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CAIE S1 2023 June Q4
9 marks Standard +0.3
4 A fair 5 -sided spinner has sides labelled 1, 2, 3, 4, 5. The spinner is spun repeatedly until a 2 is obtained on the side on which the spinner lands. The random variable \(X\) denotes the number of spins required.
  1. Find \(\mathrm { P } ( X = 4 )\).
  2. Find \(\mathrm { P } ( X < 6 )\).
    Two fair 5 -sided spinners, each with sides labelled \(1,2,3,4,5\), are spun at the same time. If the numbers obtained are equal, the score is 0 . Otherwise, the score is the higher number minus the lower number.
  3. Find the probability that the score is greater than 0 given that the score is not equal to 2 .
    The two spinners are spun at the same time repeatedly .
  4. For 9 randomly chosen spins of the two spinners, find the probability that the score is greater than 2 on at least 3 occasions.
CAIE S1 2023 June Q5
12 marks Standard +0.3
5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.
  1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
    The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
  2. Find the value of \(\sigma\).
    A random sample of 120 Eastern bluebirds is chosen.
  3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .
CAIE S1 2023 June Q6
10 marks Standard +0.3
6 In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition.
  1. Find the number of different ways in which the team of 7 can be selected if it consists of exactly 1 swimmer, at least 4 cyclists and at most 2 runners.
    For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph.
  2. How many different arrangements are there of the 9 people if the swimmers stand together, the cyclists stand together and the runners stand together?
  3. How many different arrangements are there of the 9 people if none of the cyclists stand next to 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 2023 June Q1
4 marks Moderate -0.3
1 Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. State the value of \(\mathrm { E } ( X )\).
  2. Find the probability that exactly 5 throws are required to obtain a pair of heads.
  3. Find the probability that fewer than 7 throws are required to obtain a pair of heads.
CAIE S1 2023 June Q2
5 marks Moderate -0.5
2 Anil is a candidate in an election. He received \(40 \%\) of the votes. A random sample of 120 voters is chosen. Use an approximation to find the probability that, of the 120 voters, between 36 and 54 inclusive voted for Anil.
CAIE S1 2023 June Q3
7 marks Standard +0.3
3 The random variable \(X\) takes the values \(1,2,3,4\). It is given that \(\mathrm { P } ( X = x ) = k x ( x + a )\), where \(k\) and \(a\) are constants.
  1. Given that \(\mathrm { P } ( X = 4 ) = 3 \mathrm { P } ( X = 2 )\), find the value of \(a\) and the value of \(k\).
  2. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  3. Given that \(\mathrm { E } ( X ) = 3.2\), find \(\operatorname { Var } ( X )\).
CAIE S1 2023 June Q4
8 marks Easy -1.2
4 The times taken, in minutes, to complete a cycle race by 19 cyclists from each of two clubs, the Cheetahs and the Panthers, are represented in the following back-to-back stem-and-leaf diagram.
CheetahsPanthers
9874
87320868
987917899
6533110234456
Key: 7 |9| 1 means 97 minutes for Cheetahs and 91 minutes for Panthers
  1. Find the median and the interquartile range of the times of the Cheetahs.
    The median and interquartile range for the Panthers are 103 minutes and 14 minutes respectively.
  2. Make two comparisons between the times taken by the Cheetahs and the times taken by the Panthers.
    Another cyclist, Kenny, from the Cheetahs also took part in the race. The mean time taken by the 20 cyclists from the Cheetahs was 99 minutes.
  3. Find the time taken by Kenny to complete the race.
CAIE S1 2023 June Q5
7 marks Standard +0.3
5 Jasmine throws two ordinary fair 6-sided dice at the same time and notes the numbers on the uppermost faces. The events \(A\) and \(B\) are defined as follows. \(A\) : The sum of the two numbers is less than 6 . \(B : \quad\) The difference between the two numbers is at most 2 .
  1. Determine whether or not the events \(A\) and \(B\) are independent.
  2. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
CAIE S1 2023 June Q6
10 marks Standard +0.3
6 The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On \(10 \%\) of days less than 16 kg of grapes are sold.
  1. Find the standard deviation of the mass of grapes sold per day.
    The mass of grapes sold on any day is independent of the mass sold on any other day.
  2. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
  3. In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?
CAIE S1 2023 June Q7
9 marks Standard +0.3
7
  1. Find the number of different arrangements of the 10 letters in the word CASABLANCA in which the two Cs are not together.
