Questions — CAIE (7646 questions)

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CAIE S1 2016 March Q7
11 marks Standard +0.3
7 The times taken by a garage to fit a tow bar onto a car have a normal distribution with mean \(m\) hours and standard deviation 0.35 hours. It is found that \(95 \%\) of times taken are longer than 0.9 hours.
  1. Find the value of \(m\).
  2. On one day 4 cars have a tow bar fitted. Find the probability that none of them takes more than 2 hours to fit. The times in hours taken by another garage to fit a tow bar onto a car have the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) where \(\mu = 3 \sigma\).
  3. Find the probability that it takes more than \(0.6 \mu\) hours to fit a tow bar onto a randomly chosen car at this garage.
CAIE S1 2017 March Q1
4 marks Easy -1.2
1 Twelve values of \(x\) are shown below.
1761.61758.51762.31761.41759.41759.1
1762.51761.91762.41761.91762.81761.0
Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]
CAIE S1 2017 March Q2
3 marks Moderate -0.5
2 A bag contains 10 pink balloons, 9 yellow balloons, 12 green balloons and 9 white balloons. 7 balloons are selected at random without replacement. Find the probability that exactly 3 of them are green.
CAIE S1 2017 March Q3
5 marks Moderate -0.3
3 It is found that \(10 \%\) of the population enjoy watching Historical Drama on television. Use an appropriate approximation to find the probability that, out of 160 people chosen randomly, more than 17 people enjoy watching Historical Drama on television.
CAIE S1 2017 March Q4
7 marks Easy -1.8
4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data.
7473\(( 1 )\)
748125779\(( 6 )\)
749022235556789\(( 12 )\)
750112223445677889\(( 15 )\)
7510023344455779\(( 13 )\)
75200011223444\(( 11 )\)
7532\(( 1 )\)
Key: 748 | 5 represents 0.7485 kg .
  1. On the grid, draw a box-and-whisker plot to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-05_814_1604_1336_299}
  2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.
CAIE S1 2017 March Q5
9 marks Standard +0.3
5
  1. A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes.
  2. Another plate holds 7 cup cakes, each with a different colour icing, and 4 brownies, each of a different size. Find the number of different ways these 11 cakes can be arranged in a row if no brownie is next to another brownie.
  3. A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together.
CAIE S1 2017 March Q6
9 marks Moderate -0.3
6 Pack \(A\) consists of ten cards numbered \(0,0,1,1,1,1,1,3,3,3\). Pack \(B\) consists of six cards numbered \(0,0,2,2,2,2\). One card is chosen at random from each pack. The random variable \(X\) is defined as the sum of the two numbers on the cards.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }\). \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-08_59_1569_497_328}
  2. Draw up the probability distribution table for \(X\).
  3. Given that \(X = 3\), find the probability that the card chosen from pack \(A\) is a 1 .
CAIE S1 2017 March Q7
13 marks Standard +0.3
7
  1. The lengths, in centimetres, of middle fingers of women in Raneland have a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(25 \%\) of these women have fingers longer than 8.8 cm and \(17.5 \%\) have fingers shorter than 7.7 cm .
    1. Find the values of \(\mu\) and \(\sigma\).
      The lengths, in centimetres, of middle fingers of women in Snoland have a normal distribution with mean 7.9 and standard deviation 0.44. A random sample of 5 women from Snoland is chosen.
    2. Find the probability that exactly 3 of these women have middle fingers shorter than 8.2 cm .
  2. The random variable \(X\) has a normal distribution with mean equal to the standard deviation. Find the probability that a particular value of \(X\) is less than 1.5 times the mean.
CAIE S1 2019 March Q1
5 marks Moderate -0.5
1 On each day that Tamar goes to work, he wears either a blue suit with probability 0.6 or a grey suit with probability 0.4 . If he wears a blue suit then the probability that he wears red socks is 0.2 . If he wears a grey suit then the probability that he wears red socks is 0.32 .
  1. Find the probability that Tamar wears red socks on any particular day that he is at work.
  2. Given that Tamar is not wearing red socks at work, find the probability that he is wearing a grey suit.
CAIE S1 2019 March Q2
4 marks Moderate -0.8
2 For 40 values of the variable \(x\), it is given that \(\Sigma ( x - c ) ^ { 2 } = 3099.2\), where \(c\) is a constant. The standard deviation of these values of \(x\) is 3.2 .
  1. Find the value of \(\Sigma ( x - c )\).
  2. Given that \(c = 50\), find the mean of these values of \(x\).
CAIE S1 2019 March Q3
6 marks Moderate -0.8
3 The times taken, in minutes, for trains to travel between Alphaton and Beeton are normally distributed with mean 140 and standard deviation 12.
  1. Find the probability that a randomly chosen train will take less than 132 minutes to travel between Alphaton and Beeton.
  2. The probability that a randomly chosen train takes more than \(k\) minutes to travel between Alphaton and Beeton is 0.675 . Find the value of \(k\).
