Questions S1 (2020 questions)

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CAIE S1 2021 March Q6
10 marks Standard +0.8
6
  1. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
  2. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
  3. Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.
CAIE S1 2021 March Q7
11 marks Moderate -0.3
7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table.
SwimmingCyclingRunning
Female1045066
Male315792
A student is chosen at random.
    1. Find the probability that the student prefers swimming.
    2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
    1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
    2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      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 2022 March Q1
5 marks Moderate -0.8
1 A fair red spinner has edges numbered \(1,2,2,3\). A fair blue spinner has edges numbered \(- 3 , - 2 , - 1 , - 1\). Each spinner is spun once and the number on the edge on which each spinner lands is noted. The random variable \(X\) denotes the sum of the resulting two numbers.
  1. Draw up the probability distribution table for \(X\).
  2. Given that \(\mathrm { E } ( X ) = 0.25\), find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2022 March Q2
6 marks Standard +0.3
2 In a certain country, the probability of more than 10 cm of rain on any particular day is 0.18 , independently of the weather on any other day.
  1. Find the probability that in any randomly chosen 7-day period, more than 2 days have more than 10 cm of rain.
  2. For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than 10 cm of rain.
CAIE S1 2022 March Q3
6 marks Moderate -0.8
3 At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.
Time taken, in minutes\(1 - 30\)\(31 - 45\)\(46 - 65\)\(66 - 75\)\(76 - 100\)
Frequency2130688645
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{c1bc5ac2-6b0e-48c7-92e9-9b8b56b57d90-05_1000_1198_785_516}
  2. State which class interval contains the median.
  3. Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.
CAIE S1 2022 March Q4
11 marks Standard +0.3
4 The weights of male leopards in a particular region are normally distributed with mean 55 kg and standard deviation 6 kg .
  1. Find the probability that a randomly chosen male leopard from this region weighs between 46 and 62 kg .
    The weights of female leopards in this region are normally distributed with mean 42 kg and standard deviation \(\sigma \mathrm { kg }\). It is known that \(25 \%\) of female leopards in the region weigh less than 36 kg .
  2. Find the value of \(\sigma\).
    The distributions of the weights of male and female leopards are independent of each other. A male leopard and a female leopard are each chosen at random.
  3. Find the probability that both the weights of these leopards are less than 46 kg .
CAIE S1 2022 March Q5
10 marks Standard +0.3
5 A group of 12 people consists of 3 boys, 4 girls and 5 adults.
  1. In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
  2. In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?
    The same group of 12 people stand in a line.
  3. How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?
CAIE S1 2022 March Q6
12 marks Moderate -0.3
6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio \(3 : 5 : 7\) respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.
  1. Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
  2. Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.
    'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry. Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
  3. Find the probability that none of Petra's 3 chocolates has orange flavour.
  4. Find the probability that each of Petra's 3 chocolates has a different flavour.
  5. Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.
    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 March Q1
4 marks Standard +0.3
1 A bag contains 9 blue marbles and 3 red marbles. One marble is chosen at random from the bag. If this marble is blue, it is replaced back into the bag. If this marble is red, it is not returned to the bag. A second marble is now chosen at random from the bag.
  1. Find the probability that both the marbles chosen are red.
  2. Find the probability that the first marble chosen is blue given that the second marble chosen is red.
CAIE S1 2024 March Q2
8 marks Moderate -0.5
2 Sam is a member of a soccer club. She is practising scoring goals. The probability that Sam will score a goal on any attempt is 0.7 , independently of all other attempts.
  1. Sam makes 10 attempts at scoring goals. Find the probability that Sam will score goals on fewer than 8 of these attempts.
  2. Find the probability that Sam's first successful attempt will be before her 5th attempt.
  3. Wei is a member of the same soccer club. He is also practising scoring goals. The probability that Wei will score a goal on any attempt is 0.6 , independently of all other attempts. Wei is going to keep making attempts until he scores 3 goals.
    Find the probability that he scores his third goal on his 7th attempt.
