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CAIE S1 2018 November Q4
8 marks Standard +0.8
4
  1. It is given that \(X \sim \mathrm {~N} ( 31.4,3.6 )\). Find the probability that a randomly chosen value of \(X\) is less than 29.4.
  2. The lengths of fish of a particular species are modelled by a normal distribution. A scientist measures the lengths of 400 randomly chosen fish of this species. He finds that 42 fish are less than 12 cm long and 58 are more than 19 cm long. Find estimates for the mean and standard deviation of the lengths of fish of this species.
CAIE S1 2018 November Q5
9 marks Moderate -0.8
5 At the Nonland Business College, all students sit an accountancy examination at the end of their first year of study. On average, \(80 \%\) of the students pass this examination.
  1. A random sample of 9 students who will take this examination is chosen. Find the probability that at most 6 of these students will pass the examination.
  2. A random sample of 200 students who will take this examination is chosen. Use a suitable approximate distribution to find the probability that more than 166 of them will pass the examination.
  3. Justify the use of your approximate distribution in part (ii).
CAIE S1 2018 November Q6
10 marks Easy -1.8
6 The daily rainfall, \(x \mathrm {~mm}\), in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.
Rainfall, \(x \mathrm {~mm}\)\(x \leqslant 20\)\(x \leqslant 30\)\(x \leqslant 40\)\(x \leqslant 50\)\(x \leqslant 70\)\(x \leqslant 100\)
Cumulative frequency5294142172222250
  1. On the grid, draw a cumulative frequency graph to illustrate the data.
  2. On 100 of the days, the rainfall was \(k \mathrm {~mm}\) or more. Use your graph to estimate the value of \(k\).
  3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village.
CAIE S1 2018 November Q7
10 marks Easy -1.2
7 In a group of students, the numbers of boys and girls studying Art, Music and Drama are given in the following table. Each of these 160 students is studying exactly one of these subjects.
ArtMusicDrama
Boys244032
Girls151237
  1. Find the probability that a randomly chosen student is studying Music.
  2. Determine whether the events 'a randomly chosen student is a boy' and 'a randomly chosen student is studying Music' are independent, justifying your answer.
  3. Find the probability that a randomly chosen student is not studying Drama, given that the student is a girl.
  4. Three students are chosen at random. Find the probability that exactly 1 is studying Music and exactly 2 are boys.
    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 2018 November Q1
5 marks Moderate -0.8
1
  1. How many different arrangements are there of the 11 letters in the word MISSISSIPPI?
  2. Two letters are chosen at random from the 11 letters in the word MISSISSIPPI. Find the probability that these two letters are the same.
CAIE S1 2018 November Q2
6 marks Moderate -0.8
2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\).
\(A\)\(B\)
(4)420020567(3)
(5)9850021122377(6)
(8)98753222221356689(7)
(6)8765212345788999(8)
(3)863242456788(7)
(1)0250278(4)
Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)
  1. Find the median and the interquartile range for group \(A\).
    The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
  2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid. \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}
CAIE S1 2018 November Q3
6 marks Moderate -0.3
3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.
  1. Find the probability that he will complete the puzzle at least three times over a period of five days.
    Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
  2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.
CAIE S1 2018 November Q4
6 marks Moderate -0.8
4
  1. Find the number of different ways that 5 boys and 6 girls can stand in a row if all the boys stand together and all the girls stand together.
  2. Find the number of different ways that 5 boys and 6 girls can stand in a row if no boy stands next to another boy.
CAIE S1 2018 November Q5
6 marks Moderate -0.8
5 The Quivers Archery club has 12 Junior members and 20 Senior members. For the Junior members, the mean age is 15.5 years and the standard deviation of the ages is 1.2 years. The ages of the Senior members are summarised by \(\Sigma y = 910\) and \(\Sigma y ^ { 2 } = 42850\), where \(y\) is the age of a Senior member in years.
  1. Find the mean age of all 32 members of the club.
  2. Find the standard deviation of the ages of all 32 members of the club.
CAIE S1 2018 November Q6
9 marks Moderate -0.3
6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
  3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.
