Questions — CAIE S1 (785 questions)

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CAIE S1 2011 June Q4
4
  1. Find the number of different ways that the 9 letters of the word HAPPINESS can be arranged in a line.
  2. The 9 letters of the word HAPPINESS are arranged in random order in a line. Find the probability that the 3 vowels (A, E, I) are not all next to each other.
  3. Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS which contain no Ps and either one or two Ss.
CAIE S1 2011 June Q5
5 A hotel has 90 rooms. The table summarises information about the number of rooms occupied each day for a period of 200 days.
Number of rooms occupied\(1 - 20\)\(21 - 40\)\(41 - 50\)\(51 - 60\)\(61 - 70\)\(71 - 90\)
Frequency103262502818
  1. Draw a cumulative frequency graph on graph paper to illustrate this information.
  2. Estimate the number of days when over 30 rooms were occupied.
  3. On \(75 \%\) of the days at most \(n\) rooms were occupied. Estimate the value of \(n\).
CAIE S1 2011 June Q6
6 The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution with mean \(\mu\) and variance 0.64 . It is given that \(10 \%\) of the straws are shorter than 20 cm .
  1. Find the value of \(\mu\).
  2. Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length between 21.5 cm and 22.5 cm .
CAIE S1 2011 June Q7
7 Judy and Steve play a game using five cards numbered 3, 4, 5, 8, 9. Judy chooses a card at random, looks at the number on it and replaces the card. Then Steve chooses a card at random, looks at the number on it and replaces the card. If their two numbers are equal the score is 0 . Otherwise, the smaller number is subtracted from the larger number to give the score.
  1. Show that the probability that the score is 6 is 0.08 .
  2. Draw up a probability distribution table for the score.
  3. Calculate the mean score. If the score is 0 they play again. If the score is 4 or more Judy wins. Otherwise Steve wins. They continue playing until one of the players wins.
  4. Find the probability that Judy wins with the second choice of cards.
  5. Find an expression for the probability that Judy wins with the \(n\)th choice of cards.
CAIE S1 2011 June Q1
1 Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In Fairwheel Garage, \(\Sigma x = 64\) and \(\Sigma x ^ { 2 } = 352\), where \(x\) is the age of a car in years.
  1. Find the mean age of all 24 cars.
  2. Find the standard deviation of the ages of all 24 cars.
CAIE S1 2011 June Q2
2 Fahad has 4 different coloured pairs of shoes (white, red, blue and black), 3 different coloured pairs of jeans (blue, black and brown) and 7 different coloured tee shirts (red, orange, yellow, blue, green, white and purple).
  1. Fahad chooses an outfit consisting of one pair of shoes, one pair of jeans and one tee shirt. How many different outfits can he choose?
  2. How many different ways can Fahad arrange his 3 jeans and 7 tee shirts in a row if the two blue items are not next to each other? Fahad also has 9 different books about sport. When he goes on holiday he chooses at least one of these books to take with him.
  3. How many different selections are there if he can take any number of books ranging from just one of them to all of them?
CAIE S1 2011 June Q3
3 The following cumulative frequency table shows the examination marks for 300 candidates in country \(A\) and 300 candidates in country \(B\).
Mark\(< 10\)\(< 20\)\(< 35\)\(< 50\)\(< 70\)\(< 100\)
Cumulative frequency, \(A\)2568159234260300
Cumulative frequency, \(B\)104672144198300
  1. Without drawing a graph, show that the median for country \(B\) is higher than the median for country \(A\).
  2. Find the number of candidates in country \(A\) who scored between 20 and 34 marks inclusive.
  3. Calculate an estimate of the mean mark for candidates in country \(A\).
CAIE S1 2011 June Q4
4 Tim throws a fair die twice and notes the number on each throw.
  1. Tim calculates his final score as follows. If the number on the second throw is a 5 he multiplies the two numbers together, and if the number on the second throw is not a 5 he adds the two numbers together. Find the probability that his final score is
    (a) 12,
    (b) 5 .
  2. Events \(A , B , C\) are defined as follows.
    \(A\) : the number on the second throw is 5
    \(B\) : the sum of the numbers is 6
    \(C\) : the product of the numbers is even
    By calculation find which pairs, if any, of the events \(A , B\) and \(C\) are independent.
CAIE S1 2011 June Q5
5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).
  1. Find \(\mu\).
  2. Find \(\mathrm { P } ( 10 < X < 20 )\).
  3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.
