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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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CAIE S1 2020 June Q2
4 marks Moderate -0.8
2 In a certain large college, \(22 \%\) of students own a car.
  1. 3 students from the college are chosen at random. Find the probability that all 3 students own a car.
  2. 16 students from the college are chosen at random. Find the probability that the number of these students who own a car is at least 2 and at most 4 .
CAIE S1 2020 June Q3
5 marks Moderate -0.8
3 In a certain town, the time, \(X\) hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard deviation 4.2 hours.
  1. Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.
  2. Find the value of \(k\) such that \(\mathrm { P } ( X < k ) = 0.75\).
CAIE S1 2020 June Q4
6 marks Moderate -0.8
4 A fair four-sided spinner has edges numbered 1, 2, 2, 3. A fair three-sided spinner has edges numbered \(- 2 , - 1,1\). Each spinner is spun and the number on the edge on which it comes to rest is noted. The random variable \(X\) is the sum of the two numbers that have been noted.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2020 June Q5
9 marks Standard +0.3
5 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The random variable \(X\) denotes the number of throws required to obtain a pair of tails.
  1. Find the expected value of \(X\).
  2. Find the probability that exactly 3 throws are required to obtain a pair of tails.
  3. Find the probability that fewer than 6 throws are required to obtain a pair of tails.
    On a different occasion, a pair of fair coins is thrown 80 times.
  4. Use an approximation to find the probability that a pair of tails is obtained more than 25 times.
CAIE S1 2020 June Q6
10 marks Easy -1.3
6 The annual salaries, in thousands of dollars, for 11 employees at each of two companies \(A\) and \(B\) are shown below.
Company \(A\)3032354141424749525364
Company \(B\)2647305241383542493142
  1. Represent the data by drawing a back-to-back stem-and-leaf diagram with company \(A\) on the left-hand side of the diagram.
  2. Find the median and the interquartile range of the salaries of the employees in company \(A\). [3]
    A new employee joins company \(B\). The mean salary of the 12 employees is now \(\\) 38500$.
  3. Find the salary of the new employee.
CAIE S1 2020 June Q7
10 marks Moderate -0.3
7
  1. Find the number of different possible arrangements of the 9 letters in the word CELESTIAL.
  2. Find the number of different arrangements of the 9 letters in the word CELESTIAL in which the first letter is C, the fifth letter is T and the last letter is E.
  3. Find the probability that a randomly chosen arrangement of the 9 letters in the word CELESTIAL does not have the two Es together.
    5 letters are selected at random from the 9 letters in the word CELESTIAL.
  4. Find the number of different selections if the 5 letters include at least one E and at most one L .
    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 June Q1
4 marks Moderate -0.3
1 A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are blue. How many different selections of 5 marbles contain at least 4 marbles of the same colour?
CAIE S1 2021 June Q2
5 marks Moderate -0.8
2 A company produces a particular type of metal rod. The lengths of these rods are normally distributed with mean 25.2 cm and standard deviation 0.4 cm . A random sample of 500 of these rods is chosen. How many rods in this sample would you expect to have a length that is within 0.5 cm of the mean length?
CAIE S1 2021 June Q3
8 marks Moderate -0.3
3
  1. How many different arrangements are there of the 8 letters in the word RELEASED?
  2. How many different arrangements are there of the 8 letters in the word RELEASED in which the letters LED appear together in that order?
  3. An arrangement of the 8 letters in the word RELEASED is chosen at random. Find the probability that the letters A and D are not together.
CAIE S1 2021 June Q4
7 marks Moderate -0.8
4 To gain a place at a science college, students first have to pass a written test and then a practical test.
Each student is allowed a maximum of two attempts at the written test. A student is only allowed a second attempt if they fail the first attempt. No student is allowed more than one attempt at the practical test. If a student fails both attempts at the written test, then they cannot attempt the practical test. The probability that a student will pass the written test at the first attempt is 0.8 . If a student fails the first attempt at the written test, the probability that they will pass at the second attempt is 0.6 . The probability that a student will pass the practical test is always 0.3 .
  1. Draw a tree diagram to represent this information, showing the probabilities on the branches.
  2. Find the probability that a randomly chosen student will succeed in gaining a place at the college.
    [0pt] [2]
  3. Find the probability that a randomly chosen student passes the written test at the first attempt given that the student succeeds in gaining a place at the college.
CAIE S1 2021 June Q5
8 marks Moderate -0.3
5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)
Number of players1654783220
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
  2. Calculate an estimate of the mean time taken by these 200 players.
  3. Find the greatest possible value of the interquartile range of these times.
CAIE S1 2021 June Q6
9 marks Moderate -0.8
6 In Questa, 60\% of the adults travel to work by car.
  1. A random sample of 12 adults from Questa is taken. Find the probability that the number who travel to work by car is less than 10 .
  2. A random sample of 150 adults from Questa is taken. Use an approximation to find the probability that the number who travel to work by car is less than 81 .
  3. Justify the use of your approximation in part (b).
CAIE S1 2021 June Q7
9 marks Standard +0.3
7 Sharma knows that she has 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn in her cupboard. All the tins are the same shape and size, but the labels have all been removed, so Sharma does not know what each tin contains. Sharma wants carrots for her meal, and she starts opening the tins one at a time, chosen randomly, until she opens a tin of carrots. The random variable \(X\) is the number of tins that she needs to open.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 6 } { 35 }\).
