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CAIE S1 2017 June Q4
6 marks Easy -1.2
4 The times taken, \(t\) seconds, by 1140 people to solve a puzzle are summarised in the table.
Time \(( t\) seconds \()\)\(0 \leqslant t < 20\)\(20 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 100\)\(100 \leqslant t < 140\)
Number of people320280220220100
  1. On the grid, draw a histogram to illustrate this information. \includegraphics[max width=\textwidth, alt={}, center]{7652f36c-59b5-4fcd-b17b-d796dc82aec0-05_812_1406_804_411}
  2. Calculate an estimate of the mean of \(t\).
CAIE S1 2017 June Q5
7 marks Moderate -0.3
5 Eggs are sold in boxes of 20. Cracked eggs occur independently and the mean number of cracked eggs in a box is 1.4 .
  1. Calculate the probability that a randomly chosen box contains exactly 2 cracked eggs.
  2. Calculate the probability that a randomly chosen box contains at least 1 cracked egg.
  3. A shop sells \(n\) of these boxes of eggs. Find the smallest value of \(n\) such that the probability of there being at least 1 cracked egg in each box sold is less than 0.01 .
CAIE S1 2017 June Q6
11 marks Standard +0.3
6
  1. The random variable \(X\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). You are given that \(\sigma = 0.25 \mu\) and \(\mathrm { P } ( X < 6.8 ) = 0.75\).
    1. Find the value of \(\mu\).
    2. Find \(\mathrm { P } ( X < 4.7 )\).
  2. The lengths of metal rods have a normal distribution with mean 16 cm and standard deviation 0.2 cm . Rods which are shorter than 15.75 cm or longer than 16.25 cm are not usable. Find the expected number of usable rods in a batch of 1000 rods.
CAIE S1 2017 June Q7
11 marks Standard +0.3
7
  1. Eight children of different ages stand in a random order in a line. Find the number of different ways this can be done if none of the three youngest children stand next to each other.
  2. David chooses 5 chocolates from 6 different dark chocolates, 4 different white chocolates and 1 milk chocolate. He must choose at least one of each type. Find the number of different selections he can make.
  3. A password for Chelsea's computer consists of 4 characters in a particular order. The characters are chosen from the following.
    • The 26 capital letters A to Z
    • The 9 digits 1 to 9
    • The 5 symbols \# \~{} * ? !
    The password must include at least one capital letter, at least one digit and at least one symbol. No character can be repeated. Find the number of different passwords that Chelsea can make.
CAIE S1 2017 June Q1
5 marks Easy -1.2
1 Rani and Diksha go shopping for clothes.
  1. Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \(\\) 5.50\( and the coat costs \)\\( 90\). The mean cost of Rani's 8 items is \(\\) 29\(. Find the cost of a sweater.
  2. Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \)\\( 26\) and the standard deviation is \(\\) 0\(. Explain how you can tell that Diksha spends \)\\( 104\) on shirts.
CAIE S1 2017 June Q2
6 marks Easy -1.8
2 Anabel measured the lengths, in centimetres, of 200 caterpillars. Her results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{184a04ac-4396-4a0f-8fa8-ab11a4b6df39-03_1173_1195_356_466}
  1. Estimate the median and the interquartile range of the lengths.
  2. Estimate how many caterpillars had a length of between 2 and 3.5 cm .
  3. 6\% of caterpillars were of length \(l\) centimetres or more. Estimate \(l\).
CAIE S1 2017 June Q3
6 marks Moderate -0.8
3 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x ^ { 2 }\), where \(k\) is a constant and \(x\) takes values \(- 2 , - 1,2,4\) only.
  1. Show that \(\mathrm { P } ( X = - 2 )\) has the same value as \(\mathrm { P } ( X = 2 )\).
  2. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S1 2017 June Q4
6 marks Standard +0.3
4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).
CAIE S1 2017 June Q5
9 marks Standard +0.3
5 The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. \(18 \%\) of these videos last for longer than 4.2 minutes.
  1. Find the standard deviation of the lengths of these videos.
  2. Find the probability that the length of a randomly chosen video differs from the mean by less than half a minute.
    The lengths of videos of another popular song have a normal distribution with the same mean of 3.9 minutes but the standard deviation is twice the standard deviation in part (i). The probability that the length of a randomly chosen video of this song differs from the mean by less than half a minute is denoted by \(p\).
  3. Without any further calculation, determine whether \(p\) is more than, equal to, or less than your answer to part (ii). You must explain your reasoning.
CAIE S1 2017 June Q6
9 marks Standard +0.8
6 A library contains 4 identical copies of book \(A , 2\) identical copies of book \(B\) and 5 identical copies of book \(C\). These 11 books are arranged on a shelf in the library.
  1. Calculate the number of different arrangements if the end books are either both book \(A\) or both book \(B\).
  2. Calculate the number of different arrangements if all the books \(A\) are next to each other and none of the books \(B\) are next to each other.
CAIE S1 2017 June Q7
9 marks Moderate -0.8
7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .
  1. Find the probability that Khalid hurts himself on any particular day.
  2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
  3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
  4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.
CAIE S1 2017 June Q1
4 marks Moderate -0.3
1 A biased die has faces numbered 1 to 6 . The probabilities of the die landing on 1,3 or 5 are each equal to 0.1 . The probabilities of the die landing on 2 or 4 are each equal to 0.2 . The die is thrown twice. Find the probability that the sum of the numbers it lands on is 9 .
