Questions — CAIE S1 (789 questions)

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CAIE S1 2013 November Q6
9 marks Moderate -0.3
6 The 11 letters of the word REMEMBRANCE are arranged in a line.
  1. Find the number of different arrangements if there are no restrictions.
  2. Find the number of different arrangements which start and finish with the letter M .
  3. Find the number of different arrangements which do not have all 4 vowels ( \(\mathrm { E } , \mathrm { E } , \mathrm { A } , \mathrm { E }\) ) next to each other. 4 letters from the letters of the word REMEMBRANCE are chosen.
  4. Find the number of different selections which contain no Ms and no Rs and at least 2 Es.
CAIE S1 2013 November Q7
11 marks Standard +0.3
7 Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed on them. He takes three cards at random, without replacement, and adds up the numbers on the cards.
  1. Show that P (the sum of the numbers on the three cards is \(11 ) = \frac { 1 } { 2 }\).
  2. Draw up a probability distribution table for the sum of the numbers on the three cards. Event \(R\) is 'the sum of the numbers on the three cards is 11 '. Event \(S\) is 'the number on the first card taken is a \(3 ^ { \prime }\).
  3. Determine whether events \(R\) and \(S\) are independent. Justify your answer.
  4. Determine whether events \(R\) and \(S\) are exclusive. Justify your answer.
CAIE S1 2013 November Q1
2 marks Moderate -0.8
1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table.
Distance from college \(( \mathrm { km } )\)\(1 - 3\)\(4 - 5\)\(6 - 8\)\(9 - 11\)\(12 - 16\)
Number of students18138124
Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below. \includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.
CAIE S1 2013 November Q2
5 marks Standard +0.3
2 A factory produces flower pots. The base diameters have a normal distribution with mean 14 cm and standard deviation 0.52 cm . Find the probability that the base diameters of exactly 8 out of 10 randomly chosen flower pots are between 13.6 cm and 14.8 cm .
CAIE S1 2013 November Q3
6 marks Standard +0.3
3 In a large consignment of mangoes, 15\% of mangoes are classified as small, 70\% as medium and \(15 \%\) as large.
  1. Yue-chen picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or large.
  2. Yue-chen picks \(n\) mangoes at random. The probability that none of these \(n\) mangoes is small is at least 0.1 . Find the largest possible value of \(n\).
CAIE S1 2013 November Q4
7 marks Moderate -0.3
4 Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the standard deviation is 32 g . For the lemons, the mean weight is 118 g and the standard deviation is 12 g .
  1. Find the mean weight of the 45 fruits.
  2. The individual weights of the oranges in grams are denoted by \(x _ { o }\), and the individual weights of the lemons in grams are denoted by \(x _ { l }\). By first finding \(\Sigma x _ { o } ^ { 2 }\) and \(\Sigma x _ { l } ^ { 2 }\), find the variance of the weights of the 45 fruits.
CAIE S1 2013 November Q5
7 marks Challenging +1.2
5
  1. The random variable \(X\) is normally distributed with mean 82 and standard deviation 7.4. Find the value of \(q\) such that \(\mathrm { P } ( 82 - q < X < 82 + q ) = 0.44\).
  2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(5 \mu = 2 \sigma ^ { 2 }\) and that \(\mathrm { P } \left( Y < \frac { 1 } { 2 } \mu \right) = 0.281\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 2013 November Q6
10 marks Standard +0.3
6
  1. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the first letter is \(R\).
  2. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if the 3 letters G are together, both letters A are together and both letters E are together.
  3. The letters G, R and T are consonants and the letters A and E are vowels. Find the number of different ways that the 9 letters of the word AGGREGATE can be arranged in a line if consonants and vowels occur alternately.
  4. Find the number of different selections of 4 letters of the word AGGREGATE which contain exactly 2 Gs or exactly 3 Gs.
CAIE S1 2013 November Q7
13 marks Moderate -0.3
7 Dayo chooses two digits at random, without replacement, from the 9-digit number 113333555.
  1. Find the probability that the two digits chosen are equal.
  2. Find the probability that one digit is a 5 and one digit is not a 5 .
  3. Find the probability that the first digit Dayo chose was a 5, given that the second digit he chose is not a 5 .
  4. The random variable \(X\) is the number of 5s that Dayo chooses. Draw up a table to show the probability distribution of \(X\).
CAIE S1 2014 November Q1
3 marks Easy -1.8
1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$
CAIE S1 2014 November Q2
6 marks Easy -1.3
2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.
\(x\)01234\(\geqslant 5\)
\(\mathrm { P } ( X = x )\)0.240.35\(2 k\)\(k\)0.050
  1. Find the value of \(k\).
  2. Find the mode of \(X\).
  3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.
