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CAIE S1 2013 June Q3
6 marks Moderate -0.8
3 Cans of lemon juice are supposed to contain 440 ml of juice. It is found that the actual volume of juice in a can is normally distributed with mean 445 ml and standard deviation 3.6 ml .
  1. Find the probability that a randomly chosen can contains less than 440 ml of juice. It is found that \(94 \%\) of the cans contain between \(( 445 - c ) \mathrm { ml }\) and \(( 445 + c ) \mathrm { ml }\) of juice.
  2. Find the value of \(c\).
CAIE S1 2013 June Q4
7 marks Standard +0.8
4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.
  1. Find the probability that at least 2 of the 5 integers are less than or equal to 4 . Robert now generates \(n\) random integers between 1 and 9 inclusive. The random variable \(X\) is the number of these \(n\) integers which are less than or equal to a certain integer \(k\) between 1 and 9 inclusive. It is given that the mean of \(X\) is 96 and the variance of \(X\) is 32 .
  2. Find the values of \(n\) and \(k\).
CAIE S1 2013 June Q5
9 marks Easy -1.8
5 The following are the annual amounts of money spent on clothes, to the nearest \(\\) 10$, by 27 people.
10406080100130140140140
150150150160160160160170180
180200210250270280310450570
  1. Construct a stem-and-leaf diagram for the data.
  2. Find the median and the interquartile range of the data. 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.
  3. List the outliers.
CAIE S1 2013 June Q6
10 marks Standard +0.8
6 A town council plans to plant 12 trees along the centre of a main road. The council buys the trees from a garden centre which has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees for sale.
  1. How many different selections of 12 trees can be made if there must be at least 2 of each type of tree? The council buys 4 hibiscus trees, 6 jacaranda trees and 2 oleander trees.
  2. How many different arrangements of these 12 trees can be made if the hibiscus trees have to be next to each other, the jacaranda trees have to be next to each other and the oleander trees have to be next to each other?
  3. How many different arrangements of these 12 trees can be made if no hibiscus tree is next to another hibiscus tree?
CAIE S1 2013 June Q7
11 marks Standard +0.3
7 Susan has a bag of sweets containing 7 chocolates and 5 toffees. Ahmad has a bag of sweets containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Susan's bag and put in Ahmad's bag. A sweet is then taken at random from Ahmad's bag.
  1. Find the probability that the two sweets taken are a toffee from Susan's bag and a boiled sweet from Ahmad's bag.
  2. Given that the sweet taken from Ahmad's bag is a chocolate, find the probability that the sweet taken from Susan's bag was also a chocolate.
  3. The random variable \(X\) is the number of times a chocolate is taken. State the possible values of \(X\) and draw up a table to show the probability distribution of \(X\).
CAIE S1 2013 June Q2
4 marks Standard +0.3
2 The 12 houses on one side of a street are numbered with even numbers starting at 2 and going up to 24 . A free newspaper is delivered on Monday to 3 different houses chosen at random from these 12. Find the probability that at least 2 of these newspapers are delivered to houses with numbers greater than 14.
CAIE S1 2013 June Q3
7 marks Standard +0.3
3 Buildings in a certain city centre are classified by height as tall, medium or short. The heights can be modelled by a normal distribution with mean 50 metres and standard deviation 16 metres. Buildings with a height of more than 70 metres are classified as tall.
  1. Find the probability that a building chosen at random is classified as tall.
  2. The rest of the buildings are classified as medium and short in such a way that there are twice as many medium buildings as there are short ones. Find the height below which buildings are classified as short.
CAIE S1 2013 June Q4
7 marks Standard +0.3
4 In a certain country, on average one student in five has blue eyes.
  1. For a random selection of \(n\) students, the probability that none of the students has blue eyes is less than 0.001 . Find the least possible value of \(n\).
  2. For a random selection of 120 students, find the probability that fewer than 33 have blue eyes.
CAIE S1 2013 June Q5
8 marks Moderate -0.8
5
  1. John plays two games of squash. The probability that he wins his first game is 0.3 . If he wins his first game, the probability that he wins his second game is 0.6 . If he loses his first game, the probability that he wins his second game is 0.15 . Given that he wins his second game, find the probability that he won his first game.
  2. Jack has a pack of 15 cards. 10 cards have a picture of a robot on them and 5 cards have a picture of an aeroplane on them. Emma has a pack of cards. 7 cards have a picture of a robot on them and \(x - 3\) cards have a picture of an aeroplane on them. One card is taken at random from Jack's pack and one card is taken at random from Emma's pack. The probability that both cards have pictures of robots on them is \(\frac { 7 } { 18 }\). Write down an equation in terms of \(x\) and hence find the value of \(x\).
CAIE S1 2013 June Q6
10 marks Easy -1.8
6 The weights, \(x\) kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.
Weight \(( x\) kilograms \()\)\(x < 40\)\(x < 50\)\(x < 60\)\(x < 65\)\(x < 70\)\(x < 90\)
Cumulative frequency012346492144
  1. On graph paper, draw a cumulative frequency graph to represent these results.
  2. 64 people weigh more than \(c \mathrm {~kg}\). Use your graph to find the value of \(c\).
  3. Calculate estimates of the mean and standard deviation of the weights.
CAIE S1 2013 June Q7
10 marks Standard +0.8
7 There are 10 spaniels, 14 retrievers and 6 poodles at a dog show. 7 dogs are selected to go through to the final.
  1. How many selections of 7 different dogs can be made if there must be at least 1 spaniel, at least 2 retrievers and at least 3 poodles? 2 spaniels, 2 retrievers and 3 poodles go through to the final. They are placed in a line.
  2. How many different arrangements of these 7 dogs are there if the spaniels stand together and the retrievers stand together?
