Questions — CAIE S1 (785 questions)

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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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
CAIE S1 2021 March Q6
4 marks
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
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
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
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
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
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
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
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 2023 March Q1
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
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
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
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
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
3 marks
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
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
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
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
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
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
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
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
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
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
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
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.