CAIE S1 2014 November Q3
5 marks Standard +0.3
3 Jodie tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin shows a head is \(\frac { 1 } { 3 }\). Each of the dice has four faces, numbered \(1,2,3\) and 4 . Jodie's score is calculated from the numbers on the faces that the dice land on, as follows:
  • if the coin shows a head, the two numbers from the dice are added together;
  • if the coin shows a tail, the two numbers from the dice are multiplied together.
Find the probability that the coin shows a head given that Jodie's score is 8 .
CAIE S1 2014 November Q4
7 marks Easy -1.2
4 The following back-to-back stem-and-leaf diagram shows the times to load an application on 61 smartphones of type \(A\) and 43 smartphones of type \(B\).
(7)
Type \(A\)Type \(B\)
976643321358
55442223044566667889
998887664322040112368899
655432110525669
973061389
874410757
766653321081244
86555906
Key: 3 | 2 | 1 means 0.23 seconds for type \(A\) and 0.21 seconds for type \(B\).
  1. Find the median and quartiles for smartphones of type \(A\). You are given that the median, lower quartile and upper quartile for smartphones of type \(B\) are 0.46 seconds, 0.36 seconds and 0.63 seconds respectively.
  2. Represent the data by drawing a pair of box-and-whisker plots in a single diagram on graph paper.
  3. Compare the loading times for these two types of smartphone.
CAIE S1 2014 November Q5
9 marks Standard +0.3
5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .
  1. Show that the variance of the number of faulty screws in a packet is 1.104 .
  2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
  3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.
CAIE S1 2014 November Q6
10 marks Moderate -0.3
6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .
  1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
  2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
  3. Find the value of \(\mu\).
CAIE S1 2014 November Q7
10 marks Standard +0.3
7 A committee of 6 people is to be chosen from 5 men and 8 women. In how many ways can this be done
  1. if there are more women than men on the committee,
  2. if the committee consists of 3 men and 3 women but two particular men refuse to be on the committee together? One particular committee consists of 5 women and 1 man.
  3. In how many different ways can the committee members be arranged in a line if the man is not at either end?
CAIE S1 2014 November Q1
3 marks Moderate -0.5
1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]
CAIE S1 2014 November Q2
5 marks Easy -1.2
2 A traffic camera measured the speeds, \(x\) kilometres per hour, of 8 cars travelling along a certain street, with the following results. $$\begin{array} { l l l l l l l l } 62.7 & 59.6 & 64.2 & 61.5 & 68.3 & 66.9 & 62.0 & 62.3 \end{array}$$
  1. Find \(\Sigma ( x - 62 )\).
  2. Find \(\Sigma ( x - 62 ) ^ { 2 }\).
  3. Find the mean and variance of the speeds of the 8 cars.
CAIE S1 2014 November Q3
5 marks Easy -1.2
3 The number of books read by members of a book club each year has the binomial distribution \(B ( 12,0.7 )\).
  1. State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.
  2. Find the probability that a member reads fewer than 10 books in a particular year.
CAIE S1 2014 November Q4
8 marks Easy -1.3
4 A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.
2319321425
2226364542
4728173815
4618262241
1921282430
  1. Draw a stem-and-leaf diagram to represent the data.
  2. On graph paper draw a box-and-whisker plot to represent the data.
CAIE S1 2014 November Q5
9 marks Standard +0.3
5 Gem stones from a certain mine have weights, \(X\) grams, which are normally distributed with mean 1.9 g and standard deviation 0.55 g . These gem stones are sorted into three categories for sale depending on their weights, as follows. Small: under 1.2 g Medium: between 1.2 g and 2.5 g Large: over 2.5 g
  1. Find the proportion of gem stones in each of these three categories.
  2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 2.5 ) = 0.8\).
CAIE S1 2014 November Q6
9 marks Standard +0.3
6
  1. Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
  2. Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.
CAIE S1 2014 November Q7
11 marks Standard +0.3
7 A box contains 2 green apples and 2 red apples. Apples are taken from the box, one at a time, without replacement. When both red apples have been taken, the process stops. The random variable \(X\) is the number of apples which have been taken when the process stops.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 3 }\).
  2. Draw up the probability distribution table for \(X\). Another box contains 2 yellow peppers and 5 orange peppers. Three peppers are taken at random from the box without replacement.
  3. Given that at least 2 of the peppers taken from the box are orange, find the probability that all 3 peppers are orange.
CAIE S1 2015 November Q1
4 marks Moderate -0.8
1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.
CAIE S1 2015 November Q2
4 marks Moderate -0.3
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).