  3. How many different arrangements of these 7 dogs are there if no poodle is next to another poodle?
CAIE S1 2014 June Q1
4 marks Moderate -0.8
1 The petrol consumption of a certain type of car has a normal distribution with mean 24 kilometres per litre and standard deviation 4.7 kilometres per litre. Find the probability that the petrol consumption of a randomly chosen car of this type is between 21.6 kilometres per litre and 28.7 kilometres per litre.
CAIE S1 2014 June Q2
5 marks Moderate -0.3
2 Lengths of a certain type of white radish are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm } .4 \%\) of these radishes are longer than 12 cm and \(32 \%\) are longer than 9 cm . Find \(\mu\) and \(\sigma\).
CAIE S1 2014 June Q3
5 marks Moderate -0.8
3
  1. State three conditions which must be satisfied for a situation to be modelled by a binomial distribution. George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account.
  2. Find the probability that George will buy shares in a small company in at least 3 of these 18 months.
CAIE S1 2014 June Q4
7 marks Standard +0.3
4 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.
  1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
  2. Draw up the probability distribution table for \(X\).
  3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).
CAIE S1 2014 June Q5
8 marks Moderate -0.3
5 Playground equipment consists of swings ( \(S\) ), roundabouts ( \(R\) ), climbing frames ( \(C\) ) and play-houses \(( P )\). The numbers of pieces of equipment in each of 3 playgrounds are as follows.
Playground \(X\)Playground \(Y\)Playground \(Z\)
\(3 S , 2 R , 4 P\)\(6 S , 3 R , 1 C , 2 P\)\(8 S , 3 R , 4 C , 1 P\)
Each day Nur takes her child to one of the playgrounds. The probability that she chooses playground \(X\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Y\) is \(\frac { 1 } { 4 }\). The probability that she chooses playground \(Z\) is \(\frac { 1 } { 2 }\). When she arrives at the playground, she chooses one piece of equipment at random.
  1. Find the probability that Nur chooses a play-house.
  2. Given that Nur chooses a climbing frame, find the probability that she chose playground \(Y\).
CAIE S1 2014 June Q6
10 marks Moderate -0.3
6 Find the number of different ways in which all 8 letters of the word TANZANIA can be arranged so that
  1. all the letters A are together,
  2. the first letter is a consonant ( \(\mathrm { T } , \mathrm { N } , \mathrm { Z }\) ), the second letter is a vowel ( \(\mathrm { A } , \mathrm { I }\) ), the third letter is a consonant, the fourth letter is a vowel, and so on alternately. 4 of the 8 letters of the word TANZANIA are selected. How many possible selections contain
  3. exactly 1 N and 1 A ,
  4. exactly 1 N ?
CAIE S1 2014 June Q7
11 marks Moderate -0.8
7 A typing test is taken by 111 people. The numbers of typing errors they make in the test are summarised in the table below.
Number of typing errors\(1 - 5\)\(6 - 20\)\(21 - 35\)\(36 - 60\)\(61 - 80\)
Frequency249211542
  1. Draw a histogram on graph paper to represent this information.
  2. Calculate an estimate of the mean number of typing errors for these 111 people.
  3. State which class contains the lower quartile and which class contains the upper quartile. Hence find the least possible value of the interquartile range.
CAIE S1 2014 June Q1
4 marks Moderate -0.5
1 In a certain country \(12 \%\) of houses have solar heating. 19 houses are chosen at random. Find the probability that fewer than 4 houses have solar heating.
CAIE S1 2014 June Q2
4 marks
2 A school club has members from 3 different year-groups: Year 1, Year 2 and Year 3. There are 7 members from Year 1, 2 members from Year 2 and 2 members from Year 3. Five members of the club are selected. Find the number of possible selections that include at least one member from each year-group.
CAIE S1 2014 June Q3
5 marks Standard +0.3
3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.
  1. Find the probability that there is a winner of the match after exactly two sets.
  2. Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.
CAIE S1 2014 June Q4
9 marks Moderate -0.3
4 Coin \(A\) is weighted so that the probability of throwing a head is \(\frac { 2 } { 3 }\). Coin \(B\) is weighted so that the probability of throwing a head is \(\frac { 1 } { 4 }\). Coin \(A\) is thrown twice and coin \(B\) is thrown once.
  1. Show that the probability of obtaining exactly 1 head and 2 tails is \(\frac { 13 } { 36 }\).
  2. Draw up the probability distribution table for the number of heads obtained.
  3. Find the expectation of the number of heads obtained.
CAIE S1 2014 June Q5
8 marks Standard +0.3
5 Find how many different numbers can be made from some or all of the digits of the number 1345789 if
  1. all seven digits are used, the odd digits are all together and no digits are repeated,
  2. the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
  3. the numbers made are multiples of 5 which are less than 1000 , and digits can be repeated.
CAIE S1 2014 June Q6
9 marks Moderate -0.3
6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.
Time (seconds)\(< 10.0\)\(< 10.5\)\(< 11.0\)\(< 12.0\)\(< 12.5\)\(< 13.5\)
Cumulative frequency0410404957
  1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
  2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
  3. Calculate estimates of the mean and variance of the times taken by these athletes.
CAIE S1 2014 June Q7
11 marks Standard +0.3
7 The time Rafa spends on his homework each day in term-time has a normal distribution with mean 1.9 hours and standard deviation \(\sigma\) hours. On \(80 \%\) of these days he spends more than 1.35 hours on his homework.
  1. Find the value of \(\sigma\).
  2. Find the probability that, on a randomly chosen day in term-time, Rafa spends less than 2 hours on his homework.
  3. A random sample of 200 days in term-time is taken. Use an approximation to find the probability that the number of days on which Rafa spends more than 1.35 hours on his homework is between 163 and 173 inclusive.