Questions S1 (2020 questions)

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CAIE S1 2014 June Q4
6 marks Moderate -0.8
4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.
  1. Find the mean height of the remaining group of 27 people.
  2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.
CAIE S1 2014 June Q5
8 marks Standard +0.8
5 When Moses makes a phone call, the amount of time that the call takes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.
  1. \(90 \%\) of Moses's phone calls take longer than \(t\) minutes. Find the value of \(t\).
  2. Find the probability that, in a random sample of 9 phone calls made by Moses, more than 7 take a time which is within 1 standard deviation of the mean.
CAIE S1 2014 June Q6
8 marks Standard +0.8
6 Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3 . Each game is won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion) has won two games.
  1. Find the probability that Ben becomes the champion after playing exactly 2 games.
  2. Find the probability that Ben becomes the champion.
  3. Given that Tom becomes the champion, find the probability that he won the 2nd game.
CAIE S1 2014 June Q7
11 marks Standard +0.3
7 Nine cards are numbered \(1,2,2,3,3,4,6,6,6\).
  1. All nine cards are placed in a line, making a 9-digit number. Find how many different 9-digit numbers can be made in this way
    1. if the even digits are all together,
    2. if the first and last digits are both odd.
    3. Three of the nine cards are chosen and placed in a line, making a 3-digit number. Find how many different numbers can be made in this way
      (a) if there are no repeated digits,
      (b) if the number is between 200 and 300 .
CAIE S1 2015 June Q1
4 marks Standard +0.3
1 The lengths, in metres, of cars in a city are normally distributed with mean \(\mu\) and standard deviation 0.714 . The probability that a randomly chosen car has a length more than 3.2 metres and less than \(\mu\) metres is 0.475 . Find \(\mu\).
CAIE S1 2015 June Q2
5 marks Moderate -0.8
2 The table summarises the lengths in centimetres of 104 dragonflies.
Length \(( \mathrm { cm } )\)\(2.0 - 3.5\)\(3.5 - 4.5\)\(4.5 - 5.5\)\(5.5 - 7.0\)\(7.0 - 9.0\)
Frequency825283112
  1. State which class contains the upper quartile.
  2. Draw a histogram, on graph paper, to represent the data.
CAIE S1 2015 June Q3
6 marks Standard +0.3
3 Jason throws two fair dice, each with faces numbered 1 to 6 . Event \(A\) is 'one of the numbers obtained is divisible by 3 and the other number is not divisible by 3 '. Event \(B\) is 'the product of the two numbers obtained is even'.
  1. Determine whether events \(A\) and \(B\) are independent, showing your working.
  2. Are events \(A\) and \(B\) mutually exclusive? Justify your answer.
CAIE S1 2015 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{c0c7e038-805a-4237-a579-a6571b84f337-2_451_1530_1393_303} A survey is undertaken to investigate how many photos people take on a one-week holiday and also how many times they view past photos. For a randomly chosen person, the probability of taking fewer than 100 photos is \(x\). The probability that these people view past photos at least 3 times is 0.76 . For those who take at least 100 photos, the probability that they view past photos fewer than 3 times is 0.90 . This information is shown in the tree diagram. The probability that a randomly chosen person views past photos fewer than 3 times is 0.801 .
  1. Find \(x\).
  2. Given that a person views past photos at least 3 times, find the probability that this person takes at least 100 photos.
CAIE S1 2015 June Q5
7 marks Moderate -0.8
5 The table shows the mean and standard deviation of the weights of some turkeys and geese.
Number of birdsMean (kg)Standard deviation (kg)
Turkeys97.11.45
Geese185.20.96
  1. Find the mean weight of the 27 birds.
  2. The weights of individual turkeys are denoted by \(x _ { t } \mathrm {~kg}\) and the weights of individual geese by \(x _ { g } \mathrm {~kg}\). By first finding \(\Sigma x _ { t } ^ { 2 }\) and \(\Sigma x _ { g } ^ { 2 }\), find the standard deviation of the weights of all 27 birds.
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.