Questions S1 (1967 questions)

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CAIE S1 2012 June Q2
2 The heights, \(x \mathrm {~cm}\), of a group of young children are summarised by $$\Sigma ( x - 100 ) = 72 , \quad \Sigma ( x - 100 ) ^ { 2 } = 499.2 .$$ The mean height is 104.8 cm .
  1. Find the number of children in the group.
  2. Find \(\Sigma ( x - 104.8 ) ^ { 2 }\).
CAIE S1 2012 June Q3
3
  1. In how many ways can all 9 letters of the word TELEPHONE be arranged in a line if the letters P and L must be at the ends? How many different selections of 4 letters can be made from the 9 letters of the word TELEPHONE if
  2. there are no Es,
  3. there is exactly 1 E ,
  4. there are no restrictions?
CAIE S1 2012 June Q4
4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.
  1. Find the mean and standard deviation of \(W\).
  2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
  3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).
CAIE S1 2012 June Q5
5 Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with designer labels in total.
  1. Copy and complete the table below to show the number of pairs in each category.
    Designer labelsNo designer labelsTotal
    High-heeled shoes
    Low-heeled shoes
    Sports shoes
    Total20
    Suzanne chooses 1 pair of shoes at random to wear.
  2. Find the probability that she wears the pair of low-heeled shoes with designer labels.
  3. Find the probability that she wears a pair of sports shoes.
  4. Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair of shoes with designer labels.
  5. State with a reason whether the events 'Suzanne wears a pair of shoes with designer labels' and 'Suzanne wears a pair of sports shoes' are independent. Suzanne chooses 1 pair of shoes at random each day.
  6. Find the probability that Suzanne wears a pair of shoes with designer labels on at most 4 days out of the next 7 days.
CAIE S1 2012 June Q6
6 The lengths, in cm, of trout in a fish farm are normally distributed. 96\% of the lengths are less than 34.1 cm and 70\% of the lengths are more than 26.7 cm .
  1. Find the mean and the standard deviation of the lengths of the trout. In another fish farm, the lengths of salmon, \(X \mathrm {~cm}\), are normally distributed with mean 32.9 cm and standard deviation 2.4 cm .
  2. Find the probability that a randomly chosen salmon is 34 cm long, correct to the nearest centimetre.
  3. Find the value of \(t\) such that \(\mathrm { P } ( 31.8 < X < t ) = 0.5\).
CAIE S1 2013 June Q1
1 A summary of 30 values of \(x\) gave the following information: $$\Sigma ( x - c ) = 234 , \quad \Sigma ( x - c ) ^ { 2 } = 1957.5 ,$$ where \(c\) is a constant.
  1. Find the standard deviation of these values of \(x\).
  2. Given that the mean of these values is 86 , find the value of \(c\).
CAIE S1 2013 June Q2
2 Assume that, for a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 350 randomly chosen people, at least 47 will have their next birthday on a Monday.
CAIE S1 2013 June Q3
3 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
FemalesMales
(4)\multirow{7}{*}{9}5200203
(9)8876400021007
(8)\multirow{5}{*}{}8753310022004566
(6)\multirow{4}{*}{}64210023002335677
(6)754000240112556889
(4)9500253457789
(2)5026046
Key: 2 | 20 | 3 means \\(20200 for females and
)20300 for males.
  1. Find the median and the quartiles of the females' salaries. You are given that the median salary of the males is \(
    ) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(
    ) 25300$.
  2. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
CAIE S1 2013 June Q4
4 marks
4
  1. The random variable \(Y\) is normally distributed with positive mean \(\mu\) and standard deviation \(\frac { 1 } { 2 } \mu\). Find the probability that a randomly chosen value of \(Y\) is negative.
  2. The weights of bags of rice are normally distributed with mean 2.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 8000 such bags, 253 weighed over 2.1 kg . Find the value of \(\sigma\). [4]
CAIE S1 2013 June Q5
5 Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The random variable \(X\) is the number of these 12 integers which are multiples of 5 .
  1. State the distribution of \(X\) and give its parameters.
  2. Calculate the probability that \(X\) is between 3 and 5 inclusive. Fiona now produces \(n\) random integers between 7 and 21 inclusive.
  3. Find the least possible value of \(n\) if the probability that none of these integers is a multiple of 5 is less than 0.01.
CAIE S1 2013 June Q6
6 Four families go to a theme park together. Mr and Mrs Lin take their 2 children. Mr O'Connor takes his 2 children. Mr and Mrs Ahmed take their 3 children. Mrs Burton takes her son. The 14 people all have to go through a turnstile one at a time to enter the theme park.
  1. In how many different orders can the 14 people go through the turnstile if each family stays together?
  2. In how many different orders can the 8 children and 6 adults go through the turnstile if no two adults go consecutively? Once inside the theme park, the children go on the roller-coaster. Each roller-coaster car holds 3 people.
  3. In how many different ways can the 8 children be divided into two groups of 3 and one group of 2 to go on the roller-coaster?
CAIE S1 2013 June Q7
7 Box \(A\) contains 8 white balls and 2 yellow balls. Box \(B\) contains 5 white balls and \(x\) yellow balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Box \(A\)} \includegraphics[alt={},max width=\textwidth]{60a9d5d4-0a6a-43e2-9828-03ea2a76ed8a-3_451_874_1774_639}
\end{figure}
  1. Justify the probability \(\frac { x } { x + 6 }\) on the tree diagram.
  2. Copy and complete the tree diagram.
  3. If the ball chosen from box \(A\) is white then the probability that the ball chosen from box \(B\) is also white is \(\frac { 1 } { 3 }\). Show that the value of \(x\) is 12 .
  4. Given that the ball chosen from box \(B\) is yellow, find the conditional probability that the ball chosen from box \(A\) was yellow.
CAIE S1 2013 June Q1
1 The random variable \(Y\) is normally distributed with mean equal to five times the standard deviation. It is given that \(\mathrm { P } ( Y > 20 ) = 0.0732\). Find the mean.
CAIE S1 2013 June Q2
2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \quad \text { and } \quad \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).
CAIE S1 2013 June Q3
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
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
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
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
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
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
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
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
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
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
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?