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CAIE S1 2013 November Q5
9 marks Standard +0.3
5 Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm .
  1. \(8 \%\) of carrots are shorter than \(c \mathrm {~cm}\). Find the value of \(c\).
  2. Rebekah picks 7 carrots at random. Find the probability that at least 2 of them have lengths between 15 and 16 cm .
CAIE S1 2013 November Q6
10 marks Standard +0.8
6 A shop has 7 different mountain bicycles, 5 different racing bicycles and 8 different ordinary bicycles on display. A cycling club selects 6 of these 20 bicycles to buy.
  1. How many different selections can be made if there must be no more than 3 mountain bicycles and no more than 2 of each of the other types of bicycle? The cycling club buys 3 mountain bicycles, 1 racing bicycle and 2 ordinary bicycles and parks them in a cycle rack, which has a row of 10 empty spaces.
  2. How many different arrangements are there in the cycle rack if the mountain bicycles are all together with no spaces between them, the ordinary bicycles are both together with no spaces between them and the spaces are all together?
  3. How many different arrangements are there in the cycle rack if the ordinary bicycles are at each end of the bicycles and there are no spaces between any of the bicycles?
CAIE S1 2013 November Q7
11 marks Moderate -0.8
7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable \(X\) is defined as follows.
  • If the coin shows a head then \(X\) is the sum of the scores on the two throws of the die.
  • If the coin shows a tail then \(X\) is the score on the first throw of the die only.
    1. Explain why \(X = 1\) can only be obtained by throwing a tail, and show that \(\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }\).
    2. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }\).
    3. Copy and complete the probability distribution table for \(X\).
\(x\)12345678
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 8 }\)\(\frac { 3 } { 16 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 32 }\)
Event \(Q\) is 'James throws a tail'. Event \(R\) is 'the value of \(X\) is 7'.
  • Determine whether events \(Q\) and \(R\) are exclusive. Justify your answer.
    •
    CAIE S1 2013 November Q1
    3 marks Easy -1.2
    1 It is given that \(X \sim \mathrm {~N} \left( 1.5,3.2 ^ { 2 } \right)\). Find the probability that a randomly chosen value of \(X\) is less than - 2.4 .
    CAIE S1 2013 November Q2
    5 marks Moderate -0.3
    2 On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with probability 0.35 or stays at home. If he goes shopping the probability that he spends more than \(\\) 50\( is 0.7 . If he goes to the cinema the probability that he spends more than \)\\( 50\) is 0.8 . If he stays at home he spends \(\\) 10$ on a pizza.
    1. Find the probability that Mohit will go to the cinema and spend less than \(\\) 50\(.
    2. Given that he spends less than \)\\( 50\), find the probability that he went to the cinema.
    CAIE S1 2013 November Q3
    5 marks Standard +0.3
    3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.
    1. Find the standard deviation of the amount of fibre in a packet.
    2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.
    CAIE S1 2013 November Q4
    8 marks Moderate -0.8
    4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race. \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}
    1. Show that 75 people took between 200 and 250 minutes to complete the race.
    2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
    3. Explain why your answers to part (ii) are estimates.
    CAIE S1 2013 November Q5
    9 marks Standard +0.3
    5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.
    1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
    2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.
    CAIE S1 2013 November Q6
    9 marks Moderate -0.3
    6 The 11 letters of the word REMEMBRANCE are arranged in a line.
    1. Find the number of different arrangements if there are no restrictions.
    2. Find the number of different arrangements which start and finish with the letter M .
    3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
    4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.
    CAIE S1 2013 November Q7
    11 marks Standard +0.3
    7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.
    1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
    2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
    3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
    4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.
    CAIE S1 2013 November Q1
    2 marks Moderate -0.8
    1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table.
    Distance from college \(( \mathrm { km } )\)\(1 - 3\)\(4 - 5\)\(6 - 8\)\(9 - 11\)\(12 - 16\)
    Number of students18138124
    Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below. \includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.
    CAIE S1 2013 November Q2
    5 marks Standard +0.3
    2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .
    CAIE S1 2013 November Q3
    6 marks Standard +0.3
    3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.
    1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
    2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).
    CAIE S1 2013 November Q4
    7 marks Moderate -0.3
    4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .
    1. Find the mean weight of the 45 fruits.
    2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.
    CAIE S1 2013 November Q5
    7 marks Challenging +1.2
    5
    1. The random variable \(X\) is normally distributed with mean 82 and standard deviation 7.4. Find the value of \(q\) such that \(\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44\).
    2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(5 \mu = 2 \sigma ^ { 2 }\) and that \(\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281\). Find the values of \(\mu\) and \(\sigma\).
    CAIE S1 2013 November Q6
    10 marks Standard +0.3
    6
    1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
    2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
    3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
    4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.
    CAIE S1 2013 November Q7
    13 marks Moderate -0.3
    7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.
    1. Find the probability that the two digits chosen are equal.
    2. Find the probability that one digit is a 5 and one digit is not a 5 .
    3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
    4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).
    CAIE S1 2014 November Q1
    3 marks Easy -1.8
    1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$
    CAIE S1 2014 November Q2
    6 marks Easy -1.3
    2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.
    \(x\)01234\(\geqslant 5\)
    \(\mathrm { P } ( X = x )\)0.240.35\(2 k\)\(k\)0.050
    1. Find the value of \(k\).
    2. Find the mode of \(X\).
    3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
    CAIE S1 2014 November Q3
    5 marks Standard +0.3
    3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:
    • if the coin shows a head, the two numbers from the dice are added together;
    • if the coin shows a tail, the two numbers from the dice are multiplied together.
    Find the probability that the coin shows a head given that Jodie's score is 8 .
    CAIE S1 2014 November Q4
    7 marks Easy -1.2
    4 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 \(B\).
    (7)
    Type \(A\)Type \(B\)
    976643321358
    55442223044566667889
    998887664322040112368899
    655432110525669
    973061389
    874410757
    766653321081244
    86555906
    Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).
    1. Find the median and quartiles for smartphones of type \(A\). 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.
    2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
    3. Compare the loading times for these two types of smartphone.
    CAIE S1 2014 November Q5
    9 marks Standard +0.3
    5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .
    1. Show that the variance of the number of faulty screws in a packet is 1.104 .
    2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
    3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.
    CAIE S1 2014 November Q6
    10 marks Moderate -0.3
    6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .
    1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
    2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
    3. Find the value of \(\mu\).
    CAIE S1 2014 November Q7
    10 marks Standard +0.3
    7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
    1. if there are more women than men on the committee,
    2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
    3. In how many different ways can the committee members be arranged in a line if the man is not at either end?
    CAIE S1 2014 November Q1
    3 marks Easy -1.2
    1 The 50 members of a club include both the club president and the club treasurer. All 50 members want to go on a coach tour, but the coach only has room for 45 people. In how many ways can 45 members be chosen if both the club president and the club treasurer must be included?