Questions — CAIE S1 (785 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 November Q4
4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.
  1. Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
  2. Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
  3. On 95\% of days, Raj runs for more than \(t\) minutes. Find the value of \(t\).
CAIE S1 2021 November Q5
5 A security code consists of 2 letters followed by a 4-digit number. The letters are chosen from \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) and the digits are chosen from \(\{ 1,2,3,4,5,6,7 \}\). No letter or digit may appear more than once. An example of a code is BE 3216 .
  1. How many different codes can be formed?
  2. Find the number of different codes that include the letter A or the digit 5 or both.
    A security code is formed at random.
  3. Find the probability that the code is DE followed by a number between 4500 and 5000 .
CAIE S1 2021 November Q6
6 In a game, Jim throws three darts at a board. This is called a 'turn'. The centre of the board is called the bull's-eye. The random variable \(X\) is the number of darts in a turn that hit the bull's-eye. The probability distribution of \(X\) is given in the following table.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.6\(p\)\(q\)0.05
It is given that \(\mathrm { E } ( X ) = 0.55\).
  1. Find the values of \(p\) and \(q\).
  2. Find \(\operatorname { Var } ( X )\).
    Jim is practising for a competition and he repeatedly throws three darts at the board.
  3. Find the probability that \(X = 1\) in at least 3 of 12 randomly chosen turns.
  4. Find the probability that Jim first succeeds in hitting the bull's-eye with all three darts on his 9th turn.
CAIE S1 2021 November Q7
7 Box \(A\) contains 6 red balls and 4 blue balls. Box \(B\) contains \(x\) red balls and 9 blue balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\).
  1. Complete the tree diagram below, giving the remaining four probabilities in terms of \(x\).
    \includegraphics[max width=\textwidth, alt={}, center]{217c5a58-2966-4b86-b3b6-9d1676d2979c-12_688_759_484_731}
  2. Show that the probability that both balls chosen are blue is \(\frac { 4 } { x + 10 }\).
    It is given that the probability that both balls chosen are blue is \(\frac { 1 } { 6 }\).
  3. Find the probability, correct to 3 significant figures, that the ball chosen from box \(A\) is red given that the ball chosen from box \(B\) is red.
    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 November Q1
1 The probability distribution table for a random variable \(X\) is shown below.
\(x\)- 2- 10.512
\(\mathrm { P } ( X = x )\)0.12\(p\)\(q\)0.160.3
Given that \(\mathrm { E } ( X ) = 0.28\), find the value of \(p\) and the value of \(q\).
CAIE S1 2022 November Q2
2 The residents of Persham were surveyed about the reliability of their internet service. 12\% rated the service as 'poor', \(36 \%\) rated it as 'satisfactory' and \(52 \%\) rated it as 'good'. A random sample of 8 residents of Persham is chosen.
  1. Find the probability that more than 2 and fewer than 8 of them rate their internet service as poor or satisfactory.
    A random sample of 125 residents of Persham is now chosen.
  2. Use an approximation to find the probability that more than 72 of these residents rate their internet service as good.
CAIE S1 2022 November Q3
3 The Lions and the Tigers are two basketball clubs. The heights, in cm, of the 11 players in each of their first team squads are given in the table.
Lions178186181187179190189190180169196
Tigers194179187190183201184180195191197
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Lions on the left.
  2. Find the median and the interquartile range of the heights of the Lions first team squad.
    It is given that for the Tigers, the lower quartile is 183 cm , the median is 190 cm and the upper quartile is 195 cm .
  3. Make two comparisons between the heights of the players in the Lions first team squad and the heights of the players in the Tigers first team squad.
CAIE S1 2022 November Q4
4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.
  1. Find the probability that the SBP of a randomly chosen adult is less than 132.
    The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
  2. Find the standard deviation of this distribution.
    Three adults are chosen at random from this population.
  3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.