CAIE S1 2011 June Q6
6 The probability that Sue completes a Sudoku puzzle correctly is 0.75 .
  1. Sue attempts \(n\) Sudoku puzzles. Find the least value of \(n\) for which the probability that she completes all \(n\) puzzles correctly is less than 0.06 . Sue attempts 14 Sudoku puzzles every month. The number that she completes successfully is denoted by \(X\).
  2. Find the value of \(X\) that has the highest probability. You may assume that this value is one of the two values closest to the mean of \(X\).
  3. Find the probability that in exactly 3 of the next 5 months Sue completes more than 11 Sudoku puzzles correctly.
CAIE S1 2012 June Q1
1 It is given that \(X \sim \mathrm {~N} ( 28.3,4.5 )\). Find the probability that a randomly chosen value of \(X\) lies between 25 and 30 .
CAIE S1 2012 June Q2
2 Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of 0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability of 0.25 that it will be set to station 3 . On station 1 the probability that the presenter is male is 0.1 , on station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the presenter is male is \(p\). When Maria switches on the radio, the probability that it is set to station 3 and the presenter is male is 0.075 .
  1. Show that the value of \(p\) is 0.3 .
  2. Given that Maria switches on and hears a male presenter, find the probability that the radio was set to station 2.
CAIE S1 2012 June Q3
3 A spinner has 5 sides, numbered 1, 2, 3, 4 and 5 . When the spinner is spun, the score is the number of the side on which it lands. The score is denoted by the random variable \(X\), which has the probability distribution shown in the table.
\(x\)12345
\(\mathrm { P } ( X = x )\)0.30.15\(3 p\)\(2 p\)0.05
  1. Find the value of \(p\). A second spinner has 3 sides, numbered 1, 2 and 3. The score when this spinner is spun is denoted by the random variable \(Y\). It is given that \(\mathrm { P } ( Y = 1 ) = 0.3 , \mathrm { P } ( Y = 2 ) = 0.5\) and \(\mathrm { P } ( Y = 3 ) = 0.2\).
  2. Find the probability that, when both spinners are spun together,
    (a) the sum of the scores is 4,
    (b) the product of the scores is less than 8 .
CAIE S1 2012 June Q4
4 In a certain mountainous region in winter, the probability of more than 20 cm of snow falling on any particular day is 0.21 .
  1. Find the probability that, in any 7-day period in winter, fewer than 5 days have more than 20 cm of snow falling.
  2. For 4 randomly chosen 7-day periods in winter, find the probability that exactly 3 of these periods will have at least 1 day with more than 20 cm of snow falling.
CAIE S1 2012 June Q5
5 The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most popular conventional TVs are shown below.
Flat screen :0.850.940.910.961.040.891.070.920.76
Conventional :0.690.650.850.770.740.670.710.860.75
  1. Represent this information on a back-to-back stem-and-leaf diagram.
  2. Find the median and the interquartile range of the lengths of the diagonals of the 9 conventional TVs.
  3. Find the mean and standard deviation of the lengths of the diagonals of the 9 flat screen TVs.
CAIE S1 2012 June Q6
6 The lengths of body feathers of a particular species of bird are modelled by a normal distribution. A researcher measures the lengths of a random sample of 600 body feathers from birds of this species and finds that 63 are less than 6 cm long and 155 are more than 12 cm long.
  1. Find estimates of the mean and standard deviation of the lengths of body feathers of birds of this species.
  2. In a random sample of 1000 body feathers from birds of this species, how many would the researcher expect to find with lengths more than 1 standard deviation from the mean?
CAIE S1 2012 June Q7
7
  1. Seven friends together with their respective partners all meet up for a meal. To commemorate the occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
    1. How many different arrangements are there if each friend is standing next to his or her partner?
    2. How many different arrangements are there if the 7 friends all stand together and the 7 partners all stand together?
  2. A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be chosen if
    1. both boys are in the team,
    2. the adults are either all in the team or all not in the team,
    3. at least 2 girls are in the team?
CAIE S1 2012 June Q1
1 The ages, \(x\) years, of 150 cars are summarised by \(\Sigma x = 645\) and \(\Sigma x ^ { 2 } = 8287.5\). Find \(\Sigma ( x - \bar { x } ) ^ { 2 }\), where \(\bar { x }\) denotes the mean of \(x\).