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\operatorname { Var } ( X )\).
    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 June Q1
5 marks Moderate -0.8
1 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted by the random variable \(X\).
  1. Write down the mean of \(X\).
  2. Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw.
  3. Find the probability that a 5 is first obtained in fewer than 10 throws.
CAIE S1 2021 June Q2
4 marks Standard +0.3
2 The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 2000 bags of sugar, 72 weighed more than 1.10 kg . Find the value of \(\sigma\).
CAIE S1 2021 June Q3
6 marks Moderate -0.3
3 On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are \(0.4,0.35\) and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55 . When she travels by train, the probability that she arrives late is 0.7 . When she travels by car, the probability that she arrives late is \(x\). On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late is 0.48 .
  1. Find the value of \(x\).
  2. Find the probability that Alexa travels to work by train given that she arrives late.
CAIE S1 2021 June Q4
6 marks Moderate -0.8
4 A fair spinner has sides numbered 1, 2, 2. Another fair spinner has sides numbered \(- 2,0,1\). Each spinner is spun. The number on the side on which a spinner comes to rest is noted. The random variable \(X\) is the sum of the numbers for the two spinners.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2021 June Q5
9 marks Moderate -0.8
5 Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight arrives early is 0.15 , the probability that it arrives on time is 0.55 and the probability that it arrives late is 0.3 .
  1. Find the probability that on each of 3 randomly chosen days, Richard's flight does not arrive late.
  2. Find the probability that for 9 randomly chosen days, Richard's flight arrives early at least 3 times.
  3. 60 days are chosen at random. Use an approximation to find the probability that Richard's flight arrives early at least 12 times.
CAIE S1 2021 June Q6
10 marks Standard +0.3
6
  1. Find the total number of different arrangements of the 8 letters in the word TOMORROW.
  2. Find the total number of different arrangements of the 8 letters in the word TOMORROW that have an R at the beginning and an R at the end, and in which the three Os are not all together.
    Four letters are selected at random from the 8 letters of the word TOMORROW.
  3. Find the probability that the selection contains at least one O and at least one R .
CAIE S1 2021 June Q7
10 marks Easy -1.2
7 The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below.
Amazons205198181182190215201178202196184
Giants175182184187189192193195195195204
  1. State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information.
  2. Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram.
  3. Find the interquartile range of the heights of the players in the Amazons.
    Four new players join the Amazons. The mean height of the 15 players in the Amazons in now 191.2 cm . The heights of three of the new players are \(180 \mathrm {~cm} , 185 \mathrm {~cm}\) and 190 cm .
  4. Find the height of the fourth new player.
    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 June Q1
5 marks Easy -1.8
1 The heights in cm of 160 sunflower plants were measured. The results are summarised on the following cumulative frequency curve. \includegraphics[max width=\textwidth, alt={}, center]{b72bd3eb-2c10-4d01-ab18-74d6fb812f27-02_1783_1424_404_356}
  1. Use the graph to estimate the number of plants with heights less than 100 cm .
  2. Use the graph to estimate the 65th percentile of the distribution.
  3. Use the graph to estimate the interquartile range of the heights of these plants.
CAIE S1 2021 June Q2
4 marks Moderate -0.5
2 The random variable \(X\) can take only the values \(- 2 , - 1,0,1,2\). The probability distribution of \(X\) is given in the following table.
\(x\)- 2- 1012
\(\mathrm { P } ( X = x )\)\(p\)\(p\)0.1\(q\)\(q\)
Given that \(\mathrm { P } ( X \geqslant 0 ) = 3 \mathrm { P } ( X < 0 )\), find the values of \(p\) and \(q\).
CAIE S1 2021 June Q3
5 marks Moderate -0.8
3 A sports club has a volleyball team and a hockey team. The heights of the 6 members of the volleyball team are summarised by \(\Sigma x = 1050\) and \(\Sigma x ^ { 2 } = 193700\), where \(x\) is the height of a member in cm . The heights of the 11 members of the hockey team are summarised by \(\Sigma y = 1991\) and \(\Sigma y ^ { 2 } = 366400\), where \(y\) is the height of a member in cm .
  1. Find the mean height of all 17 members of the club.
  2. Find the standard deviation of the heights of all 17 members of the club.
CAIE S1 2021 June Q4
6 marks Moderate -0.3
4 Three fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.
  1. Find the probability that the score is 4 on a single throw of the three dice.
  2. Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.
CAIE S1 2021 June Q5
9 marks Standard +0.3
5 The lengths of the leaves of a particular type of tree are modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves from this type of tree and finds that 42 are less than 4 cm long and 100 are more than 10 cm long.
  1. Find estimates for the mean and standard deviation of the lengths of leaves from this type of tree.
    The lengths, in cm , of the leaves of a different type of tree have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The scientist takes a random sample of 800 leaves from this type of tree.
  2. Find how many of these leaves the scientist would expect to have lengths, in cm , between \(\mu - 2 \sigma\) and \(\mu + 2 \sigma\).