CAIE S1 2024 March Q3
8 marks Moderate -0.8
3 The times taken, in minutes, by 150 students to complete a puzzle are summarised in the table.
Time taken
\(( t\) minutes \()\)
\(0 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 35\)\(35 \leqslant t < 40\)\(40 \leqslant t < 50\)\(50 \leqslant t < 70\)
Frequency82335522012
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d1a3524c-a3b5-45fe-86a7-5cbda087efcd-06_1193_1489_886_328}
  2. Calculate an estimate for the mean time for these students to complete the puzzle.
  3. In which class interval does the lower quartile of the times lie?
CAIE S1 2024 March Q4
12 marks Standard +0.3
4 A company sells small and large bags of rice. The masses of the small bags of rice are normally distributed with mean 1.20 kg and standard deviation 0.16 kg .
  1. In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than 1.26 kg ?
    The masses of the large bags of rice are normally distributed with mean 2.50 kg and standard deviation \(\sigma \mathrm { kg } .20 \%\) of these large bags of rice have a mass less than 2.40 kg .
  2. Find the value of \(\sigma\).
    A random sample of 80 large bags of rice is chosen.
  3. Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than 2.40 kg .
CAIE S1 2024 March Q5
8 marks Moderate -0.3
5 Anil is taking part in a tournament. In each game in this tournament, players are awarded 2 points for a win, 1 point for a draw and 0 points for a loss. For each of Anil's games, the probabilities that he will win, draw or lose are \(0.5,0.3\) and 0.2 respectively. The results of the games are all independent of each other. The random variable \(X\) is the total number of points that Anil scores in his first 3 games in the tournament.
  1. Show that \(\mathrm { P } ( X = 2 ) = 0.114\).
  2. Complete the probability distribution table for \(X\).
    \(x\)0123456
    \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\)0.1140.2070.2850.125
  3. Find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2024 March Q6
10 marks Standard +0.3
6 A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.
  1. In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?
    The 10 members of the club stand in a line for a photograph.
  2. How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?
    For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club.
  3. How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?
    If you use the following page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2020 November Q1
4 marks Moderate -0.8
1 Two ordinary fair dice, one red and the other blue, are thrown.
Event \(A\) is 'the score on the red die is divisible by 3 '.
Event \(B\) is 'the sum of the two scores is at least 9 '.
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Hence determine whether or not the events \(A\) and \(B\) are independent.
CAIE S1 2020 November Q2
5 marks Moderate -0.5
2 The probability that a student at a large music college plays in the band is 0.6. For a student who plays in the band, the probability that she also sings in the choir is 0.3 . For a student who does not play in the band, the probability that she sings in the choir is \(x\). The probability that a randomly chosen student from the college does not sing in the choir is 0.58 .
  1. Find the value of \(x\).
    Two students from the college are chosen at random.
  2. Find the probability that both students play in the band and both sing in the choir.
CAIE S1 2020 November Q3
5 marks Moderate -0.8
3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .
  1. Find the probability that Kayla takes more than 6 throws to achieve a success.
  2. Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.
CAIE S1 2020 November Q4
6 marks Standard +0.3
4 The random variable \(X\) takes each of the values \(1,2,3,4\) with probability \(\frac { 1 } { 4 }\). Two independent values of \(X\) are chosen at random. If the two values of \(X\) are the same, the random variable \(Y\) takes that value. Otherwise, the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).
  1. Draw up the probability distribution table for \(Y\).
  2. Find the probability that \(Y = 2\) given that \(Y\) is even.
CAIE S1 2020 November Q5
9 marks Moderate -0.8
5 The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.
  1. Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.
  2. On 90\% of days Davin plays on his games machine for more than \(t\) hours. Find the value of \(t\).
  3. Calculate an estimate for the number of days in a year ( 365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.
CAIE S1 2020 November Q6
10 marks Easy -1.8
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
11 marks Moderate -0.3
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
5 marks Moderate -0.8
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
7 marks Moderate -0.8
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
9 marks Moderate -0.8
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
9 marks Moderate -0.3
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.