Questions — CAIE (7659 questions)

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CAIE S1 2023 November Q5
8 marks Standard +0.3
5
  1. The heights of the members of a club are normally distributed with mean 166 cm and standard deviation 10 cm .
    1. Find the probability that a randomly chosen member of the club has height less than 170 cm .
    2. Given that \(40 \%\) of the members have heights greater than \(h \mathrm {~cm}\), find the value of \(h\) correct to 2 decimal places.
  2. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a randomly chosen value of \(X\) is positive.
CAIE S1 2023 November Q6
9 marks Standard +0.3
6 Freddie has two bags of marbles.
Bag \(X\) contains 7 red marbles and 3 blue marbles.
Bag \(Y\) contains 4 red marbles and 1 blue marble.
Freddie chooses one of the bags at random. A marble is removed at random from that bag and not replaced. A new red marble is now added to each bag. A second marble is then removed at random from the same bag that the first marble had been removed from.
  1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
  2. Find the probability that both of the marbles removed from the bag are the same colour.
  3. Find the probability that bag \(Y\) is chosen given that the marbles removed are not both the same colour.
CAIE S1 2023 November Q7
10 marks Challenging +1.2
7
  1. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which no consonant is next to another consonant. (The letters D, M, N and R are consonants and the letters A, E and O are not consonants.)
  2. Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which there is an A at each end and the Ds are not together.
    Four letters are selected at random from the 9 letters in the word ANDROMEDA.
  3. Find the probability that this selection contains at least one D and exactly one A .
    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 2023 November Q1
6 marks Moderate -0.8
1 Becky sometimes works in an office and sometimes works at home. The random variable \(X\) denotes the number of days that she works at home in any given week. It is given that $$\mathrm { P } ( X = x ) = k x ( x + 1 )$$ where \(k\) is a constant and \(x = 1,2,3\) or 4 only.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2023 November Q2
6 marks Moderate -0.3
2 The weights of large bags of pasta produced by a company are normally distributed with mean 1.5 kg and standard deviation 0.05 kg .
  1. Find the probability that a randomly chosen large bag of pasta weighs between 1.42 kg and 1.52 kg .
    The weights of small bags of pasta produced by the company are normally distributed with mean 0.75 kg and standard deviation \(\sigma \mathrm { kg }\). It is found that \(68 \%\) of these small bags have weight less than 0.9 kg .
  2. Find the value of \(\sigma\).
CAIE S1 2023 November Q3
7 marks Standard +0.3
3 Tim has two bags of marbles, \(A\) and \(B\).
Bag \(A\) contains 8 white, 4 red and 3 yellow marbles.
Bag \(B\) contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6 -sided dice. He rolls the dice. If he obtains a 1 or 2 , he chooses two marbles at random from bag \(A\), without replacement. If he obtains a \(3,4,5\) or 6 , he chooses two marbles at random from bag \(B\), without replacement.
  1. Find the probability that both marbles are white.
  2. Find the probability that the two marbles come from bag \(B\) given that one is white and one is red. [4]
CAIE S1 2023 November Q4
10 marks Easy -1.8
4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.
Weight \(( x \mathrm {~kg} )\)\(x \leqslant 40\)\(x \leqslant 60\)\(x \leqslant 65\)\(x \leqslant 70\)\(x \leqslant 85\)\(x \leqslant 100\)
Cumulative frequency0143860106120
  1. Draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
  2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
  3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]
CAIE S1 2023 November Q5
11 marks Standard +0.3
5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.
  1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
    1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
    2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
  2. Find the probability that at least three of them will pass tomorrow.
    75 people will take their advanced driving test next week.
    [0pt]
  3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]
CAIE S1 2023 November Q6
10 marks Standard +0.3
6 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.
  1. At the beginning of the party, the 12 people will stand in a line for a photograph.
    1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
    2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
  2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as each other.
    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 2024 November Q1
4 marks Moderate -0.8
1 Nicola throws an ordinary fair six-sided dice. The random variable \(X\) is the number of throws that she takes to obtain a 6.
  1. Find \(\mathrm { P } ( X < 8 )\).
  2. Find the probability that Nicola obtains a 6 for the second time on her 8th throw. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-02_2717_35_109_2012}
CAIE S1 2024 November Q2
6 marks Moderate -0.8
2 The random variable \(X\) takes the values \(- 2 , - 1,0,2,3\). It is given that \(\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)\), where \(k\) is a positive constant.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2024 November Q3
8 marks Easy -1.8
3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.
Time taken
\(( t\) minutes \()\)
\(t \leqslant 15\)\(t \leqslant 25\)\(t \leqslant 30\)\(t \leqslant 40\)\(t \leqslant 50\)\(t \leqslant 70\)
Cumulative
frequency
184688140176200
  1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
  2. Use your graph to estimate the median and the interquartile range of the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
  3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.
CAIE S1 2024 November Q4
6 marks Standard +0.3
4 Rahul has two bags, \(X\) and \(Y\). Bag \(X\) contains 4 red marbles and 2 blue marbles. Bag \(Y\) contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). Rahul throws the coin.
