Questions S1 (1967 questions)

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CAIE S1 2020 November Q6
6 The times, \(t\) minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.
Time taken \(( t\) minutes \()\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 40\)\(t \leqslant 60\)\(t \leqslant 100\)
Cumulative frequency1248106134150
  1. Draw a cumulative frequency graph to illustrate the data.
    \includegraphics[max width=\textwidth, alt={}, center]{033ceb76-8fd4-4a89-ab05-5e20039d1c8d-08_1689_1195_744_516}
  2. \(24 \%\) of the students take \(k\) minutes or longer to complete the challenge. Use your graph to estimate the value of \(k\).
  3. Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.
CAIE S1 2020 November Q7
7
  1. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that all 3 Es are together.
  2. Find the number of different ways in which the 10 letters of the word SHOPKEEPER can be arranged so that the Ps are not next to each other.
  3. Find the probability that a randomly chosen arrangement of the 10 letters of the word SHOPKEEPER has an E at the beginning and an E at the end.
    Four letters are selected from the 10 letters of the word SHOPKEEPER.
  4. Find the number of different selections if the four letters include exactly one P .
    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 2020 November Q1
1 A fair six-sided die, with faces marked \(1,2,3,4,5,6\), is thrown repeatedly until a 4 is obtained.
  1. Find the probability that obtaining a 4 requires fewer than 6 throws.
    On another occasion, the die is thrown 10 times.
  2. Find the probability that a 4 is obtained at least 3 times.
CAIE S1 2020 November Q2
2 A bag contains 5 red balls and 3 blue balls. Sadie takes 3 balls at random from the bag, without replacement. The random variable \(X\) represents the number of red balls that she takes.
  1. Show that the probability that Sadie takes exactly 1 red ball is \(\frac { 15 } { 56 }\).
  2. Draw up the probability distribution table for \(X\).
  3. Given that \(\mathrm { E } ( X ) = \frac { 15 } { 8 }\), find \(\operatorname { Var } ( X )\).
CAIE S1 2020 November Q3
3 Pia runs 2 km every day and her times in minutes are normally distributed with mean 10.1 and standard deviation 1.3.
  1. Find the probability that on a randomly chosen day Pia takes longer than 11.3 minutes to run 2 km .
  2. On \(75 \%\) of days, Pia takes longer than \(t\) minutes to run 2 km . Find the value of \(t\).
  3. On how many days in a period of 90 days would you expect Pia to take between 8.9 and 11.3 minutes to run 2 km ?
CAIE S1 2020 November Q4
4 In a certain country, the weather each day is classified as fine or rainy. The probability that a fine day is followed by a fine day is 0.75 and the probability that a rainy day is followed by a fine day is 0.4 . The probability that it is fine on 1 April is 0.8 . The tree diagram below shows the possibilities for the weather on 1 April and 2 April.
  1. Complete the tree diagram to show the probabilities. 1 April
    \includegraphics[max width=\textwidth, alt={}, center]{33c0bd01-f27b-424c-a78a-6f36178bc08c-08_601_405_706_408} 2 April Fine Rainy Fine Rainy
  2. Find the probability that 2 April is fine.
    Let \(X\) be the event that 1 April is fine and \(Y\) be the event that 3 April is rainy.
  3. Find the value of \(\mathrm { P } ( X \cap Y )\).
  4. Find the probability that 1 April is fine given that 3 April is rainy.
CAIE S1 2020 November Q5
5 The following table gives the weekly snowfall, in centimetres, for 11 weeks in 2018 at two ski resorts, Dados and Linva.
Dados68121510364228102216
Linva2111516032364010129
  1. Represent the information in a back-to-back stem-and-leaf diagram.
  2. Find the median and the interquartile range for the weekly snowfall in Dados.
  3. The median, lower quartile and upper quartile of the weekly snowfall for Linva are 12, 9 and 32 cm respectively. Use this information and your answers to part (b) to compare the central tendency and the spread of the weekly snowfall in Dados and Linva.
CAIE S1 2020 November Q6
6 Mr and Mrs Ahmed with their two children, and Mr and Mrs Baker with their three children, are visiting an activity centre together. They will divide into groups for some of the activities.
  1. In how many ways can the 9 people be divided into a group of 6 and a group of 3?
    5 of the 9 people are selected at random for a particular activity.
  2. Find the probability that this group of 5 people contains all 3 of the Baker children.
    All 9 people stand in a line.
  3. Find the number of different arrangements in which Mr Ahmed is not standing next to Mr Baker.
  4. Find the number of different arrangements in which there is exactly one person between Mr Ahmed and Mr Baker.
    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 2020 November Q1
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
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
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
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
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
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
    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
    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
    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
    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
    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
    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
    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
    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
    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
    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
    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.