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CAIE S1 2016 November Q6
10 marks Moderate -0.3
6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams
  1. Find the proportion of bananas that are small.
  2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
  3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
    (b) Calculate the expected total cost of 100 randomly chosen bananas.
CAIE S1 2016 November Q7
10 marks Standard +0.3
7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .
  1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
  2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
  3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.
CAIE S1 2017 November Q1
4 marks Moderate -0.8
1 The discrete random variable \(X\) has the following probability distribution.
\(x\)1236
\(\mathrm { P } ( X = x )\)0.15\(p\)0.4\(q\)
Given that \(\mathrm { E } ( X ) = 3.05\), find the values of \(p\) and \(q\).
CAIE S1 2017 November Q2
5 marks Easy -1.8
2 The time taken by a car to accelerate from 0 to 30 metres per second was measured correct to the nearest second. The results from 48 cars are summarised in the following table.
Time (seconds)\(3 - 5\)\(6 - 8\)\(9 - 11\)\(12 - 16\)\(17 - 25\)
Frequency10151742
  1. On the grid, draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{ee1e5987-315b-48eb-8dba-b9d4d34c87c9-03_1207_1406_897_411}
  2. 35 of these cars accelerated from 0 to 30 metres per second in a time more than \(t\) seconds. Estimate the value of \(t\).
CAIE S1 2017 November Q3
5 marks Moderate -0.8
3 An experiment consists of throwing a biased die 30 times and noting the number of 4 s obtained. This experiment was repeated many times and the average number of 4 s obtained in 30 throws was found to be 6.21.
  1. Estimate the probability of throwing a 4.
    ..................................................................................................................................... .
    \section*{Hence}
  2. find the variance of the number of 4 s obtained in 30 throws,
  3. find the probability that in 15 throws the number of 4 s obtained is 2 or more.
CAIE S1 2017 November Q4
6 marks Moderate -0.3
4 The ages of a group of 12 people at an Art class have mean 48.7 years and standard deviation 7.65 years. The ages of a group of 7 people at another Art class have mean 38.1 years and standard deviation 4.2 years.
  1. Find the mean age of all 19 people.
  2. The individual ages in years of people in the first Art class are denoted by \(x\) and those in the second Art class by \(y\). By first finding \(\Sigma x ^ { 2 }\) and \(\Sigma y ^ { 2 }\), find the standard deviation of the ages of all 19 people.
CAIE S1 2017 November Q5
7 marks Moderate -0.8
5 Over a period of time Julian finds that on long-distance flights he flies economy class on \(82 \%\) of flights. On the rest of the flights he flies first class. When he flies economy class, the probability that he gets a good night's sleep is \(x\). When he flies first class, the probability that he gets a good night's sleep is 0.9 .
  1. Draw a fully labelled tree diagram to illustrate this situation. The probability that Julian gets a good night's sleep on a randomly chosen flight is 0.285 .
  2. Find the value of \(x\).
  3. Given that on a particular flight Julian does not get a good night's sleep, find the probability that he is flying economy class.
CAIE S1 2017 November Q6
10 marks Moderate -0.8
6
  1. A village hall has seats for 40 people, consisting of 8 rows with 5 seats in each row. Mary, Ahmad, Wayne, Elsie and John are the first to arrive in the village hall and no seats are taken before they arrive.
    1. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John assuming there are no restrictions?
    2. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John if Mary and Ahmad sit together in the front row and the other three sit together in one of the other rows?
  2. In how many ways can a team of 4 people be chosen from 10 people if 2 of the people, Ross and Lionel, refuse to be in the team together?
CAIE S1 2017 November Q7
13 marks Moderate -0.3
7 The weight, in grams, of pineapples is denoted by the random variable \(X\) which has a normal distribution with mean 500 and standard deviation 91.5. Pineapples weighing over 570 grams are classified as 'large'. Those weighing under 390 grams are classified as 'small' and the rest are classified as 'medium'.
  1. Find the proportions of large, small and medium pineapples.
  2. Find the weight exceeded by the heaviest \(5 \%\) of pineapples.
  3. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 610 ) = 0.3\).
CAIE S1 2017 November Q1
3 marks Easy -1.2
1 Andy counts the number of emails, \(x\), he receives each day and notes that, over a period of \(n\) days, \(\Sigma ( x - 10 ) = 27\) and the mean number of emails is 11.5 . Find the value of \(n\).
CAIE S1 2017 November Q2
5 marks Moderate -0.8
2 The circumferences, \(c \mathrm {~cm}\), of some trees in a wood were measured. The results are summarised in the table.
Circumference \(( c \mathrm {~cm} )\)\(40 < c \leqslant 50\)\(50 < c \leqslant 80\)\(80 < c \leqslant 100\)\(100 < c \leqslant 120\)
Frequency1448708
  1. On the grid, draw a cumulative frequency graph to represent the information. \includegraphics[max width=\textwidth, alt={}, center]{9c23b94b-e573-4e13-be90-e63a0daf18e5-03_1401_1404_854_413}
  2. Estimate the percentage of trees which have a circumference larger than 75 cm .
CAIE S1 2017 November Q3
5 marks Moderate -0.3
3 A box contains 6 identical-sized discs, of which 4 are blue and 2 are red. Discs are taken at random from the box in turn and not replaced. Let \(X\) be the number of discs taken, up to and including the first blue one.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 15 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2017 November Q4
7 marks Standard +0.8
4 A fair tetrahedral die has faces numbered \(1,2,3,4\). A coin is biased so that the probability of showing a head when thrown is \(\frac { 1 } { 3 }\). The die is thrown once and the number \(n\) that it lands on is noted. The biased coin is then thrown \(n\) times. So, for example, if the die lands on 3 , the coin is thrown 3 times.
