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CAIE S1 2020 November Q1
6 marks Moderate -0.8
1 The times taken to swim 100 metres by members of a large swimming club have a normal distribution with mean 62 seconds and standard deviation 5 seconds.
  1. Find the probability that a randomly chosen member of the club takes between 56 and 66 seconds to swim 100 metres.
  2. \(13 \%\) of the members of the club take more than \(t\) minutes to swim 100 metres. Find the value of \(t\).
CAIE S1 2020 November Q2
5 marks Standard +0.3
2 An ordinary fair die is thrown until a 6 is obtained.
  1. Find the probability that obtaining a 6 takes more than 8 throws.
    Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable \(X\).
  2. Find the expected value of \(X\).
  3. Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.
CAIE S1 2020 November Q3
6 marks Standard +0.3
3 A committee of 6 people is to be chosen from 9 women and 5 men.
  1. Find the number of ways in which the 6 people can be chosen if there must be more women than men on the committee.
    The 9 women and 5 men include a sister and brother.
  2. Find the number of ways in which the committee can be chosen if the sister and brother cannot both be on the committee.
CAIE S1 2020 November Q4
8 marks Moderate -0.3
4 The 1300 train from Jahor to Keman runs every day. The probability that the train arrives late in Keman is 0.35 .
  1. For a random sample of 7 days, find the probability that the train arrives late on fewer than 3 days.
    A random sample of 142 days is taken.
  2. Use an approximation to find the probability that the train arrives late on more than 40 days.
CAIE S1 2020 November Q5
7 marks Standard +0.3
5 The 8 letters in the word RESERVED are arranged in a random order.
  1. Find the probability that the arrangement has V as the first letter and E as the last letter.
  2. Find the probability that the arrangement has both Rs together given that all three Es are together.
CAIE S1 2020 November Q6
8 marks Moderate -0.3
6 Three coins \(A , B\) and \(C\) are each thrown once.
  • Coins \(A\) and \(B\) are each biased so that the probability of obtaining a head is \(\frac { 2 } { 3 }\).
  • Coin \(C\) is biased so that the probability of obtaining a head is \(\frac { 4 } { 5 }\).
    1. Show that the probability of obtaining exactly 2 heads and 1 tail is \(\frac { 4 } { 9 }\).
The random variable \(X\) is the number of heads obtained when the three coins are thrown.
  • Draw up the probability distribution table for \(X\).
  • Given that \(\mathrm { E } ( X ) = \frac { 32 } { 15 }\), find \(\operatorname { Var } ( X )\).
    •
    CAIE S1 2020 November Q7
    10 marks Moderate -0.3
    7 A particular piece of music was played by 91 pianists and for each pianist, the number of incorrect notes was recorded. The results are summarised in the table.
    Number of incorrect notes\(1 - 5\)\(6 - 10\)\(11 - 20\)\(21 - 40\)\(41 - 70\)
    Frequency105263218
    1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{9f0f0e3c-7baf-42eb-a4fb-9ce61922c3cd-10_1488_1493_785_365}
    2. State which class interval contains the lower quartile and which class interval contains the upper quartile. Hence find the greatest possible value of the interquartile range.
    3. Calculate an estimate for the mean number of incorrect notes.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    CAIE S1 2021 November Q1
    4 marks Moderate -0.3
    1 Two fair coins are thrown at the same time. The random variable \(X\) is the number of throws of the two coins required to obtain two tails at the same time.
    1. Find the probability that two tails are obtained for the first time on the 7th throw.
    2. Find the probability that it takes more than 9 throws to obtain two tails for the first time.
    CAIE S1 2021 November Q2
    4 marks Moderate -0.8
    2 A summary of 40 values of \(x\) gives the following information: $$\Sigma ( x - k ) = 520 , \quad \Sigma ( x - k ) ^ { 2 } = 9640$$ where \(k\) is a constant.
    1. Given that the mean of these 40 values of \(x\) is 34 , find the value of \(k\).
    2. Find the variance of these 40 values of \(x\).
    CAIE S1 2021 November Q3
    5 marks Moderate -0.8
    3 For her bedtime drink, Suki has either chocolate, tea or milk with probabilities \(0.45,0.35\) and 0.2 respectively. When she has chocolate, the probability that she has a biscuit is 0.3 When she has tea, the probability that she has a biscuit is 0.6 . When she has milk, she never has a biscuit. Find the probability that Suki has tea given that she does not have a biscuit.
    CAIE S1 2021 November Q4
    6 marks Moderate -0.8
    4 A fair spinner has edges numbered \(0,1,2,2\). Another fair spinner has edges numbered \(- 1,0,1\). Each spinner is spun. The number on the edge on which a spinner comes to rest is noted. The random variable \(X\) is the sum of the numbers for the two spinners.
    1. Draw up the probability distribution table for \(X\).
    2. Find \(\operatorname { Var } ( X )\).
    CAIE S1 2021 November Q5
    10 marks Moderate -0.3
    5 Raman and Sanjay are members of a quiz team which has 9 members in total. Two photographs of the quiz team are to be taken. For the first photograph, the 9 members will stand in a line.
    1. How many different arrangements of the 9 members are possible in which Raman will be at the centre of the line?
    2. How many different arrangements of the 9 members are possible in which Raman and Sanjay are not next to each other?
      For the second photograph, the members will stand in two rows, with 5 in the back row and 4 in the front row.
    3. In how many different ways can the 9 members be divided into a group of 5 and a group of 4?
    4. For a random division into a group of 5 and a group of 4, find the probability that Raman and Sanjay are in the same group as each other.