CAIE S1 2019 March Q4
6 marks Moderate -0.8
4 The random variable \(X\) takes the values \(- 1,1,2,3\) only. The probability that \(X\) takes the value \(x\) is \(k x ^ { 2 }\), where \(k\) is a constant.
  1. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2019 March Q5
7 marks Easy -1.8
5 The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below.
Dolphins6275698263806565738272
Sharks6884597071647780667472
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Dolphins on the left-hand side of the diagram and Sharks on the right-hand side.
  2. Find the median and interquartile range for the Dolphins.
CAIE S1 2019 March Q6
11 marks Moderate -0.3
6 The results of a survey by a large supermarket show that \(35 \%\) of its customers shop online.
  1. Six customers are chosen at random. Find the probability that more than three of them shop online.
  2. For a random sample of \(n\) customers, the probability that at least one of them shops online is greater than 0.95 . Find the least possible value of \(n\).
  3. For a random sample of 100 customers, use a suitable approximating distribution to find the probability that more than 39 shop online.
CAIE S1 2019 March Q7
11 marks Standard +0.3
7 Find the number of different arrangements that can be made of all 9 letters in the word CAMERAMAN in each of the following cases.
  1. There are no restrictions.
  2. The As occupy the 1st, 5th and 9th positions.
  3. There is exactly one letter between the Ms.
    Three letters are selected from the 9 letters of the word CAMERAMAN.
  4. Find the number of different selections if the three letters include exactly one M and exactly one A.
  5. Find the number of different selections if the three letters include at least one M.
    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 2002 November Q1
4 marks Moderate -0.8
1 The discrete random variable \(X\) has the following probability distribution.
\(x\)1357
\(\mathrm { P } ( X = x )\)0.3\(a\)\(b\)0.25
  1. Write down an equation satisfied by \(a\) and \(b\).
  2. Given that \(\mathrm { E } ( X ) = 4\), find \(a\) and \(b\).
CAIE S1 2002 November Q2
6 marks Easy -1.2
2 Ivan throws three fair dice.
  1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
  2. Find the probability of Ivan obtaining a total score of 7.
CAIE S1 2002 November Q3
6 marks Moderate -0.8
3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .
  1. Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .
  2. The school gives a certificate to the \(10 \%\) of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.
CAIE S1 2002 November Q4
7 marks Moderate -0.3
4 In a certain hotel, the lock on the door to each room can be opened by inserting a key card. The key card can be inserted only one way round. The card has a pattern of holes punched in it. The card has 4 columns, and each column can have either 1 hole, 2 holes, 3 holes or 4 holes punched in it. Each column has 8 different positions for the holes. The diagram illustrates one particular key card with 3 holes punched in the first column, 3 in the second, 1 in the third and 2 in the fourth. \includegraphics[max width=\textwidth, alt={}, center]{2bcbd4d3-0d41-48fa-8f70-192b158c0bbe-2_410_214_1811_968}
  1. Show that the number of different ways in which a column could have exactly 2 holes is 28 .
  2. Find how many different patterns of holes can be punched in a column.
  3. How many different possible key cards are there?
CAIE S1 2002 November Q5
9 marks Standard +0.3
5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,
  1. find the conditional probability that Rachel wins the first game, given that she loses the second,
  2. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.
CAIE S1 2002 November Q6
9 marks Standard +0.3
6
  1. A manufacturer of biscuits produces 3 times as many cream ones as chocolate ones. Biscuits are chosen randomly and packed into boxes of 10 . Find the probability that a box contains equal numbers of cream biscuits and chocolate biscuits.
  2. A random sample of 8 boxes is taken. Find the probability that exactly 1 of them contains equal numbers of cream biscuits and chocolate biscuits.
  3. A large box of randomly chosen biscuits contains 120 biscuits. Using a suitable approximation, find the probability that it contains fewer than 35 chocolate biscuits.
CAIE S1 2002 November Q7
9 marks Moderate -0.8
7 The weights in kilograms of two groups of 17-year-old males from country \(P\) and country \(Q\) are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, ... \(4 | 7 | 1 \ldots\) denotes weights of 74 kg for a male in country \(P\) and 71 kg for a male in country \(Q\).
Country \(P\)Country \(Q\)
515
62348
9876471345677889
88665382367788
97765554290224
544311045
  1. Find the median and quartile weights for country \(Q\).
  2. You are given that the lower quartile, median and upper quartile for country \(P\) are 84,94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data.
  3. Make two comments on the weights of the two groups.
CAIE S1 2003 November Q1
4 marks Easy -1.8
1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.
CAIE S1 2003 November Q2
4 marks Easy -1.3
2 The floor areas, \(x \mathrm {~m} ^ { 2 }\), of 20 factories are as follows.
150350450578595644722798802904
1000133015331561177819602167233024333231
Represent these data by a histogram on graph paper, using intervals $$0 \leqslant x < 500,500 \leqslant x < 1000,1000 \leqslant x < 2000,2000 \leqslant x < 3000,3000 \leqslant x < 4000 .$$
CAIE S1 2003 November Q3
6 marks Standard +0.3
3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.