Questions S1 (1967 questions)

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CAIE S1 2023 June Q3
3
  1. Find the number of different arrangements of the 8 letters in the word COCOONED.
  2. Find the number of different arrangements of the 8 letters in the word COCOONED in which the first letter is O and the last letter is N .
  3. Find the probability that a randomly chosen arrangement of the 8 letters in the word COCOONED has all three Os together given that the two Cs are next to each other.
CAIE S1 2023 June Q4
4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.
  1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
    All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
  2. Find the value of \(\mu\) and the value of \(\sigma\).
CAIE S1 2023 June Q5
5 The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.
Population\(100 - 800\)\(900 - 1200\)\(1300 - 2000\)\(2100 - 3200\)\(3300 - 4800\)
Number of villages812504832
  1. Draw a histogram to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{a7157882-d87e-4efb-abc5-9c9f58197012-08_1395_1195_1043_516}
  2. Write down the class interval which contains the median for this information.
  3. Find the greatest possible value of the interquartile range for the populations of the 150 villages.
CAIE S1 2023 June Q6
6 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
  2. Complete the following probability distribution table for \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)\(\frac { 81 } { 256 }\)\(\frac { 3 } { 64 }\)\(\frac { 1 } { 256 }\)
  3. Find \(\mathrm { E } ( X )\).
    Eli throws the four dice at the same time on 96 occasions.
  4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.
CAIE S1 2023 June Q7
7 A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.
  1. Find the probability that the first time that the free gift is an elephant is before the 6th Monday.
  2. Find the probability that Sahim will get more than two leopards in the 12 magazines.
  3. Find the probability that after 5 weeks Sahim has exactly one of each animal.
    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
1 The random variable \(X\) takes the values \(- 2,2\) and 3. It is given that $$\mathrm { P } ( X = x ) = k \left( x ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
  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 June Q2
2 A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4 . On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.
  1. Find the probability that it does not rain on any of the 4 days of the event.
  2. Find the probability that the first day on which it rains during the event is Tuesday.
  3. Find the probability that it rains on exactly one of the 4 days of the event.
CAIE S1 2023 June Q3
3 The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, \(A\) and \(B\).
Company \(A\)Company \(B\)
\multirow{6}{*}{9}411025445667
72102601355799
4210271346688
54202801222
98529
1309
Key: 1 |27| 6 means \(
) 2710\( for company \)A\( and \)\\( 2760\) for company \(B\)
  1. Find the median and the interquartile range of the monthly salaries of employees in company \(A\).
    The lower quartile, median and upper quartile for company \(B\) are \(
    ) 2600 , \\( 2690\) and \(
    ) 2780\( respectively.
  2. Draw two box-and-whisker plots in a single diagram to represent the information for the salaries of employees at companies \)A\( and \)B$.
    \includegraphics[max width=\textwidth, alt={}, center]{f2666d82-4711-499a-98c0-3421e4c228fb-07_810_1406_573_411}
  3. Comment on whether the mean would be a more appropriate measure than the median for comparing the given information for the two companies.
CAIE S1 2023 June Q4
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.