Questions — CAIE S1 (785 questions)

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CAIE S1 2024 June Q1
1 Rajesh applies once every year for a ticket to a music festival. The probability that he is successful in any particular year is 0.3 , independently of other years.
  1. Find the probability that Rajesh is successful for the first time on his 7th attempt.
  2. Find the probability that Rajesh is successful for the first time before his 6th attempt.
  3. Find the probability that Rajesh is successful for the second time on his 10th attempt.
CAIE S1 2024 June Q2
2 Seva has a coin which is biased so that when it is thrown the probability of obtaining a head is \(\frac { 1 } { 3 }\). He also has a bag containing 4 red marbles and 5 blue marbles. Seva throws the coin. If he obtains a head, he selects one marble from the bag at random. If he obtains a tail, he selects two marbles from the bag at random and without replacement.
  1. Find the probability that Seva selects at least one red marble.
  2. Find the probability that Seva obtains a head given that he selects no red marbles.
CAIE S1 2024 June Q3
3 The weights of oranges can be modelled by a normal distribution with mean 131 grams and standard deviation 54 grams. Oranges are classified as small, medium or large. A large orange weighs at least 184 grams and 20\% of oranges are classified as small.
  1. Find the percentage of oranges that are classified as large.
  2. Find the greatest possible weight of a small orange.
CAIE S1 2024 June Q4
4 The back-to-back stem-and-leaf diagram shows the annual salaries of 19 employees at each of two companies, Petral and Ravon.
PetralRavon
\multirow{7}{*}{99}3003026
82213115
554032002
753330489
103411346
353
83679
Key: 2 | 31 | 5 means \\(31 200 for a Petral employee and
)31500 for a Ravon employee.
  1. Find the median and the interquartile range of the salaries of the Petral employees.
    The median salary of the Ravon employees is \(
    ) 33800\(, the lower quartile is \)\\( 32000\) and the upper quartile is \(
    ) 34400$.
  2. Represent the data shown in the back-to-back stem-and-leaf diagram by a pair of box-and-whisker plots in a single diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{f979a442-da05-410b-84dc-3da3286514a0-07_707_1395_477_335}
  3. Comment on whether the mean or the median would be a better representation of the data for the employees at Petral.
CAIE S1 2024 June Q5
5 Jasmine has one \(
) 5\( coin, two \)\\( 2\) coins and two \(
) 1\( coins. She selects two of these coins at random. The random variable \)X$ is the total value, in dollars, of these two coins.
  1. Show that \(\mathrm { P } ( X = 7 ) = 0.2\).
  2. Draw up the probability distribution table for \(X\).
  3. Find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2024 June Q6
6 The residents of Mahjing were asked to classify their local bus service:
  • \(25 \%\) of residents classified their service as good.
  • \(60 \%\) of residents classified their service as satisfactory.
  • \(15 \%\) of residents classified their service as poor.
    1. A random sample of 110 residents of Mahjing is chosen.
Use a suitable approximation to find the probability that fewer than 22 residents classified their bus service as good.
  • For a random sample of 10 residents of Mahjing, find the probability that fewer than 8 classified their bus service as good or satisfactory.
  • Three residents of Mahjing are selected at random. Find the probability that one resident classified the bus service as good, one as satisfactory and one as poor.
    •
    CAIE S1 2024 June Q7
    5 marks
    7
    1. How many different arrangements are there of the 10 letters in the word REGENERATE?
    2. How many different arrangements are there of the 10 letters in the word REGENERATE in which the 4 Es are together and the 2 Rs have exactly 3 letters in between them?
    3. Find the probability that a randomly chosen arrangement of the 10 letters in the word REGENERATE is one in which the consonants ( \(\mathrm { G } , \mathrm { N } , \mathrm { R } , \mathrm { R } , \mathrm { T }\) ) and vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { E } , \mathrm { E }\) ) alternate, so that no two consonants are next to each other and no two vowels are next to each other. [5]
      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 June Q1
    1 The numbers on the faces of a fair six-sided dice are \(1,2,2,3,3,3\). The random variable \(X\) is the total score when the dice is rolled twice.
    1. Draw up the probability distribution table for \(X\).
    2. Find the value of \(\operatorname { Var } ( X )\).
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-02_2714_34_143_2012}
    3. Find the probability that \(X\) is even given that \(X > 3\).
    CAIE S1 2024 June Q2
    2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .
    1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m .
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012}
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
    2. Find the mean and the standard deviation of this distribution.
    CAIE S1 2024 June Q3
    4 marks
    3 Box \(A\) contains 6 green balls and 3 yellow balls.
    Box \(B\) contains 4 green balls and \(x\) yellow 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. Draw a tree diagram to represent this information, showing the probability on each of the branches.
      [0pt] [4]
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-06_2727_38_132_2010}
      The probability that both the balls chosen are the same colour is \(\frac { 8 } { 15 }\).
