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CAIE S1 2023 March Q1
8 marks Easy -1.3
1 Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.
\(x\)\(30 \leqslant x < 60\)\(60 \leqslant x < 90\)\(90 \leqslant x < 110\)\(110 \leqslant x < 140\)\(140 \leqslant x < 180\)\(180 \leqslant x \leqslant 240\)
Number
of years
48142572
  1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{2577c597-0a04-4909-ad71-1347aacec6d9-02_1395_1397_881_415}
  2. Use your graph to estimate the 70th percentile of the data.
  3. Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years.
CAIE S1 2023 March Q2
7 marks Moderate -0.3
2 Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6 . The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.
  1. Show that the probability of obtaining exactly one head is 0.225 .
  2. Complete the following probability distribution table for \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)0.050.2250.075
  3. Given that \(\mathrm { E } ( X ) = 2.1\), find the value of \(\operatorname { Var } ( X )\). \(380 \%\) of the residents of Kinwawa are in favour of a leisure centre being built in the town.
    20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.
  4. Find the probability that more than 17 of these residents are in favour of the leisure centre.
  5. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre.
  6. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre.
CAIE S1 2023 March Q4
3 marks Standard +0.3
4 The probability that it will rain on any given day is \(x\). If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3 . Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4 . If he does not wear a hat, the probability that he wears a scarf is 0.1 . The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36 . Find the value of \(x\).
CAIE S1 2023 March Q5
3 marks Challenging +1.2
5 Marco has four boxes labelled \(K , L , M\) and \(N\). He places them in a straight line in the order \(K , L , M\), \(N\) with \(K\) on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events \(A\) and \(B\) are defined as follows. \(A\) : The white marble is in either box \(L\) or box \(M\). \(B\) : The red marble is to the left of both the green marble and the yellow marble.
Determine whether or not events \(A\) and \(B\) are independent.
CAIE S1 2023 March Q6
11 marks Standard +0.3
6 In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.
  1. Find the probability that a randomly chosen cyclist has a time less than 74 minutes.
  2. Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes.
    In a different cycling event, the times can also be modelled by a normal distribution. \(23 \%\) of the cyclists have times less than 36 minutes and \(10 \%\) of the cyclists have times greater than 54 minutes.
  3. Find estimates for the mean and standard deviation of this distribution.
CAIE S1 2023 March Q7
12 marks Standard +0.8
7
  1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other.
  2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds.
    Five letters are selected from the 9 letters in the word DELIVERED.
    [0pt]
  3. Find the number of different selections if the 5 letters include at least one D and at least one E . [3]
    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.
CAIE S1 2020 November Q5
9 marks Easy -1.2
5 The following table gives the weekly snowfall, in centimetres, for 11 weeks in 2018 at two ski resorts, Dados and Linva.
Dados68121510364228102216
Linva2111516032364010129
  1. Represent the information in a back-to-back stem-and-leaf diagram.
  2. Find the median and the interquartile range for the weekly snowfall in Dados.
  3. The median, lower quartile and upper quartile of the weekly snowfall for Linva are 12, 9 and 32 cm respectively. Use this information and your answers to part (b) to compare the central tendency and the spread of the weekly snowfall in Dados and Linva.
CAIE S1 2020 November Q6
11 marks Standard +0.3
6 Mr and Mrs Ahmed with their two children, and Mr and Mrs Baker with their three children, are visiting an activity centre together. They will divide into groups for some of the activities.
  1. In how many ways can the 9 people be divided into a group of 6 and a group of 3?
    5 of the 9 people are selected at random for a particular activity.
  2. Find the probability that this group of 5 people contains all 3 of the Baker children.
    All 9 people stand in a line.
  3. Find the number of different arrangements in which Mr Ahmed is not standing next to Mr Baker.
  4. Find the number of different arrangements in which there is exactly one person between Mr Ahmed and Mr Baker.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.