Questions — CAIE (7646 questions)

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CAIE S1 2015 June Q6
10 marks Standard +0.3
6
  1. In a certain country, \(68 \%\) of households have a printer. Find the probability that, in a random sample of 8 households, 5, 6 or 7 households have a printer.
  2. Use an approximation to find the probability that, in a random sample of 500 households, more than 337 households have a printer.
  3. Justify your use of the approximation in part (ii).
CAIE S1 2015 June Q7
11 marks Moderate -0.3
7
  1. Find how many different numbers can be made by arranging all nine digits of the number 223677888 if
    1. there are no restrictions,
    2. the number made is an even number.
  2. Sandra wishes to buy some applications (apps) for her smartphone but she only has enough money for 5 apps in total. There are 3 train apps, 6 social network apps and 14 games apps available. Sandra wants to have at least 1 of each type of app. Find the number of different possible selections of 5 apps that Sandra can choose.
CAIE S1 2015 June Q1
3 marks Moderate -0.8
1 The weights, in grams, of onions in a supermarket have a normal distribution with mean \(\mu\) and standard deviation 22. The probability that a randomly chosen onion weighs more than 195 grams is 0.128 . Find the value of \(\mu\).
CAIE S1 2015 June Q2
5 marks Moderate -0.8
2 When Joanna cooks, the probability that the meal is served on time is \(\frac { 1 } { 5 }\). The probability that the kitchen is left in a mess is \(\frac { 3 } { 5 }\). The probability that the meal is not served on time and the kitchen is not left in a mess is \(\frac { 3 } { 10 }\). Some of this information is shown in the following table.
Kitchen left in a messKitchen not left in a messTotal
Meal served on time\(\frac { 1 } { 5 }\)
Meal not served on time\(\frac { 3 } { 10 }\)
Total1
  1. Copy and complete the table.
  2. Given that the kitchen is left in a mess, find the probability that the meal is not served on time.
CAIE S1 2015 June Q3
5 marks Moderate -0.8
3 On a production line making cameras, the probability of a randomly chosen camera being substandard is 0.072 . A random sample of 300 cameras is checked. Find the probability that there are fewer than 18 cameras which are substandard.
CAIE S1 2015 June Q4
6 marks Moderate -0.8
4 A pet shop has 9 rabbits for sale, 6 of which are white. A random sample of two rabbits is chosen without replacement.
  1. Show that the probability that exactly one of the two rabbits in the sample is white is \(\frac { 1 } { 2 }\).
  2. Construct the probability distribution table for the number of white rabbits in the sample.
  3. Find the expected value of the number of white rabbits in the sample.
CAIE S1 2015 June Q5
9 marks Standard +0.3
5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard deviation 6.5. A book with a height of more than 29 cm is classified as 'large'.
  1. Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.
  2. \(n\) books are chosen at random. The probability of there being at least 1 large book is more than 0.98 . Find the least possible value of \(n\).
CAIE S1 2015 June Q6
11 marks Easy -1.8
6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample. The cumulative frequency distribution is shown in the table below.
Nitrogen content\(\leqslant 3.5\)\(\leqslant 3.8\)\(\leqslant 4.0\)\(\leqslant 4.2\)\(\leqslant 4.5\)\(\leqslant 4.8\)
Cumulative frequency0618416270
  1. On graph paper draw a cumulative frequency graph to represent the data.
  2. Estimate the percentage of samples with a nitrogen content greater than 4.4.
  3. Estimate the median.
  4. Construct the frequency table for these results and draw a histogram on graph paper.
CAIE S1 2015 June Q7
11 marks Moderate -0.3
7 Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.
  1. She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make? Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if
  2. she has a duck at each end of the row and no ducks anywhere else,
  3. she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between.
CAIE S1 2016 June Q1
3 marks Moderate -0.3
1 The height of maize plants in Mpapwa is normally distributed with mean 1.62 m and standard deviation \(\sigma \mathrm { m }\). The probability that a randomly chosen plant has a height greater than 1.8 m is 0.15 . Find the value of \(\sigma\).
CAIE S1 2016 June Q2
5 marks Standard +0.3
2 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The random variable \(X\) is the score when the die is thrown. The following is the probability distribution table for \(X\).
\(x\)123456
\(\mathrm { P } ( X = x )\)\(p\)\(p\)\(p\)\(p\)0.20.2
The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.
CAIE S1 2016 June Q3
5 marks Moderate -0.8
3 The probability that the school bus is on time on any particular day is 0.6 . If the bus is on time the probability that Sam the driver gets a cup of coffee is 0.9 . If the bus is not on time the probability that Sam gets a cup of coffee is 0.3 .