  2. Find the number of different arrangements of the 10 letters in the word CASABLANCA which have an A at the beginning, an A at the end and exactly 3 letters between the 2 Cs .
    Five letters are selected from the 10 letters in the word CASABLANCA.
  3. Find the number of different selections in which the five letters include at least two As and at most one C.
    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
4 marks Standard +0.3
1 A summary of 20 values of \(x\) gives $$\Sigma ( x - 30 ) = 439 , \quad \Sigma ( x - 30 ) ^ { 2 } = 12405 .$$ A summary of another 25 values of \(x\) gives $$\sum ( x - 30 ) = 470 , \quad \sum ( x - 30 ) ^ { 2 } = 11346 .$$
  1. Find the mean of all 45 values of \(x\).
  2. Find the standard deviation of all 45 values of \(x\).
CAIE S1 2024 June Q2
7 marks Moderate -0.3
2 The lengths of the tails of adult raccoons of a certain species are normally distributed with mean 28 cm and standard deviation 3.3 cm .
  1. Find the probability that a randomly chosen adult raccoon of this species has a tail length between 23 cm and 35 cm .
    The masses of adult raccoons of this species are normally distributed with mean 8.5 kg and standard deviation \(\sigma \mathrm { kg } .75 \%\) of adult raccoons of this species have mass greater than 7.6 kg .
  2. Find the value of \(\sigma\).
CAIE S1 2024 June Q3
6 marks Moderate -0.3
3 The heights, in cm, of 200 adults in Barimba are summarised in the following table.
Height \(( h \mathrm {~cm} )\)\(130 \leqslant h < 150\)\(150 \leqslant h < 160\)\(160 \leqslant h < 170\)\(170 \leqslant h < 175\)\(175 \leqslant h < 195\)
Frequency1632766412
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-04_1397_1495_762_287}
  2. The interquartile range is \(R \mathrm {~cm}\). Show that \(R\) is not greater than 15 .
CAIE S1 2024 June Q4
6 marks Standard +0.3
4 A game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn. A player who obtains a 4 in their turn scores 1 point.
  1. Show that the probability that a player obtains a 4 in one turn is \(\frac { 37 } { 64 }\).
    Xeno and Yao play this game.
  2. Find the probability that neither Xeno nor Yao score any points in their first two turns.
  3. Xeno and Yao each have three turns. Find the probability that Xeno scores 2 more points than Yao. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1548_376_349} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1566_466_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_77_1570_913_324} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1570_1279_324} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . .......................................................................................................................................... .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_76_1570_2083_324} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_2359_324} ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2542_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2631_324}
CAIE S1 2024 June Q5
10 marks Standard +0.3
5 In a certain area in the Arctic the probability that it snows on any given day is 0.7 , independent of all other days.
  1. Find the probability that in a week (7 days) it snows on at least five days.
    A week in which it snows on at least five days out of seven is called a 'white' week.
  2. Find the probability that in three randomly chosen weeks at least one is a white week.
    In a different area in the Arctic, the probability that a week is a white week is 0.8 .
  3. Use a suitable approximation to find the probability that in 60 randomly chosen weeks fewer than 47 are white weeks.
CAIE S1 2024 June Q6
7 marks Standard +0.3
6 Harry has three coins:
  • One coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 3 }\).
  • The second coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\).
  • The third coin is biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 5 }\).
Harry throws the three coins. The random variable \(X\) is the number of heads that he obtains.
  1. Draw up the probability distribution table for \(X\).
    Harry has two other coins, each of which is biased so that the probability of obtaining a head when it is thrown is \(p\). He throws all five coins at the same time. The random variable \(Y\) is the number of heads that he obtains.
  2. Given that \(\mathrm { P } ( Y = 0 ) = 6 \mathrm { P } ( Y = 5 )\), find the value of \(p\).
CAIE S1 2024 June Q7
10 marks Standard +0.3
7 The eight digits \(1,2,2,3,4,4,4,5\) are arranged in a line.
  1. How many different arrangements are there of these 8 digits?
  2. Find the number of different arrangements of the 8 digits in which there is a 2 at the beginning, a 2 at the end and the three 4 s are not all together.
    Three digits are selected at random from the eight digits \(1,2,2,3,4,4,4,5\).
  3. Find the probability that the three digits are all different.
    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
5 marks Moderate -0.8
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
5 marks Standard +0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.8
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
7 marks Moderate -0.8
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
10 marks Moderate -0.8
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
    10 marks Standard +0.8
    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
    8 marks Moderate -0.8
    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 \(X > 3\).