CAIE S1 2022 November Q5
5 A game is played with an ordinary fair 6-sided die. A player throws the die once. If the result is \(2,3,4\) or 5 , that result is the player's score and the player does not throw the die again. If the result is 1 or 6 , the player throws the die a second time and the player's score is the sum of the two numbers from the two throws.
  1. Draw a fully labelled tree diagram to represent this information. Events \(A\) and \(B\) are defined as follows.
    \(A\) : the player's score is \(5,6,7,8\) or 9
    \(B\) : the player has two throws
  2. Show that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\).
  3. Determine whether or not events \(A\) and \(B\) are independent.
  4. Find \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
CAIE S1 2022 November Q6
6 A Social Club has 15 members, of whom 8 are men and 7 are women. The committee of the club consists of 5 of its members.
  1. Find the number of different ways in which the committee can be formed from the 15 members if it must include more men than women.
    The 15 members are having their photograph taken. They stand in three rows, with 3 people in the front row, 5 people in the middle row and 7 people in the back row.
  2. In how many different ways can the 15 members of the club be divided into a group of 3, a group of 5 and a group of 7 ?
    In one photograph Abel, Betty, Cally, Doug, Eve, Freya and Gino are the 7 members in the back row.
  3. In how many different ways can these 7 members be arranged so that Abel and Betty are next to each other and Freya and Gino are not 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 2022 November Q1
1 On any day, Kino travels to school by bus, by car or on foot with probabilities 0.2, 0.1 and 0.7 respectively. The probability that he is late when he travels by bus is \(x\). The probability that he is late when he travels by car is \(2 x\) and the probability that he is late when he travels on foot is 0.25 . The probability that, on a randomly chosen day, Kino is late is 0.235 .
  1. Find the value of \(x\).
  2. Find the probability that, on a randomly chosen day, Kino travels to school by car given that he is not late.
CAIE S1 2022 November Q2
2 The lengths of the rods produced by a company are normally distributed with mean 55.6 mm and standard deviation 1.2 mm .
  1. In a random sample of 400 of these rods, how many would you expect to have length less than 54.8 mm ?
  2. Find the probability that a randomly chosen rod produced by this company has a length that is within half a standard deviation of the mean.
CAIE S1 2022 November Q3
3 Three fair 6-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time repeatedly. The score on each throw is the sum of the numbers on the uppermost faces.
  1. Find the probability that a score of 17 or more is first obtained on the 6th throw.
  2. Find the probability that a score of 17 or more is obtained in fewer than 8 throws.
CAIE S1 2022 November Q4
4 The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.
Time taken \(( t\) minutes \()\)\(0 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 50\)\(50 \leqslant t < 60\)\(60 \leqslant t < 100\)
Frequency3246965224
  1. Draw a histogram to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{3e74785d-5981-480c-a0fd-f43d5d227f2d-06_1201_1198_1050_516} From the data, the estimate of the mean time taken by these 250 employees is 43.2 minutes.
  2. Calculate an estimate for the standard deviation of these times.
CAIE S1 2022 November Q5
5 Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac { 1 } { 4 }\), independently of all other throws. Eric throws all three coins at the same time. Events \(A\) and \(B\) are defined as follows.
\(A\) : all three coins show the same result
\(B\) : at least one of the biased coins shows a head
  1. Show that \(\mathrm { P } ( B ) = \frac { 7 } { 16 }\).
  2. Find \(\mathrm { P } ( A \mid B )\).
    The random variable \(X\) is the number of heads obtained when Eric throws the three coins.
  3. Draw up the probability distribution table for \(X\).
CAIE S1 2022 November Q6
6 At a company's call centre, \(90 \%\) of callers are connected immediately to a representative.
A random sample of 12 callers is chosen.
  1. Find the probability that fewer than 10 of these callers are connected immediately.
    A random sample of 80 callers is chosen.
