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CAIE S1 2008 June Q5
8 marks Moderate -0.8
5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.
Time spent \(( t\) hours \()\)\(0.1 \leqslant t \leqslant 0.5\)\(0.6 \leqslant t \leqslant 1.0\)\(1.1 \leqslant t \leqslant 2.0\)\(2.1 \leqslant t \leqslant 3.0\)\(3.1 \leqslant t \leqslant 4.5\)
Frequency1115183021
  1. Draw, on graph paper, a histogram to illustrate this information.
  2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.
CAIE S1 2008 June Q6
9 marks Moderate -0.8
6 Every day Eduardo tries to phone his friend. Every time he phones there is a \(50 \%\) chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.
  1. Draw a tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)
  3. Calculate the expected number of unanswered phone calls on a day.
CAIE S1 2008 June Q7
11 marks Standard +0.3
7 A die is biased so that the probability of throwing a 5 is 0.75 and the probabilities of throwing a 1,2 , 3 , 4 or 6 are all equal.
  1. The die is thrown three times. Find the probability that the result is a 1 followed by a 5 followed by any even number.
  2. Find the probability that, out of 10 throws of this die, at least 8 throws result in a 5 .
  3. The die is thrown 90 times. Using an appropriate approximation, find the probability that a 5 is thrown more than 60 times.
CAIE S1 2009 June Q1
5 marks Standard +0.3
1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.
  1. Calculate the value of \(\mu\).
  2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.
CAIE S1 2009 June Q2
6 marks Moderate -0.3
2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
  2. The table below shows the probability distribution of \(X\).
    \(x\)234567
    \(\mathrm { P } ( X = x )\)\(\frac { 5 } { 16 }\)\(\frac { 1 } { 16 }\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 16 }\)
    Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2009 June Q3
8 marks Standard +0.3
3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.
  1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
  2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.
CAIE S1 2009 June Q4
8 marks Standard +0.8
4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.
  1. In how many different ways can the group be chosen?
  2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
  3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?
CAIE S1 2009 June Q5
9 marks Moderate -0.8
5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.
  1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
  2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
  3. Find the probability that he is frightened.
  4. Given that Ravi is not frightened, find the probability that he went on the camel ride.
CAIE S1 2009 June Q6
14 marks Moderate -0.8
6 During January the numbers of people entering a store during the first hour after opening were as follows.
Time after opening,
\(x\) minutes
Frequency
Cumulative
frequency
\(0 < x \leqslant 10\)210210
\(10 < x \leqslant 20\)134344
\(20 < x \leqslant 30\)78422
\(30 < x \leqslant 40\)72\(a\)
\(40 < x \leqslant 60\)\(b\)540
  1. Find the values of \(a\) and \(b\).
  2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
  3. Use your graph to estimate the median time after opening that people entered the store.
  4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
  5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.
CAIE S1 2010 June Q1
4 marks Moderate -0.8
1 The probability distribution of the discrete random variable \(X\) is shown in the table below.
\(x\)- 3- 104
\(\mathrm { P } ( X = x )\)\(a\)\(b\)0.150.4
Given that \(\mathrm { E } ( X ) = 0.75\), find the values of \(a\) and \(b\).
CAIE S1 2010 June Q2
7 marks Easy -2.0
2 The numbers of people travelling on a certain bus at different times of the day are as follows.
17522316318
22142535172712
623192123826
  1. Draw a stem-and-leaf diagram to illustrate the information given above.
  2. Find the median, the lower quartile, the upper quartile and the interquartile range.
  3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.
CAIE S1 2010 June Q3
5 marks Standard +0.3
3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).
CAIE S1 2010 June Q4
7 marks Moderate -0.8
4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.
Roller
coaster
Water
slide
Revolving
drum
Fei420
Graeme136
  1. The mean cost of Fei's rides is \(\\) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(\\) 2.50\( per ride. [2]
  2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.
