Questions — CAIE (7659 questions)

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CAIE S1 2011 November Q6
9 marks Moderate -0.3
6
  1. Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be arranged
    1. if there are no restrictions,
    2. if the 4 vowels \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { I }\) must all be together.
    1. 4 astronauts are chosen from a certain number of candidates. If order of choosing is not taken into account, the number of ways the astronauts can be chosen is 3876 . How many ways are there if order of choosing is taken into account?
    2. 4 astronauts are chosen to go on a mission. Each of these astronauts can take 3 personal possessions with him. How many different ways can these 12 possessions be arranged in a row if each astronaut's possessions are kept together?
CAIE S1 2011 November Q7
9 marks Standard +0.8
7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.
  1. Find the probability that exactly two of the selected balls have the same number.
  2. Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2 .
  3. Event \(X\) is 'exactly two of the selected balls have the same number'. Event \(Y\) is 'the ball selected from bag \(A\) has number 2'. Showing your working, determine whether events \(X\) and \(Y\) are independent or not.
CAIE S1 2011 November Q1
3 marks Easy -1.8
1 The following are the times, in minutes, taken by 11 runners to complete a 10 km run. \(\begin{array} { l l l l l l l l l l l } 48.3 & 55.2 & 59.9 & 67.7 & 60.5 & 75.6 & 62.5 & 57.4 & 53.4 & 49.2 & 64.1 \end{array}\) Find the mean and standard deviation of these times.
CAIE S1 2011 November Q2
3 marks Easy -1.2
2 Twelve coins are tossed and placed in a line. Each coin can show either a head or a tail.
  1. Find the number of different arrangements of heads and tails which can be obtained.
  2. Find the number of different arrangements which contain 7 heads and 5 tails.
CAIE S1 2011 November Q3
7 marks Challenging +1.2
3
  1. Geoff wishes to plant 25 flowers in a flower-bed. He can choose from 15 different geraniums, 10 different roses and 8 different lilies. He wants to have at least 11 geraniums and also to have the same number of roses and lilies. Find the number of different selections of flowers he can make.
  2. Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged if exactly two of the Gs are next to each other.
CAIE S1 2011 November Q4
8 marks Easy -1.3
4 The weights of 220 sausages are summarised in the following table.
Weight (grams)\(< 20\)\(< 30\)\(< 40\)\(< 45\)\(< 50\)\(< 60\)\(< 70\)
Cumulative frequency02050100160210220
  1. State which interval the median weight lies in.
  2. Find the smallest possible value and the largest possible value for the interquartile range.
  3. State how many sausages weighed between 50 g and 60 g .
  4. On graph paper, draw a histogram to represent the weights of the sausages.
CAIE S1 2011 November Q5
9 marks Standard +0.3
5 A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.
  1. Show that the probability that the spinner lands on the blue side is \(\frac { 1 } { 8 }\).
  2. The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.
  3. The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.
CAIE S1 2011 November Q6
9 marks Moderate -0.8
6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.
  1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
  2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
  3. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
  4. Find the proportion of winter days on which the minimum temperature is below zero.
  5. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
  6. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).
CAIE S1 2011 November Q1
6 marks Standard +0.3
1 The random variable \(X\) is normally distributed and is such that the mean \(\mu\) is three times the standard deviation \(\sigma\). It is given that \(\mathrm { P } ( X < 25 ) = 0.648\).
  1. Find the values of \(\mu\) and \(\sigma\).
  2. Find the probability that, from 6 random values of \(X\), exactly 4 are greater than 25 .
CAIE S1 2011 November Q2
6 marks Moderate -0.8
2 In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.
  1. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.
  2. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.
CAIE S1 2011 November Q3
9 marks Standard +0.3
3 A factory makes a large number of ropes with lengths either 3 m or 5 m . There are four times as many ropes of length 3 m as there are ropes of length 5 m .
  1. One rope is chosen at random. Find the expectation and variance of its length.
  2. Two ropes are chosen at random. Find the probability that they have different lengths.
  3. Three ropes are chosen at random. Find the probability that their total length is 11 m .
CAIE S1 2011 November Q4
9 marks Moderate -0.8
4 Mary saves her digital images on her computer in three separate folders named 'Family', 'Holiday' and 'Friends'. Her family folder contains 3 images, her holiday folder contains 4 images and her friends folder contains 8 images. All the images are different.
  1. Find in how many ways she can arrange these 15 images in a row across her computer screen if she keeps the images from each folder together.
  2. Find the number of different ways in which Mary can choose 6 of these images if there are 2 from each folder.
  3. Find the number of different ways in which Mary can choose 6 of these images if there are at least 3 images from the friends folder and at least 1 image from each of the other two folders.
CAIE S1 2011 November Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573} The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.
  1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
  2. Comment on the salaries of the people in this sample.
  3. 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.
    (a) How high must a salary be in order to be classified as an outlier?
    (b) Show that none of the salaries is low enough to be classified as an outlier.
CAIE S1 2011 November Q6
11 marks Standard +0.3
6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).
  1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
  2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.
CAIE S1 2012 November Q1
4 marks Moderate -0.8
1 Ashok has 3 green pens and 7 red pens. His friend Rod takes 3 of these pens at random, without replacement. Draw up a probability distribution table for the number of green pens Rod takes.
CAIE S1 2012 November Q2
5 marks Moderate -0.8
2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
CAIE S1 2012 November Q3
6 marks Standard +0.3
3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .
  1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
  2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .
CAIE S1 2012 November Q4
7 marks Moderate -0.8
4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)
  1. Represent these prices by a stem-and-leaf diagram.
  2. Write down the lower quartile of the prices of the caravans in the showroom.
  3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.
CAIE S1 2012 November Q5
7 marks Standard +0.3
5 A company set up a display consisting of 20 fireworks. For each firework, the probability that it fails to work is 0.05 , independently of other fireworks.
  1. Find the probability that more than 1 firework fails to work. The 20 fireworks cost the company \(\\) 24\( each. 450 people pay the company \)\\( 10\) each to watch the display. If more than 1 firework fails to work they get their money back.
  2. Calculate the expected profit for the company.
CAIE S1 2012 November Q6
9 marks Standard +0.3
6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.
  1. Find the probability that she is on time on fewer than 20 of the next 96 days.
  2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.
CAIE S1 2012 November Q7
12 marks Standard +0.3
7
  1. In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together.
  2. Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.
    1. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account? The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.
    2. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen?
  3. 9 different fruit pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done.
CAIE S2 2014 June Q1
3 marks Easy -1.2
1 The weights, in grams, of a random sample of 8 packets of cereal are as follows. $$\begin{array} { l l l l l l l l } 250 & 248 & 255 & 244 & 259 & 250 & 242 & 258 \end{array}$$ Calculate unbiased estimates of the population mean and variance.
CAIE S2 2014 June Q2
5 marks Standard +0.3
2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]
CAIE S2 2014 June Q3
5 marks Standard +0.8
3 The number of calls per day to an enquiry desk has a Poisson distribution. In the past the mean has been 5 . In order to test whether the mean has changed, the number of calls on a random sample of 10 days was recorded. The total number of calls was found to be 61 . Use an approximate distribution to test at the 10\% significance level whether the mean has changed.
CAIE S2 2014 June Q4
8 marks Standard +0.3
4
  1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
  2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
  3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).