Working with Distributions (Edexcel A Level Maths: Statistics): Exam Questions

George throws a ball at a target 15 times.

Each time George throws the ball, the probability of the ball hitting the target is 0.48.

The random variable $X$ represents the number of times George hits the target in 15 throws.

Find

(i) $\mathrm{P}\left(X=3\right)$

(ii) $\mathrm{P}\left(X\ge 5\right)$

George now throws the ball at the target 250 times.

Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

A fair spinner has 8 sectors labelled 1 to 8. The random variable $Y$ is the number of times the spinner lands on a prime number when it is spun 12 times.

State an assumption required to model $Y$ with a binomial distribution $\text{B}(n,p)$.

The random variable $W$ is the number of times the spinner must be spun until it first lands on a '7'. The random variable $L$ is the number of wins when the spinner is spun 10 times, where a 'win' on the first spin is scored if the spinner lands on an even number, and a 'win' on subsequent spins is scored if the spinner lands on the same number as the previous spin, or on a factor of it.

For each of $W$ and $L$, give a reason why a binomial distribution would not be an appropriate model.

The random variable $S$ is the number shown when the spinner is spun once. Name the probability distribution that would be appropriate for $S$.

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

(i) A student cuts a 1-metre length of rope into two pieces at a random point. The random variable $A$ is the length of the shorter piece.

(ii) You survey a sample of students about their preferences for after-school activities. The random variable $B$ is the number of students who prefer lawn bowling.

Three histograms show the distributions of the random variables $D$, $E$ and $F$.

For each of $D$, $E$ and $F$, state with a reason whether a normal distribution would be an appropriate model.

For each of the following binomial random variables $X$:

• state, with reasons, whether $X$ can be approximated by a normal distribution

• if appropriate, write down the normal approximation to $X$ in the form $\text{N}(\mu ,{\sigma }^2)$, giving the values of $\mu$ and ${\sigma }^2$.

(i) $X~\text{B}(10,0.5)$

(ii) $X~\text{B}(80,0.54)$

On a casino roulette wheel, the probability of the ball landing on a black number is $\frac{9}{19}$. The wheel is spun 30 times. The random variable $X$ represents the number of times the ball lands on a black number.

Find $\text{P}(X=14)$.

In a separate experiment, the wheel is spun 1000 times. The random variable $Y$ represents the number of times the ball lands on a black number.

(i) State, giving reasons, why a normal approximation is appropriate for $Y$.

(ii) Write down the normal distribution that could be used to approximate $Y$.

Use the distribution from part (b)(ii) to estimate the probability that the ball lands on a black number at least 500 times in the 1000 spins.

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. The random variable $X$ represents the number of packets containing a prize.

Find the value of $\text{P}(X=49)$.

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

Using the normal distribution from part (b), estimate $\text{P}(X=49)$. You must state the continuity correction used.

The random variable $X~\text{B}(100,0.36)$ can be approximated by $Y~\text{N}(\mu ,{\sigma }^2)$.

Find the value of $\mu$ and show that $\sigma =4.8$.

Explain why a continuity correction must be applied when using this approximation.

Using a continuity correction, find the value of $k$ in each of the following:

(i) $\text{P}(X\le 30)\approx \text{P}(Y<k)$

(ii) $\text{P}(X<30)\approx \text{P}(Y<k)$

(iii) $\text{P}(X\ge 30)\approx \text{P}(Y>k)$

(iv) $\text{P}(X>30)\approx \text{P}(Y>k)$

The random variable $X~\text{B}(150,0.6)$ is to be approximated by $Y~\text{N}(\mu ,{\sigma }^2)$.

(i) State two reasons why a normal approximation is suitable.

(ii) Find the values of $\mu$ and $\sigma$.

Use a normal approximation to estimate $\text{P}(87\le X\le 101)$. Clearly state the continuity correction used.

A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, $D$ ml, follows a normal distribution with mean 25 ml.

Given that 15% of bottles contain less than 24.63 ml, find, to 2 decimal places, the value of $k$ such that $\mathrm{P}(24.63<D<k)=0.45$

A random sample of 200 bottles is taken.

Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and $k$ ml.

A type of seed has a germination rate of 55% when planted under standard conditions. A researcher plants 60 seeds and the random variable $Y$ represents the number that germinate.

Write down the normal distribution that can be used to approximate $Y$, giving the values of $\mu$ and ${\sigma }^2$. State two reasons why this approximation is valid.

Use your normal approximation to estimate the probability that at least 40 seeds germinate.

A factory quality control team monitors a production line. Records show that 60% of components pass the quality check on the first attempt. In a sample of 1200 components, the random variable $W$ is the number of components that pass on the first attempt.

Explain why $W$ can be approximated by a normal distribution. Give your reasons in the context of the problem.

Use a normal approximation to estimate the probability that fewer than 700 components pass the quality check on the first attempt, applying a continuity correction.

On a European-style casino roulette wheel, the probability of the ball landing on a red number is $\frac{18}{37}$.

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

Find $\text{P}(17<X\le 18).$

In a separate experiment, the wheel is spun 1000 times. Let $Y$ be the number of times the ball lands on a red number.

Use a normal approximation to estimate the probability that the ball lands on red more than half of the time. Give your answer to 3 significant figures.

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 3 significant figures.

