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CAIE S1 2022 November Q5
10 marks Moderate -0.3
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
10 marks Standard +0.8
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
5 marks Standard +0.3
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
7 marks Moderate -0.8
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
5 marks Standard +0.3
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
7 marks Moderate -0.8
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
7 marks Standard +0.3
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
9 marks Moderate -0.8
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
10 marks Standard +0.3
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
5 marks Moderate -0.8
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
7 marks Moderate -0.8
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
8 marks Moderate -0.3
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
9 marks Moderate -0.3
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
10 marks Standard +0.8
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
8 marks Standard +0.8
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
4 marks Easy -1.8
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
7 marks Moderate -0.3
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.
CAIE S1 2023 November Q3
11 marks Standard +0.3
3 A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight. A small egg weighs less than 76 grams and \(40 \%\) of the eggs are classified as medium.
  1. Find the percentage of eggs that are classified as small.
  2. Find the least possible weight of an egg classified as large.
    150 of the eggs for sale last week were weighed.
  3. Use an approximation to find the probability that more than 68 of these eggs were classified as medium.
CAIE S1 2023 November Q4
9 marks Moderate -0.8
4 The times, to the nearest minute, of 150 athletes taking part in a charity run are recorded. The results are summarised in the table.
Time in minutes\(101 - 120\)\(121 - 130\)\(131 - 135\)\(136 - 145\)\(146 - 160\)
Frequency1848343218
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-08_1493_1397_936_415}
  2. Calculate estimates for the mean and standard deviation of the times taken by the athletes.
CAIE S1 2023 November Q5
9 marks Standard +0.3
5 A red spinner has four sides labelled \(1,2,3,4\). When the spinner is spun, the score is the number on the side on which it lands. The random variable \(X\) denotes this score. The probability distribution table for \(X\) is given below.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.28\(p\)\(2 p\)\(3 p\)
  1. Show that \(p = 0.12\).
    A fair blue spinner and a fair green spinner each have four sides labelled 1, 2, 3, 4. All three spinners (red, blue and green) are spun at the same time.
  2. Find the probability that the sum of the three scores is 4 or less.
  3. Find the probability that the product of the three scores is 4 or less given that \(X\) is odd.
CAIE S1 2023 November Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-12_291_809_255_667} In a restaurant, the tables are rectangular. Each table seats four people: two along each of the longer sides of the table (see diagram). Eight friends have booked two tables, \(X\) and \(Y\). Rajid, Sue and Tan are three of these friends.
  1. The eight friends will be divided into two groups of 4, one group for table \(X\) and one group for table \(Y\). Find the number of ways in which this can be done if Rajid and Sue must sit at the same table as each other and Tan must sit at the other table.
    When the friends arrive at the restaurant, Rajid and Sue now decide to sit at table \(X\) on the same side as each other. Tan decides that he does not mind at which table he sits.
  2. Find the number of different seating arrangements for the 8 friends.
    As they leave the restaurant, the 8 friends stand in a line for a photograph.
  3. Find the number of different arrangements if Rajid and Sue stand next to each other, but neither is at an end of the line.
    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
5 marks Moderate -0.8
1 A competitor in a throwing event has three attempts to throw a ball as far as possible. The random variable \(X\) denotes the number of throws that exceed 30 metres. The probability distribution table for \(X\) is shown below.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.4\(p\)\(r\)0.15
  1. Given that \(\mathrm { E } ( X ) = 1.1\), find the value of \(p\) and the value of \(r\).
  2. Find the numerical value of \(\operatorname { Var } ( X )\).
CAIE S1 2023 November Q2
5 marks Moderate -0.8
2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.
  1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
    George spins the spinner 10 times.
  2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.
CAIE S1 2023 November Q3
5 marks Moderate -0.8
3 A factory produces a certain type of electrical component. It is known that \(15 \%\) of the components produced are faulty. A random sample of 200 components is chosen. Use an approximation to find the probability that more than 40 of these components are faulty.
CAIE S1 2023 November Q4
8 marks Easy -1.8
4 The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.
Aces180174169182181166173182168171164
Jets175174188168166174181181170188190
  1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
  2. Find the median and the interquartile range of the heights of the players in the Aces.
  3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.