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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.
CAIE S1 2003 November Q4
6 marks Moderate -0.8
4 Single cards, chosen at random, are given away with bars of chocolate. Each card shows a picture of one of 20 different football players. Richard needs just one picture to complete his collection. He buys 5 bars of chocolate and looks at all the pictures. Find the probability that
  1. Richard does not complete his collection,
  2. he has the required picture exactly once,
  3. he completes his collection with the third picture he looks at.
CAIE S1 2003 November Q5
6 marks Moderate -0.8
5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.
CAIE S1 2003 November Q6
8 marks Easy -1.2
6
  1. A collection of 18 books contains one Harry Potter book. Linda is going to choose 6 of these books to take on holiday.
    1. In how many ways can she choose 6 books?
    2. How many of these choices will include the Harry Potter book?
  2. In how many ways can 5 boys and 3 girls stand in a straight line
    1. if there are no restrictions,
    2. if the boys stand next to each other?
CAIE S1 2003 November Q7
8 marks Standard +0.3
7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.
  1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
  2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
  3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
    \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).
CAIE S1 2003 November Q8
8 marks Easy -1.3
8 A discrete random variable \(X\) has the following probability distribution.
\(x\)1234
\(\mathrm { P } ( X = x )\)\(3 c\)\(4 c\)\(5 c\)\(6 c\)
  1. Find the value of the constant \(c\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
CAIE S1 2004 November Q1
5 marks Moderate -0.3
1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.
  1. Find the number of different arrangements using all nine letters.
  2. How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?
CAIE S1 2004 November Q2
6 marks Easy -1.8
2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.
Length of car \(( x\) metres \()\)FrequencyFrequency density
\(2.80 \leqslant x < 3.00\)1785
\(3.00 \leqslant x < 3.10\)24240
\(3.10 \leqslant x < 3.20\)19190
\(3.20 \leqslant x < 3.40\)8\(a\)
  1. Find the value of \(a\).
  2. Draw a histogram on graph paper to represent the data.
  3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.
CAIE S1 2004 November Q3
6 marks Moderate -0.8
3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company \(A , 30 \%\) to \(B\) and \(20 \%\) to \(C\). A taxi from company \(A\) arrives late \(4 \%\) of the time, a taxi from company \(B\) arrives late \(6 \%\) of the time and a taxi from company \(C\) arrives late \(17 \%\) of the time.
  1. Find the probability that, when Andrea rings for a taxi, it arrives late.
  2. Given that Andrea's taxi arrives late, find the conditional probability that she rang company \(B\).