    CAIE S1 2021 November Q6
    10 marks Easy -1.3
    6 The weights, in kg, of 15 rugby players in the Rebels club and 15 soccer players in the Sharks club are shown below.
    Rebels7578798082828384858689939599102
    Sharks666871727475757678838384858692
    1. Represent the data by drawing a back-to-back stem-and-leaf diagram with Rebels on the left-hand side of the diagram.
    2. Find the median and the interquartile range for the Rebels.
      A box-and-whisker plot for the Sharks is shown below. \includegraphics[max width=\textwidth, alt={}, center]{a2709c37-6e81-4873-8f38-94cb9f3c3252-09_533_1246_388_445}
    3. On the same diagram, draw a box-and-whisker plot for the Rebels.
    4. Make one comparison between the weights of the players in the Rebels club and the weights of the players in the Sharks club.
    CAIE S1 2021 November Q7
    11 marks Moderate -0.3
    7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.
      1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
      2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
    2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    CAIE S1 2021 November Q1
    2 marks Easy -1.2
    1 The 26 members of the local sports club include Mr and Mrs Khan and their son Abad. The club is holding a party to celebrate Abad's birthday, but there is only room for 20 people to attend. In how many ways can the 20 people be chosen from the 26 members of the club, given that Mr and Mrs Khan and Abad must be included?
    CAIE S1 2021 November Q2
    6 marks Easy -1.8
    2 Lakeview and Riverside are two schools. The pupils at both schools took part in a competition to see how far they could throw a ball. The distances thrown, to the nearest metre, by 11 pupils from each school are shown in the following table.
    Lakeview1014192226272830323341
    Riverside2336211837251820243025
    1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Lakeview on the left-hand side.
    2. Find the interquartile range of the distances thrown by the 11 pupils at Lakeview school.
    CAIE S1 2021 November Q3
    6 marks Moderate -0.8
    3 The times taken, in minutes, by 360 employees at a large company to travel from home to work are summarised in the following table.
    Time, \(t\) minutes\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 50\)
    Frequency231021357624
    1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-04_1198_1200_836_516}
    2. Calculate an estimate of the mean time taken by an employee to travel to work.
    CAIE S1 2021 November Q4
    8 marks Moderate -0.8
    4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.
    1. Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
    2. Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
    3. On 95\% of days, Raj runs for more than \(t\) minutes. Find the value of \(t\).
    CAIE S1 2021 November Q5
    8 marks Moderate -0.5
    5 A security code consists of 2 letters followed by a 4-digit number. The letters are chosen from \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) and the digits are chosen from \(\{ 1,2,3,4,5,6,7 \}\). No letter or digit may appear more than once. An example of a code is BE 3216 .
    1. How many different codes can be formed?
    2. Find the number of different codes that include the letter A or the digit 5 or both.
      A security code is formed at random.
    3. Find the probability that the code is DE followed by a number between 4500 and 5000 .
    CAIE S1 2021 November Q6
    10 marks Moderate -0.3
    6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table.
    \(x\)0123
    \(\mathrm { P } ( X = x )\)0.6\(p\)\(q\)0.05
    It is given that \(\mathrm { E } ( X ) = 0.55\).
    1. Find the values of \(p\) and \(q\).
    2. Find \(\operatorname { Var } ( X )\).
      Jim is practising for a competition and he repeatedly throws three darts at the board.
    3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
    4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.
    CAIE S1 2021 November Q7
    10 marks Standard +0.3
    7 Box \(A\) contains 6 red balls and 4 blue balls. Box \(B\) contains \(x\) red balls and 9 blue balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\).
    1. Complete the tree diagram below, giving the remaining four probabilities in terms of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-12_688_759_484_731}
    2. Show that the probability that both balls chosen are blue is \(\frac { 4 } { x + 10 }\).
      It is given that the probability that both balls chosen are blue is \(\frac { 1 } { 6 }\).
    3. Find the probability, correct to 3 significant figures, that the ball chosen from box \(A\) is red given that the ball chosen from box \(B\) is red.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    CAIE S1 2022 November Q1
    4 marks Moderate -0.3
    1 The probability distribution table for a random variable \(X\) is shown below.
    \(x\)- 2- 10.512
    \(\mathrm { P } ( X = x )\)0.12\(p\)\(q\)0.160.3
    Given that \(\mathrm { E } ( X ) = 0.28\), find the value of \(p\) and the value of \(q\).
    CAIE S1 2022 November Q2
    8 marks Standard +0.3
    2 The residents of Persham were surveyed about the reliability of their internet service. 12\% rated the service as 'poor', \(36 \%\) rated it as 'satisfactory' and \(52 \%\) rated it as 'good'. A random sample of 8 residents of Persham is chosen.
    1. Find the probability that more than 2 and fewer than 8 of them rate their internet service as poor or satisfactory.
      A random sample of 125 residents of Persham is now chosen.
    2. Use an approximation to find the probability that more than 72 of these residents rate their internet service as good.
    CAIE S1 2022 November Q3
    9 marks Easy -1.3
    3 The Lions and the Tigers are two basketball clubs. The heights, in cm, of the 11 players in each of their first team squads are given in the table.
    Lions178186181187179190189190180169196
    Tigers194179187190183201184180195191197
    1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Lions on the left.
    2. Find the median and the interquartile range of the heights of the Lions first team squad.
      It is given that for the Tigers, the lower quartile is 183 cm , the median is 190 cm and the upper quartile is 195 cm .
    3. Make two comparisons between the heights of the players in the Lions first team squad and the heights of the players in the Tigers first team squad.
    CAIE S1 2022 November Q4
    9 marks Standard +0.3
    4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.
    1. Find the probability that the SBP of a randomly chosen adult is less than 132.
      The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
    2. Find the standard deviation of this distribution.
      Three adults are chosen at random from this population.
    3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.