Use a normal approximation to estimate the probability that exactly 45 of the 100 figures are 'double left' figures.

Give your answer to 3 significant figures.

Using the normal approximation from part (b), find the probability that fewer than 40 of the 100 figures are 'double left' figures.

Give your answer to 3 significant figures.

The random variable $W~\text{B}(500,0.4)$.

Give two reasons why a normal distribution can be used to approximate $W$.

Find, using the normal approximation:

(i) $\text{P}(189\le W\le 211)$

(ii) $\text{P}(W>220)$

Using the normal approximation, find the largest integer value of $w$ such that $\text{P}(W\le w)<0.1$.

A medical researcher is studying the number of hours, $T$, a patient stays in hospital following a particular operation.

The histogram summarises the results for a random sample of 90 patients.

Use the histogram to estimate $\mathrm{P}\left(10<T<30\right)$.

For these 90 patients the time spent in hospital following the operation had

• a mean of 14.9 hours

• a standard deviation of 9.3 hours

Tomas suggests that $T$ can be modelled by $\mathrm{N}\left(14.9,9.3^2\right)$

With reference to the histogram, state, giving a reason, whether or not Tomas’ model could be suitable.

Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation

$y=kx{\mathrm{e}}^{-x}0\le x\le 4$

where

• $x$ is measured in tens of hours

• $k$ is a constant

Use algebraic integration to show that

${\int }_0^{n}x{\mathrm{e}}^{-x}dx=1-\left(n+1\right){\mathrm{e}}^{-n}$

Show that, for Xiang’s model, $k=99$ to the nearest integer.

Estimate $\mathrm{P}\left(10<T<30\right)$ using Tomas' model of $T~\mathrm{N}\left(14.9,9.3^2\right)$.

Estimate $\mathrm{P}\left(10<T<30\right)$ using Xiang’s curve with equation $y=99x{\mathrm{e}}^{-x}$ and the answer to part (c).

The researcher decides to use Xiang’s curve to model $\mathrm{P}\left(a<T<b\right)$.

State one limitation of Xiang’s model.

A chocolate factory produces boxes of chocolates. 44% of the chocolates produced contain a caramel centre.

A box contains 80 chocolates. The random variable $X$ represents the number of chocolates in the box with a caramel centre.

State, giving a reason, a suitable normal distribution that can be used to approximate $X$. Give your answer in the form $\text{N}(\mu ,{\sigma }^2)$.

Using your approximation from part (a), estimate the probability that at least 40 of the 80 chocolates in the box contain a caramel centre.

Give your answer to 3 significant figures.

The same factory also produces boxes of premium chocolates. Only 2% of these premium chocolates contain a rare pistachio filling.

A box of premium chocolates contains 100 chocolates. The random variable $Y$ represents the number of these chocolates with a pistachio filling. A quality controller suggests that $Y$ can be approximated by a normal distribution with mean $np$ and variance $np(1-p)$.

(i) Write down this proposed normal approximation for $Y$, in the form $\text{N}(\mu ,{\sigma }^2)$.

(ii) By calculating $\text{P}(Y<0)$ for this proposed approximation, explain why it is a poor model for the number of premium chocolates containing a pistachio filling.

A manufacturer produces ceramic tiles. Historically, 40% of the tiles produced have a slight colour variation.

A random sample of 980 tiles is selected. The random variable $W$ represents the number of tiles in the sample with a colour variation.

State, giving reasons, why the distribution of $W$ can be approximated by a normal distribution.

Use a normal approximation to estimate the probability that at least 387 and no more than 397 of the tiles in the sample have a colour variation.

Give your answer to 3 significant figures.

Use the normal approximation to estimate the probability that at least 400 of the tiles in the sample have a colour variation.

Give your answer to 3 significant figures.

Using the normal approximation, find the largest integer $w$ such that the probability of obtaining fewer than $w$ tiles with a colour variation is less than 0.05.

On a casino roulette wheel, the probability of the ball landing on red is $\frac{6}{13}$.

The wheel is spun 50 times. The random variable $X$ represents the number of times the ball lands on red.

Find $\text{P}(X\ge 25)$.

Give your answer to 3 significant figures.

The wheel is now spun 1000 times. The random variable $Y$ represents the number of times the ball lands on red.

State, giving reasons, why the distribution of $Y$ can be approximated by a normal distribution.

Use a normal approximation to estimate the probability that in 1000 spins the ball lands on red at least 450 times but no more than 475 times.

Give your answer to 3 significant figures.

A local bakery finds that 58% of customers who buy a pastry also buy a coffee.

A random sample of 15 customers who bought a pastry is taken. The random variable $X$ represents the number of these customers who also bought a coffee.

Find $\text{P}(X=10)$.

Give your answer to 3 significant figures.

On a busy Saturday, a random sample of 150 customers who bought a pastry is taken. The random variable $Y$ represents the number of these customers who also bought a coffee.

State, giving reasons, why the distribution of $Y$ can be approximated by a normal distribution.

Use a normal approximation to estimate the probability that between 80 and 90 (inclusive) of the 150 customers in the sample also bought a coffee.

Give your answer to 3 significant figures.

A student attempts to use the normal approximation from part (c) to estimate the probability that more than 150 of the customers in the sample bought a coffee.

Explain why this is not a sensible use of the normal approximation.