  1. Find the probability that Sam gets a cup of coffee.
  2. Given that Sam does not get a cup of coffee, find the probability that the bus is not on time.
CAIE S1 2016 June Q4
6 marks Moderate -0.8
4 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2016 June Q5
9 marks Standard +0.3
5 Plastic drinking straws are manufactured to fit into drinks cartons which have a hole in the top. A straw fits into the hole if the diameter of the straw is less than 3 mm . The diameters of the straws have a normal distribution with mean 2.6 mm and standard deviation 0.25 mm .
  1. A straw is chosen at random. Find the probability that it fits into the hole in a drinks carton.
  2. 500 straws are chosen at random. Use a suitable approximation to find the probability that at least 480 straws fit into the holes in drinks cartons.
  3. Justify the use of your approximation.
CAIE S1 2016 June Q6
11 marks Moderate -0.3
6
    1. Find how many numbers there are between 100 and 999 in which all three digits are different.
    2. Find how many of the numbers in part (i) are odd numbers greater than 700 .
  2. A bunch of flowers consists of a mixture of roses, tulips and daffodils. Tom orders a bunch of 7 flowers from a shop to give to a friend. There must be at least 2 of each type of flower. The shop has 6 roses, 5 tulips and 4 daffodils, all different from each other. Find the number of different bunches of flowers that are possible.
CAIE S1 2016 June Q7
11 marks Easy -1.3
7 The amounts spent by 160 shoppers at a supermarket are summarised in the following table.
Amount spent \(( \\) x )\(\)0 < x \leqslant 30\(\)30 < x \leqslant 50\(\)50 < x \leqslant 70\(\)70 < x \leqslant 90\(\)90 < x \leqslant 140$
Number of shoppers1640482630
  1. Draw a cumulative frequency graph of this distribution.
  2. Estimate the median and the interquartile range of the amount spent.
  3. Estimate the number of shoppers who spent more than \(\\) 115$.
  4. Calculate an estimate of the mean amount spent.
CAIE S1 2016 June Q1
5 marks Moderate -0.8
1 Ayman's breakfast drink is tea, coffee or hot chocolate with probabilities \(0.65,0.28,0.07\) respectively. When he drinks tea, the probability that he has milk in it is 0.8 . When he drinks coffee, the probability that he has milk in it is 0.5 . When he drinks hot chocolate he always has milk in it.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Ayman's breakfast drink is coffee, given that his drink has milk in it.
CAIE S1 2016 June Q2
6 marks Standard +0.3
2 When visiting the dentist the probability of waiting less than 5 minutes is 0.16 , and the probability of waiting less than 10 minutes is 0.88 .
  1. Find the probability of waiting between 5 and 10 minutes. A random sample of 180 people who visit the dentist is chosen.
  2. Use a suitable approximation to find the probability that more than 115 of these people wait between 5 and 10 minutes.
CAIE S1 2016 June Q3
6 marks Easy -1.2
3 A particular type of bird lays 1,2,3 or 4 eggs in a nest each year. The probability of \(x\) eggs is equal to \(k x\), where \(k\) is a constant.
  1. Draw up a probability distribution table, in terms of \(k\), for the number of eggs laid in a year and find the value of \(k\).
  2. Find the mean and variance of the number of eggs laid in a year by this type of bird.
CAIE S1 2016 June Q4
6 marks Standard +0.3
4 When people visit a certain large shop, on average \(34 \%\) of them do not buy anything, \(53 \%\) spend less than \(\\) 50\( and \)13 \%\( spend at least \)\\( 50\).
  1. 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.
  2. \(n\) people visiting the shop are chosen at random. The probability that none of them spends at least \(\\) 50\( is less than 0.04 . Find the smallest possible value of \)n$.
CAIE S1 2016 June Q5
9 marks Easy -1.8
5 The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.
Bronlea21456332733214282413172522
Rogate754152371113261823161034
  1. Draw a back-to-back stem-and-leaf diagram to represent this information.
  2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
  3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.
CAIE S1 2016 June Q6
9 marks Moderate -0.8
6 The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean 9.5 and standard deviation 1.3.
  1. Find the probability that on a randomly chosen day Peter takes longer than 10.2 minutes.
  2. On \(90 \%\) of days he takes longer than \(t\) minutes. Find the value of \(t\).
  3. Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than 8.8 minutes to walk to the shop and buy a newspaper.
CAIE S1 2016 June Q7
9 marks Standard +0.3
7
  1. Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if
    1. there are no restrictions,
    2. there are exactly six letters between the two Ws.
  2. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.
CAIE S1 2016 June Q1
5 marks Moderate -0.8
1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .
  1. Copy and complete the following table to show the number of adults in each category.
    Wears spectaclesDoes not wear spectaclesTotal
    Right-handed
    Not right-handed
    Total30
    An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
  2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.
CAIE S1 2016 June Q2
5 marks Easy -1.8
2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.
MinimumLower quartileMedianUpper quartileMaximum
Girls55.57913
Boys468.51116
  1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
  2. State two comparisons of the times taken by girls and boys.