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Cambridge International AS and A Level

Name of student HENRYTAIGO Date
Adm. number Year/grade HenryTaigo Stream HenryTaigo
Subject Probability & Statistics 1 (S1) Variant(s) P61, P62, P63
Start time Duration Stop time

Qtn No. 1 2 3 4 5 6 Total
Marks 7 9 6 7 8 8 45

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Question 1 Code: 9709/61/M/J/14/4, Topic: Discrete random variables

A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable $X$ represents the number of paperback books she chooses.

$\text{(i)}$ Show that the probability that she chooses exactly 2 paperback books is $\frac{3}{14}$. $[2]$

$\text{(ii)}$ Draw up the probability distribution table for $X$. $[3]$

$\text{(iii)}$ You are given that $\mathrm{E}(X)=3$. Find $\operatorname{Var}(X)$. $[2]$

Question 2 Code: 9709/62/M/J/14/4, Topic: Probability

Coin $A$ is weighted so that the probability of throwing a head is $\frac{2}{3}$. Coin $B$ is weighted so that the probability of throwing a head is $\frac{1}{4}$. Coin $A$ is thrown twice and coin $B$ is thrown once.

$\text{(i)}$ Show that the probability of obtaining exactly 1 head and 2 tails is $\frac{13}{36}$. $[3]$

$\text{(ii)}$ Draw up the probability distribution table for the number of heads obtained. $[4]$

$\text{(iii)}$ Find the expectation of the number of heads obtained. $[2]$

Question 3 Code: 9709/63/M/J/14/4, Topic: Representation of data

The heights, $x \mathrm{~cm}$, of a group of 28 people were measured. The mean height was found to be $172.6 \mathrm{~cm}$ and the standard deviation was found to be $4.58 \mathrm{~cm}$. A person whose height was $161.8 \mathrm{~cm}$ left the group.

$\text{(i)}$ Find the mean height of the remaining group of 27 people. $[2]$

$\text{(ii)}$ Find $\Sigma x^{2}$ for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people. $[4]$

Question 4 Code: 9709/61/O/N/14/4, Topic: Representation of data

The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type $A$ and 43 smartphones of type $B$.


$\text{(i)}$ Find the median and quartiles for smartphones of type $A$. $[3]$

You are given that the median, lower quartile and upper quartile for smartphones of type $B$ are $0.46$ seconds, $0.36$ seconds and $0.63$ seconds respectively.

$\text{(ii)}$ Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper. $[3]$

$\text{(iii)}$ Compare the loading times for these two types of smartphone. $[1]$

Question 5 Code: 9709/62/O/N/14/4, Topic: Probability

Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

$\text{(i)}$ Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. $[4]$

$\text{(ii)}$ The random variable $X$ is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for $X$ and find $\mathrm{E}(X)$. $[4]$

Question 6 Code: 9709/63/O/N/14/4, Topic: Representation of data

A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.

$$ \begin{array}{lllll} 23 & 19 & 32 & 14 & 25 \\ 22 & 26 & 36 & 45 & 42 \\ 47 & 28 & 17 & 38 & 15 \\ 46 & 18 & 26 & 22 & 41 \\ 19 & 21 & 28 & 24 & 30 \end{array} $$

$\text{(i)}$ Draw a stem-and-leaf diagram to represent the data. $[3]$

$\text{(ii)}$ On graph paper draw a box-and-whisker plot to represent the data. $[5]$

Worked solutions: P1, P3 & P6 (S1)

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