  • If he obtains a head, he chooses at random a marble from bag \(X\). He notes the colour and replaces the marble in bag \(X\). He then chooses at random a second marble from bag \(X\).
  • If he obtains a tail, he chooses at random a marble from bag \(Y\). He notes the colour and discards the marble. He then chooses at random a second marble from bag \(Y\).
    1. Find the probability that the two marbles that Rahul chooses are the same colour. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-07_2725_35_99_20}
    2. Find the probability that the two marbles that Rahul chooses are both from bag \(Y\) given that both marbles are blue.
CAIE S1 2024 November Q5
9 marks Moderate -0.3
5 The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams.
  1. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-09_2717_29_105_22}
  2. The shop also sells red apples. \(60 \%\) of the red apples sold by the shop weigh more than 80 grams. 160 red apples are chosen at random from the shop. Use a suitable approximation to find the probability that fewer than 105 of the chosen red apples weigh more than 80 grams.
CAIE S1 2024 November Q6
8 marks Moderate -0.3
6 The heights of the female students at Breven college are normally distributed:
  • \(90 \%\) of the female students have heights less than 182.7 cm .
  • \(40 \%\) of the female students have heights less than 162.5 cm .
    1. Find the mean and the standard deviation of the heights of the female students at Breven college. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-10_2715_41_110_2008} \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-11_2723_35_101_20}
Ten female students are chosen at random from those at Breven college.
  • Find the probability that fewer than 8 of these 10 students have heights more than 162.5 cm .
    •
    CAIE S1 2024 November Q7
    9 marks Standard +0.3
    7
    1. How many different arrangements are there of the 9 letters in the word INTELLECT in which the two Ts are together?
    2. How many different arrangements are there of the 9 letters in the word INTELLECT in which there is a T at each end and the two Es are not next to each other?
      Four letters are selected at random from the 9 letters in the word INTELLECT.
      [0pt]
    3. Find the percentage of the possible selections which contain at least one E and exactly one T. [4]
      If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-14_2715_31_106_2016}
    CAIE S1 2024 November Q1
    6 marks Easy -1.3
    1 At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports.
    TennisHockeyNetball
    Year 1162212
    Year 2241828
    One student is chosen at random from the 120 students. Events \(X\) and \(N\) are defined as follows: \(X\) : the student is in Year 1 \(N\) : the student plays netball.
    1. Find \(\mathrm { P } ( X \mid N )\).
    2. Find \(\mathrm { P } ( N \mid X )\).
    3. Determine whether or not \(X\) and \(N\) are independent events.
      One of the students who plays netball takes 8 shots at goal. On each shot, the probability that she will succeed is 0.15 , independently of all other shots.
    4. Find the probability that she succeeds on fewer than 3 of these shots.
    CAIE S1 2024 November Q2
    4 marks Challenging +1.2
    2
    1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
    2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}
    CAIE S1 2024 November Q3
    6 marks Moderate -0.3
    3 A fair coin and an ordinary fair six-sided dice are thrown at the same time.The random variable \(X\) is defined as follows.
    -If the coin shows a tail,\(X\) is twice the score on the dice.
    -If the coin shows a head,\(X\) is the score on the dice if the score is even and \(X\) is 0 otherwise.
    1. Draw up the probability distribution table for \(X\) .
    2. Find \(\operatorname { Var } ( X )\) .
    CAIE S1 2024 November Q4
    7 marks Standard +0.3
    4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.
    1. How many of these trees would you expect to have height less than 18.2 metres?
      The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
    2. Find the value of \(\sigma\).
    CAIE S1 2024 November Q5
    6 marks Moderate -0.3
    5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.
    1. In how many ways can the group of 7 be chosen?
      On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
    2. In how many ways can the group of 5 be chosen?
    CAIE S1 2024 November Q6
    10 marks Easy -1.2
    6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.
    Falcons383942444648505152565859646976
    Kites324040454748525458595960616365
    1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
    2. Find the median and the interquartile range of the times for the Falcons.
      Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
    3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.
    CAIE S1 2024 November Q7
    11 marks Moderate -0.3
    7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.
    1. Find \(\mathrm { P } ( X = 5 )\) .
      ............................................................................................................................................
    2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
    3. Find the probability that Leno scores his second goal on or before his 5th attempt. \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017} \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
    4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
      If you use the following page to complete the answer to any question, the question number must be clearly shown.
    CAIE S1 2024 November Q2
    4 marks Moderate -0.8
    2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled \(1,1,2,2,3,3\). Both dice are thrown. The random variable \(X\) is the product of the scores on the two dice.
    1. Draw up the probability distribution table for \(X\).
    2. Find \(\mathrm { E } ( X )\).
    CAIE S1 2024 November Q3
    7 marks Moderate -0.8
    3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .
    1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
      60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
    2. Find the value of \(k\), giving your answer correct to 1 decimal place.