    2. Find the value of \(x\).
    CAIE S1 2024 June Q4
    4 The times taken, in seconds, by 15 members of each of two swimming clubs, the Penguins and the Dolphins, to swim 50 metres are shown in the following table.
    Penguins353942444545485056585961666872
    Dolphins364143484949505154565660616471
    1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Penguins on the left-hand side.
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_2720_33_141_20} The diagram shows a box-and-whisker plot representing the times for the Penguins.
    2. On the same diagram, draw a box-and-whisker plot to represent the times for the Dolphins.
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_719_1219_424_424}
    3. Hence state one difference between the distributions of the times for the Penguins and the Dolphins.
    CAIE S1 2024 June Q5
    4 marks
    5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
    [0pt]
    1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
    2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
    3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3]
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
    4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.
    CAIE S1 2024 June Q6
    6
    1. How many different arrangements are there of the 9 letters in the word RECORDERS?
    2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together?
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
      The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
    3. Find the probability that the three Rs are in the same group.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
      \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}
    CAIE S1 2020 March Q1
    1 The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car. In how many ways can the members who will travel in the coach be chosen?
    CAIE S1 2020 March Q2
    2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.
    1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
      On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
    2. Draw up the probability distribution table for \(X\).
    3. Find \(\mathrm { E } ( X )\).
    CAIE S1 2020 March Q3
    3 The weights of apples of a certain variety are normally distributed with mean 82 grams. \(22 \%\) of these apples have a weight greater than 87 grams.
    1. Find the standard deviation of the weights of these apples.
    2. Find the probability that the weight of a randomly chosen apple of this variety differs from the mean weight by less than 4 grams.
    CAIE S1 2020 March Q4
    4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.
    1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
    2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.
    CAIE S1 2020 March Q5
    3 marks
    5 In Greenton, 70\% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
    [0pt]
    1. Find the probability that the number of adults in this sample who own a car is less than 6 . [3]
      A random sample of 120 adults from Greenton is now chosen.
    2. Use an approximation to find the probability that more than 75 of them own a car.
    CAIE S1 2020 March Q6
    6 Box \(A\) contains 7 red balls and 1 blue ball. Box \(B\) contains 9 red balls and 5 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 \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen.
    1. Complete the tree diagram to show the probabilities. Box \(A\)
      \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-08_624_428_667_621} \section*{Box \(B\)} Red Blue Red Blue
    2. Find the probability that the two balls chosen are not the same colour.
    3. Find the probability that the ball chosen from box \(A\) is blue given that the ball chosen from box \(B\) is blue.
    CAIE S1 2020 March Q7
    7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.
    Length (cm)\(0 - 9\)\(10 - 14\)\(15 - 19\)\(20 - 30\)
    Frequency15486621
    1. Draw a cumulative frequency graph to illustrate the data.
      \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
    2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
      The mean length of these 150 fish is 15.295 cm .
    3. Calculate an estimate for the variance of the lengths of the fish.
      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 March Q1
    1 A fair spinner with 5 sides numbered 1,2,3,4,5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.
    1. Find the probability that a score of 3 is obtained for the first time on the 8th spin.
    2. Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.
    CAIE S1 2021 March Q2
    2 Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these scarves. The probabilities for the three colours are \(0.2,0.45\) and 0.35 respectively. When she wears a red scarf, she always wears a hat. When she wears a blue scarf, she wears a hat with probability 0.4 . When she wears a yellow scarf, she wears a hat with probability 0.3 .
    1. Find the probability that on a randomly chosen day Georgie wears a hat.
    2. Find the probability that on a randomly chosen day Georgie wears a yellow scarf given that she does not wear a hat.
    CAIE S1 2021 March Q3
    3 The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes and standard deviation 18 minutes.
    1. Find the probability that a shopper chosen at random spends between 85 and 100 minutes in the shopping centre.
      \(88 \%\) of shoppers spend more than \(t\) minutes in the shopping centre.
    2. Find the value of \(t\).
    CAIE S1 2021 March Q4
    4 The random variable \(X\) takes the values \(1,2,3,4\) only. The probability that \(X\) takes the value \(x\) is \(k x ( 5 - x )\), where \(k\) is a constant.
    1. Draw up the probability distribution table for \(X\), in terms of \(k\).
    2. Show that \(\operatorname { Var } ( X ) = 1.05\).
    CAIE S1 2021 March Q5
    5 A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km , are summarised in the following table.
    Distance \(( \mathrm { km } )\)\(0 - 4\)\(5 - 10\)\(11 - 20\)\(21 - 30\)\(31 - 40\)\(41 - 60\)
    Frequency12163266204
    1. Draw a cumulative frequency graph to illustrate the data.
      \includegraphics[max width=\textwidth, alt={}, center]{3f05dc2a-b466-40bc-9f5f-0fd2bff120c8-06_1593_1397_852_415}
    2. For 30\% of these journeys the distance travelled is \(d \mathrm {~km}\) or more. Use your graph to estimate the value of \(d\).
    3. Calculate an estimate of the mean distance travelled for the 150 journeys.