Questions — CAIE (7659 questions)

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CAIE S1 2015 June Q7
11 marks Moderate -0.3
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
3 marks Moderate -0.3
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
5 marks Standard +0.3
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
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
9 marks Standard +0.3
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
11 marks Moderate -0.3
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
11 marks Easy -1.3
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
5 marks Moderate -0.8
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
6 marks Standard +0.3
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
6 marks Easy -1.2
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.
CAIE S1 2016 June Q4
6 marks Standard +0.3
4 When people visit a certain large shop, on average \(34 \%\) of them do not buy anything, \(53 \%\) spend less than \(\\) 50\( and \)13 \%\( spend at least \)\\( 50\).
  1. 15 people visiting the shop are chosen at random. Calculate the probability that at least 14 of them buy something.
  2. \(n\) people visiting the shop are chosen at random. The probability that none of them spends at least \(\\) 50\( is less than 0.04 . Find the smallest possible value of \)n$.
CAIE S1 2016 June Q5
9 marks Easy -1.8
5 The following are the maximum daily wind speeds in kilometres per hour for the first two weeks in April for two towns, Bronlea and Rogate.
Bronlea21456332733214282413172522
Rogate754152371113261823161034
  1. Draw a back-to-back stem-and-leaf diagram to represent this information.
  2. Write down the median of the maximum wind speeds for Bronlea and find the interquartile range for Rogate.
  3. Use your diagram to make one comparison between the maximum wind speeds in the two towns.
CAIE S1 2016 June Q6
9 marks Moderate -0.8
6 The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally distributed with mean 9.5 and standard deviation 1.3.
  1. Find the probability that on a randomly chosen day Peter takes longer than 10.2 minutes.
  2. On \(90 \%\) of days he takes longer than \(t\) minutes. Find the value of \(t\).
  3. Calculate an estimate of the number of days in a year ( 365 days) on which Peter takes less than 8.8 minutes to walk to the shop and buy a newspaper.
CAIE S1 2016 June Q7
9 marks Standard +0.3
7
  1. Find the number of different arrangements which can be made of all 10 letters of the word WALLFLOWER if
    1. there are no restrictions,
    2. there are exactly six letters between the two Ws.
  2. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at least 1 from each activity and there must be more athletes than cyclists. Find the number of different ways in which the team can be chosen.
CAIE S1 2016 June Q1
5 marks Moderate -0.8
1 In a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed and do not wear spectacles is 19 .
  1. Copy and complete the following table to show the number of adults in each category.
    Wears spectaclesDoes not wear spectaclesTotal
    Right-handed
    Not right-handed
    Total30
    An adult is chosen at random from the group. Event \(X\) is 'the adult chosen is right-handed'; event \(Y\) is 'the adult chosen wears spectacles'.
  2. Determine whether \(X\) and \(Y\) are independent events, justifying your answer.
CAIE S1 2016 June Q2
5 marks Easy -1.8
2 A group of children played a computer game which measured their time in seconds to perform a certain task. A summary of the times taken by girls and boys in the group is shown below.
MinimumLower quartileMedianUpper quartileMaximum
Girls55.57913
Boys468.51116
  1. On graph paper, draw two box-and-whisker plots in a single diagram to illustrate the times taken by girls and boys to perform this task.
  2. State two comparisons of the times taken by girls and boys.
CAIE S1 2016 June Q3
6 marks Moderate -0.8
3 Two ordinary fair dice are thrown. The resulting score is found as follows.
  • If the two dice show different numbers, the score is the smaller of the two numbers.
  • If the two dice show equal numbers, the score is 0 .
    1. Draw up the probability distribution table for the score.
    2. Calculate the expected score.
CAIE S1 2016 June Q4
6 marks Moderate -0.8
4 The monthly rental prices, \(\\) x$, for 9 apartments in a certain city are listed and are summarised as follows. $$\Sigma ( x - c ) = 1845 \quad \Sigma ( x - c ) ^ { 2 } = 477450$$ The mean monthly rental price is \(\\) 2205$.
  1. Find the value of the constant \(c\).
  2. Find the variance of these values of \(x\).
  3. Another apartment is added to the list. The mean monthly rental price is now \(\\) 2120.50$. Find the rental price of this additional apartment.
CAIE S1 2016 June Q5
8 marks Standard +0.3
5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .
  1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
  2. Calculate an estimate of the number of these desks that are comfortable for Jodu.
CAIE S1 2016 June Q6
9 marks Moderate -0.8
6 Find the number of ways all 9 letters of the word EVERGREEN can be arranged if
  1. there are no restrictions,
  2. the first letter is R and the last letter is G ,
  3. the Es are all together. Three letters from the 9 letters of the word EVERGREEN are selected.
  4. Find the number of selections which contain no Es and exactly 1 R .
  5. Find the number of selections which contain no Es.
CAIE S1 2016 June Q7
11 marks Standard +0.3
7 Passengers are travelling to Picton by minibus. The probability that each passenger carries a backpack is 0.65 , independently of other passengers. Each minibus has seats for 12 passengers.
  1. Find the probability that, in a full minibus travelling to Picton, between 8 passengers and 10 passengers inclusive carry a backpack.
  2. Passengers get on to an empty minibus. Find the probability that the fourth passenger who gets on to the minibus will be the first to be carrying a backpack.
  3. Find the probability that, of a random sample of 250 full minibuses travelling to Picton, more than 54 will contain exactly 7 passengers carrying backpacks.
CAIE S1 2017 June Q1
4 marks Moderate -0.8
1 Kadijat noted the weights, \(x\) grams, of 30 chocolate buns. Her results are summarised by $$\Sigma ( x - k ) = 315 , \quad \Sigma ( x - k ) ^ { 2 } = 4022$$ where \(k\) is a constant. The mean weight of the buns is 50.5 grams.
  1. Find the value of \(k\).
  2. Find the standard deviation of \(x\).
CAIE S1 2017 June Q2
5 marks Moderate -0.5
2 Ashfaq throws two fair dice and notes the numbers obtained. \(R\) is the event 'The product of the two numbers is 12 '. \(T\) is the event 'One of the numbers is odd and one of the numbers is even'. By finding appropriate probabilities, determine whether events \(R\) and \(T\) are independent.
CAIE S1 2017 June Q3
6 marks Standard +0.3
3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is \(\frac { 3 } { 5 }\), with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is \(\frac { 7 } { 10 }\) and the probability that they lose the second match is \(\frac { 1 } { 10 }\). If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is \(\frac { 3 } { 10 }\) and the probability that they draw the second match is \(\frac { 1 } { 20 }\).
  1. Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.
  2. Given that Redbury United win the second match, find the probability that they lose the first match.