  2. Use an approximation to find the probability that more than 69 of these callers are connected immediately.
  3. Justify the use of your approximation in part (b).
CAIE S1 2022 November Q7
7
  1. Find the number of different arrangements of the 9 letters in the word ALLIGATOR in which the two As are together and the two Ls are together.
  2. The 9 letters in the word ALLIGATOR are arranged in a random order. Find the probability that the two Ls are together and there are exactly 6 letters between the two As.
  3. Find the number of different selections of 5 letters from the 9 letters in the word ALLIGATOR which contain at least one A 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 2022 November Q2
2 In a large college, \(32 \%\) of the students have blue eyes. A random sample of 80 students is chosen. Use an approximation to find the probability that fewer than 20 of these students have blue eyes.
CAIE S1 2022 November Q3
3 The times, \(t\) minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.
Time taken \(( t\) minutes \()\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 35\)\(t \leqslant 40\)\(t \leqslant 50\)\(t \leqslant 60\)
Cumulative frequency3266112178228250
  1. Draw a cumulative frequency graph to illustrate the data.
    \includegraphics[max width=\textwidth, alt={}, center]{1eb957f4-5088-4991-aa8a-f895d55d2bcf-04_1395_1298_705_466}
  2. Use your graph to estimate the 60th percentile of the data.
    It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.
  3. Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.
CAIE S1 2022 November Q4
4 Three fair 4-sided spinners each have sides labelled 1,2,3,4. The spinners are spun at the same time and the number on the side on which each spinner lands is recorded. The random variable \(X\) denotes the highest number recorded.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 64 }\).
  2. Complete the probability distribution table for \(X\).
    \(x\)1234
    \(\mathrm { P } ( X = x )\)\(\frac { 7 } { 64 }\)\(\frac { 19 } { 64 }\)
    On another occasion, one of the fair 4 -sided spinners is spun repeatedly until a 3 is obtained. The random variable \(Y\) is the number of spins required to obtain a 3 .
  3. Find \(\mathrm { P } ( Y = 6 )\).
  4. Find \(\mathrm { P } ( Y > 4 )\).
CAIE S1 2022 November Q5
5 Company \(A\) produces bags of sugar. An inspector finds that on average \(10 \%\) of the bags are underweight. 10 of the bags are chosen at random.
  1. Find the probability that fewer than 3 of these bags are underweight.
    The weights of the bags of sugar produced by company \(B\) are normally distributed with mean 1.04 kg and standard deviation 0.06 kg .
  2. Find the probability that a randomly chosen bag produced by company \(B\) weighs more than 1.11 kg .
    \(81 \%\) of the bags of sugar produced by company \(B\) weigh less than \(w \mathrm {~kg}\).
  3. Find the value of \(w\).
CAIE S1 2022 November Q6
6
  1. Find the number of different arrangements of the 9 letters in the word ACTIVATED.
  2. Find the number of different arrangements of the 9 letters in the word ACTIVATED in which there are at least 5 letters between the two As.
    Five letters are selected at random from the 9 letters in the word ACTIVATED.
  3. Find the probability that the selection does not contain more Ts than As.
CAIE S1 2022 November Q7
7 Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game. Sam removes the first disc.
  1. Find the probability that Tom removes a red disc on his first turn.
  2. Find the probability that Tom wins the game on his second turn.
  3. Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.
    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 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-03_1372_1194_260_479} The times taken by 120 children to complete a particular puzzle are represented in the cumulative frequency graph.
  1. Use the graph to estimate the interquartile range of the data.
    35\% of the children took longer than \(T\) seconds to complete the puzzle.
  2. Use the graph to estimate the value of \(T\).
CAIE S1 2023 November Q2
2 Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.
  1. Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more.
  2. Find the probability that it takes no more than 4 throws of the two dice for Hazeem to obtain a score of 8 or more.
  3. For 8 randomly chosen throws of the two dice, find the probability that Hazeem obtains a score of 8 or more on fewer than 3 occasions.