CAIE S1 2017 June Q2
5 marks Standard +0.3
2 The probability that George goes swimming on any day is \(\frac { 1 } { 3 }\). Use an approximation to calculate the probability that in 270 days George goes swimming at least 100 times.
CAIE S1 2017 June Q3
5 marks Moderate -0.8
3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, \(65 \%\) are Café Premium and \(35 \%\) are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.
  1. Find the probability that the jar is small.
  2. Find the probability that the jar is Café Standard given that it is large.
CAIE S1 2017 June Q4
6 marks Standard +0.3
4
  1. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(\mu = 1.5 \sigma\). A random value of \(X\) is chosen. Find the probability that this value of \(X\) is greater than 0 .
  2. The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation \(s\) hours. It is known that \(87.5 \%\) of these batteries last longer than 70 hours. Find the value of \(s\).
CAIE S1 2017 June Q5
8 marks Moderate -0.8
5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).
  1. State two conditions that are required for \(X\) to have a binomial distribution.
    On average, Hebe completes 7 out of 10 of these puzzles.
  2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
  3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.
CAIE S1 2017 June Q6
11 marks Moderate -0.8
6
  1. Find how many numbers between 3000 and 5000 can be formed from the digits \(1,2,3,4\) and 5,
    1. if digits are not repeated,
    2. if digits can be repeated and the number formed is odd.
  2. A box of 20 biscuits contains 4 different chocolate biscuits, 2 different oatmeal biscuits and 14 different ginger biscuits. 6 biscuits are selected from the box at random.
    1. Find the number of different selections that include the 2 oatmeal biscuits.
    2. Find the probability that fewer than 3 chocolate biscuits are selected.
CAIE S1 2017 June Q7
11 marks Moderate -0.8
7 The following histogram represents the lengths of worms in a garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-10_789_1195_301_466}
  1. Calculate the frequencies represented by each of the four histogram columns.
  2. On the grid on the next page, draw a cumulative frequency graph to represent the lengths of worms in the garden. \includegraphics[max width=\textwidth, alt={}, center]{67412184-38f6-4b37-afe3-4a149a2e0586-11_1111_1409_251_408}
  3. Use your graph to estimate the median and interquartile range of the lengths of worms in the garden.
  4. Calculate an estimate of the mean length of worms in the garden.
    \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE S1 2018 June Q1
3 marks Easy -1.2
1 In a statistics lesson 12 people were asked to think of a number, \(x\), between 1 and 20 inclusive. From the results Tom found that \(\Sigma x = 186\) and that the standard deviation of \(x\) is 4.5. Assuming that Tom's calculations are correct, find the values of \(\Sigma ( x - 10 )\) and \(\Sigma ( x - 10 ) ^ { 2 }\).
CAIE S1 2018 June Q2
7 marks Easy -1.2
2 In a survey 55 students were asked to record, to the nearest kilometre, the total number of kilometres they travelled to school in a particular week. The results are shown below.
5591013131315151515
1618181819192020202021
2121212325252727293033
3538394042454850505151
5255575760616465666970
  1. On the grid, draw a box-and-whisker plot to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{246c92f4-7603-43ff-8533-042a4be99a69-04_512_1596_900_262} An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
  2. Show that there are no outliers.
CAIE S1 2018 June Q3
6 marks Moderate -0.8
3 Andy has 4 red socks and 8 black socks in his drawer. He takes 2 socks at random from his drawer.
  1. Find the probability that the socks taken are of different colours.
    The random variable \(X\) is the number of red socks taken.
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S1 2018 June Q4
8 marks Standard +0.8
4
  1. The distance that car tyres of a certain make can travel before they need to be replaced has a normal distribution. A survey of a large number of these tyres found that the probability of this distance being more than 36800 km is 0.0082 and the probability of this distance being more than 31000 km is 0.6915 . Find the mean and standard deviation of the distribution.
  2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\), where \(3 \sigma = 4 \mu\) and \(\mu \neq 0\). Find \(\mathrm { P } ( X < 3 \mu )\). [3]
CAIE S1 2018 June Q5
8 marks Moderate -0.3
5 In Pelmerdon 22\% of families own a dishwasher.
  1. Find the probability that, of 15 families chosen at random from Pelmerdon, between 4 and 6 inclusive own a dishwasher.
  2. A random sample of 145 families from Pelmerdon is chosen. Use a suitable approximation to find the probability that more than 26 families own a dishwasher.
CAIE S1 2018 June Q6
8 marks Standard +0.3
6 Vehicles approaching a certain road junction from town \(A\) can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town \(A\), \(55 \%\) turn left, \(15 \%\) turn right and \(30 \%\) go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.
  1. Find the probability that, of the next three vehicles approaching the junction from town \(A\), one goes straight on and the other two either both turn left or both turn right.
  2. Three vehicles approach the junction from town \(A\). Given that all three drivers choose the same direction at the junction, find the probability that they all go straight on.
CAIE S1 2018 June Q7
10 marks Standard +0.8
7 Find the number of different ways in which all 9 letters of the word MINCEMEAT can be arranged in each of the following cases.
  1. There are no restrictions.
  2. No vowel (A, E, I are vowels) is next to another vowel.
    5 of the 9 letters of the word MINCEMEAT are selected.
  3. Find the number of possible selections which contain exactly 1 M and exactly 1 E .
  4. Find the number of possible selections which contain at least 1 M and at least 1 E .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.