  1. Find the probability that the die lands on 4 and the number of times the coin shows heads is 2 .
  2. Find the probability that the die lands on 3 and the number of times the coin shows heads is 3 .
  3. Find the probability that the number the die lands on is the same as the number of times the coin shows heads.
CAIE S1 2017 November Q5
8 marks Standard +0.3
5 Blank CDs are packed in boxes of 30 . The probability that a blank CD is faulty is 0.04 . A box is rejected if more than 2 of the blank CDs are faulty.
  1. Find the probability that a box is rejected.
  2. 280 boxes are chosen randomly. Use an approximation to find the probability that at least 30 of these boxes are rejected.
CAIE S1 2017 November Q6
10 marks Moderate -0.8
6
  1. Find the number of different 3-digit numbers greater than 300 that can be made from the digits \(1,2,3,4,6,8\) if
    1. no digit can be repeated,
    2. a digit can be repeated and the number made is even.
  2. A team of 5 is chosen from 6 boys and 4 girls. Find the number of ways the team can be chosen if
    1. there are no restrictions,
    2. the team contains more boys than girls.
CAIE S1 2017 November Q7
12 marks Standard +0.3
7 In Jimpuri the weights, in kilograms, of boys aged 16 years have a normal distribution with mean 61.4 and standard deviation 12.3.
  1. Find the probability that a randomly chosen boy aged 16 years in Jimpuri weighs more than 65 kilograms.
  2. For boys aged 16 years in Jimpuri, \(25 \%\) have a weight between 65 kilograms and \(k\) kilograms, where \(k\) is greater than 65 . Find \(k\).
    In Brigville the weights, in kilograms, of boys aged 16 years have a normal distribution. \(99 \%\) of the boys weigh less than 97.2 kilograms and \(33 \%\) of the boys weigh less than 55.2 kilograms.
  3. Find the mean and standard deviation of the weights of boys aged 16 years in Brigville.
CAIE S1 2017 November Q1
2 marks Easy -1.2
1 A statistics student asks people to complete a survey. The probability that a randomly chosen person agrees to complete the survey is 0.2 . Find the probability that at least one of the first three people asked agrees to complete the survey.
CAIE S1 2017 November Q2
3 marks Easy -1.2
2 Tien measured the arm lengths, \(x \mathrm {~cm}\), of 20 people in his class. He found that \(\Sigma x = 1218\) and the standard deviation of \(x\) was 4.2. Calculate \(\Sigma ( x - 45 )\) and \(\Sigma ( x - 45 ) ^ { 2 }\).
CAIE S1 2017 November Q3
6 marks Moderate -0.8
3 At the end of a revision course in mathematics, students have to pass a test to gain a certificate. The probability of any student passing the test at the first attempt is 0.85 . Those students who fail are allowed to retake the test once, and the probability of any student passing the retake test is 0.65 .
  1. Draw a fully labelled tree diagram to show all the outcomes.
  2. Given that a student gains the certificate, find the probability that this student fails the test on the first attempt.
CAIE S1 2017 November Q4
6 marks Moderate -0.8
4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.
  1. Draw up a table to show the probability distribution of \(X\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2017 November Q5
9 marks Easy -1.3
5 The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.
1048882654438353428
281818171714131312
1210101096522
  1. Draw a stem-and-leaf diagram to illustrate the data.
  2. Find the median and quartiles and draw a box-and-whisker plot on the grid. \includegraphics[max width=\textwidth, alt={}, center]{4c2afa86-960c-473e-970c-ed16c8434fec-07_1006_1406_1007_411}
CAIE S1 2017 November Q6
12 marks Moderate -0.3
6 A car park has spaces for 18 cars, arranged in a line. On one day there are 5 cars, of different makes, parked in randomly chosen positions and 13 empty spaces.
  1. Find the number of possible arrangements of the 5 cars in the car park.
  2. Find the probability that the 5 cars are not all next to each other.
    On another day, 12 cars of different makes are parked in the car park. 5 of these cars are red, 4 are white and 3 are black. Elizabeth selects 3 of these cars.
    [0pt]
  3. Find the number of selections Elizabeth can make that include cars of at least 2 different colours. [5]
CAIE S1 2017 November Q7
12 marks Standard +0.3
7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).
  1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
    On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
  2. Find the value of \(x\).
  3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
  4. Find the probability that Josie misses the bus.
CAIE S1 2018 November Q2
3 marks Easy -1.3
2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.
\(x\)- 10124
\(\mathrm { P } ( X = x )\)\(p\)\(p\)\(2 p\)\(2 p\)0.1
  1. Find the value of \(p\).
  2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).
CAIE S1 2018 November Q3
7 marks Standard +0.3
3 In an orchestra, there are 11 violinists, 5 cellists and 4 double bass players. A small group of 6 musicians is to be selected from these 20.
  1. How many different selections of 6 musicians can be made if there must be at least 4 violinists, at least 1 cellist and no more than 1 double bass player?
    The small group that is selected contains 4 violinists, 1 cellist and 1 double bass player. They sit in a line to perform a concert.
    [0pt]
  2. How many different arrangements are there of these 6 musicians if the violinists must sit together? [3]