Questions — CAIE S1 (785 questions)

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CAIE S1 2015 June Q5
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
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
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
1 A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3 and 5 inclusive. 2120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.
Time (seconds)\(1 - 25\)\(26 - 35\)\(36 - 45\)\(46 - 55\)\(56 - 90\)
Number of people424383420
Calculate estimates of the mean and standard deviation of the reading times.
CAIE S1 2015 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{b4bd1629-e2ae-4395-9773-4b14ce428ca6-2_1147_1182_884_477} In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.
  1. On graph paper draw a box-and-whisker plot to summarise this information. 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.
    \includegraphics[max width=\textwidth, alt={}, center]{b4bd1629-e2ae-4395-9773-4b14ce428ca6-3_451_1561_258_292} Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with probability 0.3 , or a handbag. The probability that her mother will like the choice of scarf is 0.72 . The probability that her mother will like the choice of handbag is \(x\). This information is shown on the tree diagram. The probability that Nikita's mother likes the present that Nikita buys is 0.783 .
  3. Find \(x\).
CAIE S1 2015 June Q5
5 A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement, and notes the numbers on the discs.
  1. Find the probability that the numbers on the 3 discs are two even numbers and one odd number. The smallest of the numbers on the 3 discs taken is denoted by the random variable \(S\).
  2. By listing all possible selections (126, 246 and so on) draw up the probability distribution table for \(S\).
CAIE S1 2015 June Q6
6
  1. Find the number of different ways the 7 letters of the word BANANAS can be arranged
    1. if the first letter is N and the last letter is B ,
    2. if all the letters A are next to each other.
  2. Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot both be in the group together.
CAIE S1 2015 June Q7
7
  1. Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
    1. Find the expected number of days during a year ( 52 weeks) for which Zak takes less than 30 minutes for his run.
    2. The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148 . Find the value of \(t\).
  2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 7 ) = 0.2119\) and \(\mathrm { P } ( X < 10 ) = 0.6700\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 2015 June Q1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.