CAIE S1 2018 November Q7
12 marks Standard +0.3
7
  1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
    1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
    2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
    3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
  2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
    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 2018 November Q1
3 marks Moderate -0.5
1 A group consists of 5 men and 2 women. Find the number of different ways that the group can stand in a line if the women are not next to each other.
CAIE S1 2018 November Q2
6 marks Moderate -0.3
2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).
  1. Draw up a table showing the probability distribution of \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2018 November Q3
7 marks Moderate -0.8
3 A box contains 3 red balls and 5 blue balls. One ball is taken at random from the box and not replaced. A yellow ball is then put into the box. A second ball is now taken at random from the box.
  1. Complete the tree diagram to show all the outcomes and the probability for each branch. First ball
    Second ball \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_655_392_688_474} \includegraphics[max width=\textwidth, alt={}, center]{7dc85f33-2647-4f73-8093-524b70f99767-04_785_387_703_1110}
  2. Find the probability that the two balls taken are the same colour.
  3. Find the probability that the first ball taken is red, given that the second ball taken is blue.
CAIE S1 2018 November Q4
7 marks Moderate -0.3
4 Out of a class of 8 boys and 4 girls, a group of 7 people is chosen at random.
  1. Find the probability that the group of 7 includes one particular boy.
  2. Find the probability that the group of 7 includes at least 2 girls.
CAIE S1 2018 November Q5
8 marks Moderate -0.3
5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.
  1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
  2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]
CAIE S1 2018 November Q6
8 marks Standard +0.3
6 The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation \(\sigma\) hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96 .
  1. Find the value of \(\sigma\).
    New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.
  2. For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.
  3. Justify the use of your approximate distribution in part (ii).
CAIE S1 2018 November Q7
11 marks Easy -1.2
7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.
Anvils173158180196175165170169181184172
Brecons166170171172172178181182183183192
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
  2. Find the median and the interquartile range for the heights of the Anvils.
    The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
  3. Find the standard deviation of the heights of all 14 members of the Anvils.
    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 2019 November Q1
3 marks Moderate -0.8
1 When Shona goes to college she either catches the bus with probability 0.8 or she cycles with probability 0.2 . If she catches the bus, the probability that she is late is 0.4 . If she cycles, the probability that she is late is \(x\). The probability that Shona is not late for college on a randomly chosen day is 0.63 . Find the value of \(x\).
CAIE S1 2019 November Q2
6 marks Moderate -0.3
2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that \(42 \%\) of customers rated the logo as good.
  1. A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
  2. On another occasion, a random sample of \(n\) customers of the company is chosen. Find the smallest value of \(n\) for which the probability that at least one person rates the logo as good is greater than 0.995 .
CAIE S1 2019 November Q3
7 marks Moderate -0.8
3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.
  1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
    Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
  2. Find the mean and standard deviation of all these 30 values of \(x\).
CAIE S1 2019 November Q4
7 marks Easy -1.3
4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(p\)\(p\)\(\frac { 3 } { 8 }\)\(4 p\)
  1. Find the value of \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .
CAIE S1 2019 November Q5
8 marks Easy -1.2
5 Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-08_1090_1424_404_356}
  1. Estimate how many leaves have a length between 14 and 24 centimetres.
  2. \(10 \%\) of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).
  3. Estimate the median and the interquartile range of the lengths.
    Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-09_540_1287_1181_424}
  4. Compare the central tendency and the spread of the two sets of data.
CAIE S1 2019 November Q6
9 marks Moderate -0.3
6
  1. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that all four Es are together.
  2. Find the number of different ways in which all 12 letters of the word STEEPLECHASE can be arranged so that the Ss are not next to each other.
    Four letters are selected from the 12 letters of the word STEEPLECHASE.
  3. Find the number of different selections if the four letters include exactly one \(S\).
CAIE S1 2019 November Q7
10 marks Moderate -0.3
7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.
  1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
  2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
    Three athletes from the club are chosen at random.
  3. Find the probability that exactly 2 have PBs of less than 46 seconds.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.