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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.
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.