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Cambridge International AS and A Level

Name of student CHRISWEK Date
Adm. number Year/grade 1990 Stream Chriswek
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 11 11 11 10 9 11 63

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Question 1 Code: 9709/61/M/J/14/7, Topic: Representation of data

A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.

$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Number of typing errors} & 1-5 & 6-20 & 21-35 & 36-60 & 61-80 \\ \hline \text{Frequency} & 24 & 9 & 21 & 15 & 42 \\ \hline \end{array} $$

$\text{(i)}$ Draw a histogram on graph paper to represent this information. $[5]$

$\text{(ii)}$ Calculate an estimate of the mean number of typing errors for these 111 people. $[3]$

$\text{(iii)}$ State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range. $[3]$

Question 2 Code: 9709/62/M/J/14/7, Topic: The normal distribution

The time Rafa spends on his homework each day in term-time has a normal distribution with mean $1.9$ hours and standard deviation $\sigma$ hours. On $80 \%$ of these days he spends more than $1.35$ hours on his homework.

$\text{(i)}$ Find the value of $\sigma$. $[3]$

$\text{(ii)}$ Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework. $[2]$

$\text{(iii)}$ A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than $1.35$ hours on his homework is between 163 and 173 inclusive. $[6]$

Question 3 Code: 9709/63/M/J/14/7, Topic: Permutations and combinations

Nine cards are numbered $1,2,2,3,3,4,6,6,6$

$\text{(i)}$ All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way

$\text{(a)}$ if the even digits are all together, $[4]$

$\text{(b)}$ if the first and last digits are both odd. $[3]$

$\text{(ii)}$ Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way

$\text{(a)}$ if there are no repeated digits, $[2]$

$\text{(b)}$ if the number is between 200 and 300. $[2]$

Question 4 Code: 9709/61/O/N/14/7, Topic: Permutations and combinations

A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done

$\text{(i)}$ if there are more women than men on the committee, $[4]$

$\text{(ii)}$ if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? $[3]$

One particular committee consists of 5 women and 1 man

$\text{(iii)}$ In how many different ways can the committee members be arranged in a line if the man is not at either end? $[3]$

Question 5 Code: 9709/62/O/N/14/7, Topic: Discrete random variables

In Marumbo, three quarters of the adults own a cell phone.

$\text{(i)}$ A random sample of 8 adults from Marumbo is taken. Find the probability that the number of adults who own a cell phone is between 4 and 6 inclusive. $[3]$

$\text{(ii)}$ A random sample of 160 adults from Marumbo is taken. Use an approximation to find the probability that more than 114 of them own a cell phone. $[5]$

$\text{(iii)}$ Justify the use of your approximation in part $\text{(ii)}$. $[1]$

Question 6 Code: 9709/63/O/N/14/7, Topic: Probability

A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable $X$ is the number of apples which have been taken when the process stops.

$\text{(i)}$ Show that $\mathrm{P}(X=3)=\frac{1}{3}$. $[3]$

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

Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.

$\text{(iii)}$ Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange. $[5]$

Worked solutions: P1, P3 & P6 (S1)

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