Questions — CAIE S1 (789 questions)

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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.
    {www.cie.org.uk} after the live examination series. }
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.
CAIE S1 2018 June Q1
4 marks Easy -1.8
1 Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows. $$\begin{array} { l l l l l l l l l l } 37 & 40 & 45 & 38 & 36 & 38 & 42 & 38 & 40 & 39 \end{array}$$
  1. Find the mode.
  2. Find the median and the interquartile range.
CAIE S1 2018 June Q2
6 marks Moderate -0.3
2 In a group of students, \(\frac { 3 } { 4 }\) are male. The proportion of male students who like their curry hot is \(\frac { 3 } { 5 }\) and the proportion of female students who like their curry hot is \(\frac { 4 } { 5 }\). One student is chosen at random.
  1. Find the probability that the student chosen is either female, or likes their curry hot, or is both female and likes their curry hot.
  2. Showing your working, determine whether the events 'the student chosen is male' and 'the student chosen likes their curry hot' are independent.
CAIE S1 2018 June Q3
6 marks Moderate -0.3
3
  1. The volume of soup in Super Soup cartons has a normal distribution with mean \(\mu\) millilitres and standard deviation 9 millilitres. Tests have shown that \(10 \%\) of cartons contain less than 440 millilitres of soup. Find the value of \(\mu\).
  2. A food retailer orders 150 Super Soup cartons. Calculate the number of these cartons for which you would expect the volume of soup to be more than 1.8 standard deviations above the mean.
CAIE S1 2018 June Q4
6 marks Moderate -0.8
4 Mrs Rupal chooses 3 animals at random from 5 dogs and 2 cats. The random variable \(X\) is the number of cats chosen.
  1. Draw up the probability distribution table for \(X\).
  2. You are given that \(\mathrm { E } ( X ) = \frac { 6 } { 7 }\). Find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2018 June Q5
7 marks Easy -1.2
5 The lengths, \(t\) minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.
Length of phone
call \(( t\) minutes \()\)
\(0 < t \leqslant 1\)\(1 < t \leqslant 2\)\(2 < t \leqslant 5\)\(5 < t \leqslant 10\)\(10 < t \leqslant 30\)
Frequency1446102\(a\)40
  1. Find the value of \(a\).
  2. Calculate an estimate of the mean length of these phone calls.
  3. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}
CAIE S1 2018 June Q6
10 marks Moderate -0.8
6
  1. Find the number of ways in which all 9 letters of the word AUSTRALIA can be arranged in each of the following cases.
    1. All the vowels (A, I, U are vowels) are together.
    2. The letter T is in the central position and each end position is occupied by one of the other consonants (R, S, L).
  2. Donna has 2 necklaces, 8 rings and 4 bracelets, all different. She chooses 4 pieces of jewellery. How many possible selections can she make if she chooses at least 1 necklace and at least 1 bracelet?
CAIE S1 2018 June Q7
11 marks Standard +0.3
7 In a certain country, \(60 \%\) of mobile phones sold are made by Company \(A , 35 \%\) are made by Company \(B\) and 5\% are made by other companies.
  1. Find the probability that, out of a random sample of 13 people who buy a mobile phone, fewer than 11 choose a mobile phone made by Company \(A\).
  2. Use a suitable approximation to find the probability that, out of a random sample of 130 people who buy a mobile phone, at least 50 choose a mobile phone made by Company \(B\).
  3. A random sample of \(n\) mobile phones sold is chosen. The probability that at least one of these phones is made by Company \(B\) is more than 0.98 . Find the least possible value of \(n\).
    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 June Q1
5 marks Moderate -0.8
1 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.
Mass (kg)\(10 - 14\)\(15 - 19\)\(20 - 24\)\(25 - 34\)\(35 - 59\)
Frequency61214108
  1. Find which class interval contains the lower quartile.
  2. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}