CAIE S1 2012 June Q2
2 The random variable \(X\) has the probability distribution shown in the table.
\(x\)246
\(\mathrm { P } ( X = x )\)0.50.40.1
Two independent values of \(X\) are chosen at random. The random variable \(Y\) takes the value 0 if the two values of \(X\) are the same. Otherwise the value of \(Y\) is the larger value of \(X\) minus the smaller value of \(X\).
  1. Draw up the probability distribution table for \(Y\).
  2. Find the expected value of \(Y\).
CAIE S1 2012 June Q3
3 In Restaurant Bijoux 13\% of customers rated the food as 'poor', 22\% of customers rated the food as 'satisfactory' and \(65 \%\) rated it as 'good'. A random sample of 12 customers who went for a meal at Restaurant Bijoux was taken.
  1. Find the probability that more than 2 and fewer than 12 of them rated the food as 'good'. On a separate occasion, a random sample of \(n\) customers who went for a meal at the restaurant was taken.
  2. Find the smallest value of \(n\) for which the probability that at least 1 person will rate the food as 'poor' is greater than 0.95.
CAIE S1 2012 June Q4
4 The back-to-back stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).
\(A\)\(B\)\multirow{3}{*}{(4)}
310151335
41162234457778
8331701333466799(11)
98865543211018247(3)
998865421915(2)
98710204(1)
Key: \(4 | 16 | 7\) means \(A = 0.164\) and \(B = 0.167\).
  1. Find the median and the interquartile range for variable \(A\).
  2. You are given that, for variable \(B\), the median is 0.171 , the upper quartile is 0.179 and the lower quartile is 0.164 . Draw box-and-whisker plots for \(A\) and \(B\) in a single diagram on graph paper.
CAIE S1 2012 June Q5
5 An English examination consists of 8 questions in Part \(A\) and 3 questions in Part \(B\). Candidates must choose 6 questions. The order in which questions are chosen does not matter. Find the number of ways in which the 6 questions can be chosen in each of the following cases.
  1. There are no restrictions on which questions can be chosen.
  2. Candidates must choose at least 4 questions from Part \(A\).
  3. Candidates must either choose both question 1 and question 2 in Part \(A\), or choose neither of these questions.
CAIE S1 2012 June Q6
6 A box of biscuits contains 30 biscuits, some of which are wrapped in gold foil and some of which are unwrapped. Some of the biscuits are chocolate-covered. 12 biscuits are wrapped in gold foil, and of these biscuits, 7 are chocolate-covered. There are 17 chocolate-covered biscuits in total.
  1. Copy and complete the table below to show the number of biscuits in each category.
    Wrapped in gold foilUnwrappedTotal
    Chocolate-covered
    Not chocolate-covered
    Total30
    A biscuit is selected at random from the box.
  2. Find the probability that the biscuit is wrapped in gold foil. The biscuit is returned to the box. An unwrapped biscuit is then selected at random from the box.
  3. Find the probability that the biscuit is chocolate-covered. The biscuit is returned to the box. A biscuit is then selected at random from the box.
  4. Find the probability that the biscuit is unwrapped, given that it is chocolate-covered. The biscuit is returned to the box. Nasir then takes 4 biscuits without replacement from the box.
  5. Find the probability that he takes exactly 2 wrapped biscuits.
CAIE S1 2012 June Q7
7 The times taken to play Beethoven's Sixth Symphony can be assumed to have a normal distribution with mean 41.1 minutes and standard deviation 3.4 minutes. Three occasions on which this symphony is played are chosen at random.
  1. Find the probability that the symphony takes longer than 42 minutes to play on exactly 1 of these occasions. The times taken to play Beethoven's Fifth Symphony can also be assumed to have a normal distribution. The probability that the time is less than 26.5 minutes is 0.1 , and the probability that the time is more than 34.6 minutes is 0.05 .
  2. Find the mean and standard deviation of the times to play this symphony.
  3. Assuming that the times to play the two symphonies are independent of each other, find the probability that, when both symphonies are played, both of the times are less than 34.6 minutes.
CAIE S1 2012 June Q1
1 Ashfaq and Kuljit have done a school statistics project on the prices of a particular model of headphones for MP3 players. Ashfaq collected prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.
  1. Name a suitable statistical diagram for Ashfaq to represent his data, together with a reason for choosing this particular diagram.
  2. Name a suitable statistical diagram for Kuljit to represent her data, together with a reason for choosing this particular diagram.