CAIE S1 2010 June Q5
8 marks Moderate -0.8
5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.
  1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
  2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
  3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.
CAIE S1 2010 June Q6
9 marks Moderate -0.3
6
  1. Find the number of different ways that a set of 10 different mugs can be shared between Lucy and Monica if each receives an odd number of mugs.
  2. Another set consists of 6 plastic mugs each of a different design and 3 china mugs each of a different design. Find in how many ways these 9 mugs can be arranged in a row if the china mugs are all separated from each other.
  3. Another set consists of 3 identical red mugs, 4 identical blue mugs and 7 identical yellow mugs. These 14 mugs are placed in a row. Find how many different arrangements of the colours are possible if the red mugs are kept together.
CAIE S1 2010 June Q7
10 marks Standard +0.3
7 In a television quiz show Peter answers questions one after another, stopping as soon as a question is answered wrongly.
  • The probability that Peter gives the correct answer himself to any question is 0.7 .
  • The probability that Peter gives a wrong answer himself to any question is 0.1 .
  • The probability that Peter decides to ask for help for any question is 0.2 .
On the first occasion that Peter decides to ask for help he asks the audience. The probability that the audience gives the correct answer to any question is 0.95 . This information is shown in the tree diagram below. \includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_394_649_1779_386} \includegraphics[max width=\textwidth, alt={}, center]{e7e0fcbe-ab96-4292-b3ad-c57b74f15301-3_270_743_2010_1023}
  1. Show that the probability that the first question is answered correctly is 0.89 . On the second occasion that Peter decides to ask for help he phones a friend. The probability that his friend gives the correct answer to any question is 0.65 .
  2. Find the probability that the first two questions are both answered correctly.
  3. Given that the first two questions were both answered correctly, find the probability that Peter asked the audience.
CAIE S1 2010 June Q1
3 marks Moderate -0.5
1 A bottle of sweets contains 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are selected at random. Find the probability that exactly 3 of them are red.
CAIE S1 2010 June Q2
4 marks Easy -1.2
2 The heights, \(x \mathrm {~cm}\), of a group of 82 children are summarised as follows. $$\Sigma ( x - 130 ) = - 287 , \quad \text { standard deviation of } x = 6.9 .$$
  1. Find the mean height.
  2. Find \(\Sigma ( x - 130 ) ^ { 2 }\).
CAIE S1 2010 June Q3
5 marks Moderate -0.3
3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.
  1. Find the probability that they go to the park on more than 5 of the next 7 days.
  2. Find the probability that the dog barks on any particular day.
  3. Find the variance of the number of times they go to the park in 30 days.
CAIE S1 2010 June Q4
8 marks Standard +0.3
4 Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are arranged in a row. Calculate the number of arrangements if
  1. the first and last cans in the row are the same type of drink,
  2. the 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.
CAIE S1 2010 June Q5
9 marks Standard +0.3
5 Set \(A\) consists of the ten digits \(0,0,0,0,0,0,2,2,2,4\).
Set \(B\) consists of the seven digits \(0,0,0,0,2,2,2\).
One digit is chosen at random from each set. The random variable \(X\) is defined as the sum of these two digits.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 3 } { 7 }\).
  2. Tabulate the probability distribution of \(X\).
  3. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  4. Given that \(X = 2\), find the probability that the digit chosen from set \(A\) was 2 .
CAIE S1 2010 June Q6
10 marks Moderate -0.3
6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results.
Country \(X\)Country \(Y\)
(10)976664443280
(18)888776655544333220811122333556789(13)
(16)999887765532210082001233394566788(15)
(16)87655533222111008301224444556677789(17)
(11)8765544331184001244556677789(15)
85\(12 r 335566788\)(12)
8601223555899(11)
Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .
  1. Find the median and interquartile range of the lengths of the insects from country \(X\).
  2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
  3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
  4. Compare the lengths of the insects from the two countries.
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.