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Qtn No.	1	2	3	4	5	6	Total
Marks	8	6	7	8	6	7	42
Score

Question 1 Code: 9709/61/M/J/18/4, Topic: The normal distribution

$\text{(a)}$ The distance that car tyres of a certain make can travel before they need to be replaced has a normal distribution. A survey of a large number of these tyres found that the probability of this distance being more than $36800 \mathrm{~km}$ is $0.0082$ and the probability of this distance being more than $31000 \mathrm{~km}$ is $0.6915.$ Find the mean and standard deviation of the distribution. $[5]$

$\text{(b)}$ The random variable $X$ has the distribution $\mathrm{N}\left(\mu, \sigma^{2}\right)$, where $3 \sigma=4 \mu$ and $\mu \neq 0$. Find $\mathrm{P}(X<3 \mu)$. $[3]$

Question 2 Code: 9709/62/M/J/18/4, Topic: Discrete random variables

Mrs Rupal chooses 3 animals at random from 5 dogs and 2 cats. The random variable $X$ is the number of cats chosen.

$\text{(i)}$ Draw up the probability distribution table for $X$. $[4]$

$\text{(ii)}$ You are given that $\mathrm{E}(X)=\frac{6}{7}$. Find the value of $\operatorname{Var}(X)$. $[2]$

Question 3 Code: 9709/63/M/J/18/4, Topic: Representation of data

Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

$$ \begin{array}{|l|c|c|c|} \hline & \text{Number of holidays} & \text{Mean price (\$)} & \text{Standard deviation (\$)} \\ \hline \text{Farfield Travel} & 30 & 1500 & 230 \\ \hline \text{Lacket Travel} & 21 & 2400 & 160 \\ \hline \end{array} $$

$\text{(i)}$ Calculate the mean price of all 51 holidays.

$\text{(ii)}$ The prices of individual holidays with Farfield Travel are denoted by $\$ x_{F}$ and the prices of individual holidays with Lacket Travel are denoted by $\$ x_{L}.$ By first finding $\Sigma x_{F}^{2}$ and $\Sigma x_{L}^{2}$, find the standard deviation of the prices of all 51 holidays. $[5]$

Question 4 Code: 9709/61/O/N/18/4, Topic: The normal distribution

$\text{(a)}$ It is given that $X \sim \mathrm{N}(31.4,3.6)$. Find the probability that a randomly chosen value of $X$ is less than $29.4$. $[3]$

$\text{(b)}$ The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than $12 \mathrm{~cm}$ long and 58 are more than $19 \mathrm{~cm}$ long. Find estimates for the mean and standard deviation of the lengths of fish of this species. $[5]$

Question 5 Code: 9709/62/O/N/18/4, Topic: Permutations and combinations

$\text{(i)}$ Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together. $[3]$

$\text{(ii)}$ Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy. $[3]$

Question 6 Code: 9709/63/O/N/18/4, Topic: Permutations and combinations

Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.

$\text{(i)}$ Find the probability that the group of 7 includes one particular boy. $[3]$

$\text{(ii)}$ Find the probability that the group of 7 includes at least 2 girls. $[4]$