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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
5 marks Moderate -0.8
1 The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$\begin{array} { l l l l l l l } 15 & 10 & 48 & 10 & 19 & 14 & 16 \end{array}$$
  1. Find the mean and standard deviation of these times.
  2. State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case.
  3. For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate.
CAIE S1 2010 June Q2
5 marks Moderate -0.8
2 The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm .
  1. Find the probability that a pencil chosen at random has a length greater than 10.9 cm .
  2. Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm .
    \includegraphics[max width=\textwidth, alt={}]{7b97cfbe-9960-4f26-8be5-ed393feeb8ae-2_1207_1642_1251_255}
    The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies.
CAIE S1 2010 June Q4
6 marks Standard +0.3
4 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Given that \(5 \sigma = 3 \mu\), find \(\mathrm { P } ( X < 2 \mu )\).
  2. With a different relationship between \(\mu\) and \(\sigma\), it is given that \(\mathrm { P } \left( X < \frac { 1 } { 3 } \mu \right) = 0.8524\). Express \(\mu\) in terms of \(\sigma\).
CAIE S1 2010 June Q5
8 marks Moderate -0.3
5 Two fair twelve-sided dice with sides marked \(1,2,3,4,5,6,7,8,9,10,11,12\) are thrown, and the numbers on the sides which land face down are noted. Events \(Q\) and \(R\) are defined as follows. \(Q\) : the product of the two numbers is 24 . \(R\) : both of the numbers are greater than 8 .
  1. Find \(\mathrm { P } ( Q )\).
  2. Find \(\mathrm { P } ( R )\).
  3. Are events \(Q\) and \(R\) exclusive? Justify your answer.
  4. Are events \(Q\) and \(R\) independent? Justify your answer.
CAIE S1 2010 June Q6
10 marks Standard +0.3
6 A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable \(X\) represents the number of geese chosen.
  1. Draw up the probability distribution of \(X\).
  2. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 7 }\) and calculate \(\operatorname { Var } ( X )\).
  3. When the farmer's dog is let loose, it chases either the ducks with probability \(\frac { 3 } { 5 }\) or the geese with probability \(\frac { 2 } { 5 }\). If the dog chases the ducks there is a probability of \(\frac { 1 } { 10 }\) that they will attack the dog. If the dog chases the geese there is a probability of \(\frac { 3 } { 4 }\) that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese.
CAIE S1 2010 June Q7
10 marks Moderate -0.3
7 Nine cards, each of a different colour, are to be arranged in a line.
  1. How many different arrangements of the 9 cards are possible? The 9 cards include a pink card and a green card.
  2. How many different arrangements do not have the pink card next to the green card? Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.
  3. How many different arrangements in total of 3 cards are possible?
  4. How many of the arrangements of 3 cards in part (iii) contain the pink card?
  5. How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card?
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.