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CAIE S1 2010 June Q7
11 marks Standard +0.3
7 The heights that children of a particular age can jump have a normal distribution. On average, 8 children out of 10 can jump a height of more than 127 cm , and 1 child out of 3 can jump a height of more than 135 cm .
  1. Find the mean and standard deviation of the heights the children can jump.
  2. Find the probability that a randomly chosen child will not be able to jump a height of 145 cm .
  3. Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height of more than 135 cm .
CAIE S1 2011 June Q1
4 marks Moderate -0.3
1 Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken, independently of other biscuits. The mean number of broken biscuits in a packet has been found to be 2.7 . Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits.
CAIE S1 2011 June Q2
4 marks Moderate -0.8
2 When Ted is looking for his pen, the probability that it is in his pencil case is 0.7 . If his pen is in his pencil case he always finds it. If his pen is somewhere else, the probability that he finds it is 0.2 . Given that Ted finds his pen when he is looking for it, find the probability that it was in his pencil case.
CAIE S1 2011 June Q3
6 marks Standard +0.3
3 The possible values of the random variable \(X\) are the 8 integers in the set \(\{ - 2 , - 1,0,1,2,3,4,5 \}\). The probability of \(X\) being 0 is \(\frac { 1 } { 10 }\). The probabilities for all the other values of \(X\) are equal. Calculate
  1. \(\mathrm { P } ( X < 2 )\),
  2. the variance of \(X\),
  3. the value of \(a\) for which \(\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }\).
CAIE S1 2011 June Q4
8 marks Challenging +1.2
4 A cricket team of 11 players is to be chosen from 21 players consisting of 10 batsmen, 9 bowlers and 2 wicketkeepers. The team must include at least 5 batsmen, at least 4 bowlers and at least 1 wicketkeeper.
  1. Find the number of different ways in which the team can be chosen. Each player in the team is given a present. The presents consist of 5 identical pens, 4 identical diaries and 2 identical notebooks.
  2. Find the number of different arrangements of the presents if they are all displayed in a row.
  3. 10 of these 11 presents are chosen and arranged in a row. Find the number of different arrangements that are possible.
CAIE S1 2011 June Q5
8 marks Challenging +1.2
5
  1. The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(3 \mu = 7 \sigma ^ { 2 }\) and that \(\mathrm { P } ( X > 2 \mu ) = 0.1016\). Find \(\mu\) and \(\sigma\).
  2. It is given that \(Y \sim \mathrm {~N} ( 33,21 )\). Find the value of \(a\) given that \(\mathrm { P } ( 33 - a < Y < 33 + a ) = 0.5\).
CAIE S1 2011 June Q6
10 marks Easy -1.8
6 There are 5000 schools in a certain country. The cumulative frequency table shows the number of pupils in a school and the corresponding number of schools.
Number of pupils in a school\(\leqslant 100\)\(\leqslant 150\)\(\leqslant 200\)\(\leqslant 250\)\(\leqslant 350\)\(\leqslant 450\)\(\leqslant 600\)
Cumulative frequency20080016002100410047005000
  1. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of pupils in a school.
  2. \(80 \%\) of the schools have more than \(n\) pupils. Estimate the value of \(n\) correct to the nearest ten.
  3. Find how many schools have between 201 and 250 (inclusive) pupils.
  4. Calculate an estimate of the mean number of pupils per school.
CAIE S1 2011 June Q7
10 marks Moderate -0.3
7
    1. Find the probability of getting at least one 3 when 9 fair dice are thrown.
    2. When \(n\) fair dice are thrown, the probability of getting at least one 3 is greater than 0.9. Find the smallest possible value of \(n\).
  2. A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement. The winner of the game is the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie wins the game.
CAIE S1 2011 June Q1
4 marks Standard +0.3
1 A biased die was thrown 20 times and the number of 5 s was noted. This experiment was repeated many times and the average number of 5 s was found to be 4.8 . Find the probability that in the next 20 throws the number of 5 s will be less than three.
CAIE S1 2011 June Q2
5 marks Moderate -0.8
2 In Scotland, in November, on average \(80 \%\) of days are cloudy. Assume that the weather on any one day is independent of the weather on other days.
  1. Use a normal approximation to find the probability of there being fewer than 25 cloudy days in Scotland in November (30 days).
  2. Give a reason why the use of a normal approximation is justified.
CAIE S1 2011 June Q3
7 marks Moderate -0.3
3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).
  1. Find the mean and standard deviation of the 36 values.
  2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.
CAIE S1 2011 June Q4
8 marks Moderate -0.3
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
8 marks Easy -1.3
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
9 marks Standard +0.3
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
9 marks Standard +0.3
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
6 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.8
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
9 marks Standard +0.8
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
11 marks Standard +0.8
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
11 marks Standard +0.3
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
3 marks Easy -1.2
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
5 marks Moderate -0.8
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
7 marks Moderate -0.8
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
7 marks Standard +0.8
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.