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CAIE S1 2014 June Q1
5 marks Easy -1.3
1 Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.
Adults:555867746361637156535478736462
Children:869589726184779281544368626783
  1. Draw a back-to-back stem-and-leaf diagram to represent the data.
  2. Make two comparisons between the estimates of the adults and the children.
CAIE S1 2014 June Q2
6 marks Moderate -0.3
2 There is a probability of \(\frac { 1 } { 7 }\) that Wenjie goes out with her friends on any particular day. 252 days are chosen at random.
  1. Use a normal approximation to find the probability that the number of days on which Wenjie goes out with her friends is less than than 30 or more than 44.
  2. Give a reason why the use of a normal approximation is justified.
CAIE S1 2014 June Q3
6 marks Moderate -0.3
3 A pet shop has 6 rabbits and 3 hamsters. 5 of these pets are chosen at random. The random variable \(X\) represents the number of hamsters chosen.
  1. Show that the probability that exactly 2 hamsters are chosen is \(\frac { 10 } { 21 }\).
  2. Draw up the probability distribution table for \(X\).
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
    (a) if the even digits are all together,
    (b) if the first and last digits are both odd.
  2. 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 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
6 marks Easy -1.2
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
8 marks Moderate -0.8
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
9 marks Moderate -0.8
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
12 marks Standard +0.3
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
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.