The number of chocolate bars in the box that have a mass of less than 81 g is denoted by $X$.

Justify the use of a Poisson distribution as an approximating distribution for $X$.

Use a suitable approximating distribution to find the probability that more than one chocolate bar in the box has a mass of less than 81 g.

The random variable $X$ has the distribution $\text{Po}\left(\lambda \right)$, where $\lambda$ is large. Using a suitable approximating distribution, it is found that $\text{P}(X\ge 100)=0.999$.

Find the value of $\lambda$.

The random variable $Y$ has the distribution $\text{B}(n,0.00025)$, where $n$ is large. It is given that $\text{P}(Y>0)=0.847$, correct to 3 decimal places.

Use a suitable approximating distribution to find the value of $n$.

The random variable $X$ has the distribution $\text{B}(n,p)$.

Use the formulae for $\text{E}\left(X\right)$ and $\text{Var}\left(X\right)$ to explain why it is necessary for $p$ to be close to 0 for a Poisson distribution to be a reasonable approximation for $X$.

Data from previous years show that the probability that a student at a large college completes their assignments on time is 0.99.

A random sample of 250 students at the college is chosen. By first defining a suitable binomial distribution with a small value of $p$, use a Poisson approximation to find the probability that at least 249 of these students complete their assignments on time.

The random variable $W$ has the distribution $\text{Po}\left(105\right)$.

State why a normal distribution can be used as an approximation for $W$.

Use a suitable approximating distribution to find

(i) $\text{P}(80<W<100)$

(ii) the probability that $W$ differs from 105 by at least 15.

Use a suitable approximating distribution to find the smallest integer value of $w$ such that $\text{P}(10+w<W<200-w)<0.9$.

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

Use a suitable approximating distribution to show that the probability that Alzena searches 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 shifts, each of 8 hours. You may assume that these shifts are independent of each other and that the probability that Alzena searches for a new job after a shift is 0.002.

Use a suitable approximating distribution to find the probability that Alzena searches for a new job after more than 3 shifts in a year.

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

Mick runs a business selling cages for chinchillas. Orders from customers occur at random and at a constant average rate of 6 per week.

At the beginning of a particular week, Mick has 3 cages in stock and is unable to obtain any more that week. Find the probability that he is able to fulfil all the orders from customers that week.

Mick visits his supplier to buy more cages. His supplier tells Mick that this will be his last chance to buy cages for 40 days. Mick receives orders on all seven days of the week.

Use a suitable approximating distribution to find the smallest number of cages that Mick should buy so that there is at least a 90% chance that he will be able to fulfil all the orders over the next 40 days.

During a storm, raindrops fall randomly on a flat area of ground. A researcher draws a circle of radius 1 m on the ground and finds that raindrops land inside it at a constant average rate of 12 per second. The mean number of raindrops landing inside any circle drawn on the ground is proportional to the area of the circle.

Use a suitable approximating distribution to find the radius of the circle that should be drawn so that there is a 99% chance that more than 1000 raindrops land inside it in one second.