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CAIE S1 2008 November Q1
3 marks Easy -1.8
1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.
CAIE S1 2008 November Q2
5 marks Moderate -0.8
2 On a production line making toys, the probability of any toy being faulty is 0.08 . A random sample of 200 toys is checked. Use a suitable approximation to find the probability that there are at least 15 faulty toys.
CAIE S1 2008 November Q3
6 marks Moderate -0.8
3
  1. The daily minimum temperature in degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ) in January in Ottawa is a random variable with distribution \(\mathrm { N } ( - 15.1,62.0 )\). Find the probability that a randomly chosen day in January in Ottawa has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\).
  2. In another city the daily minimum temperature in \({ } ^ { \circ } \mathrm { C }\) in January is a random variable with distribution \(\mathrm { N } ( \mu , 40.0 )\). In this city the probability that a randomly chosen day in January has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\) is 0.8888 . Find the value of \(\mu\).
CAIE S1 2008 November Q4
7 marks Moderate -0.8
4 A builder is planning to build 12 houses along one side of a road. He will build 2 houses in style \(A\), 2 houses in style \(B , 3\) houses in style \(C , 4\) houses in style \(D\) and 1 house in style \(E\).
  1. Find the number of possible arrangements of these 12 houses.
  2. Road
    \(\square \square \square \square \square \square \square \square \square\)\(\square \square \square\)
    The 12 houses will be in two groups of 6 (see diagram). Find the number of possible arrangements if all the houses in styles \(A\) and \(D\) are in the first group and all the houses in styles \(B , C\) and \(E\) are in the second group.
  3. Four of the 12 houses will be selected for a survey. Exactly one house must be in style \(B\) and exactly one house in style \(C\). Find the number of ways in which these four houses can be selected.
CAIE S1 2008 November Q5
8 marks Easy -1.2
5 The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.
115120158132125
104142160145104
162117109124134
  1. Draw a stem-and-leaf diagram to represent the data.
  2. Find the median and the quartiles.
  3. On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.
CAIE S1 2008 November Q6
10 marks Moderate -0.3
6 There are three sets of traffic lights on Karinne's journey to work. The independent probabilities that Karinne has to stop at the first, second and third set of lights are \(0.4,0.8\) and 0.3 respectively.
  1. Draw a tree diagram to show this information.
  2. Find the probability that Karinne has to stop at each of the first two sets of lights but does not have to stop at the third set.
  3. Find the probability that Karinne has to stop at exactly two of the three sets of lights.
  4. Find the probability that Karinne has to stop at the first set of lights, given that she has to stop at exactly two sets of lights.
CAIE S1 2008 November Q7
11 marks Moderate -0.3
7 A fair die has one face numbered 1, one face numbered 3, two faces numbered 5 and two faces numbered 6 .
  1. Find the probability of obtaining at least 7 odd numbers in 8 throws of the die. The die is thrown twice. Let \(X\) be the sum of the two scores. The following table shows the possible values of \(X\). \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Second throw}
    135566
    \cline { 2 - 8 }1246677
    3468899
    First56810101111
    throw56810101111
    67911111212
    67911111212
    \end{table}
  2. Draw up a table showing the probability distribution of \(X\).
  3. Calculate \(\mathrm { E } ( X )\).
  4. Find the probability that \(X\) is greater than \(\mathrm { E } ( X )\).
CAIE S1 2009 November Q1
5 marks Moderate -0.3
1 The mean number of defective batteries in packs of 20 is 1.6 . Use a binomial distribution to calculate the probability that a randomly chosen pack of 20 will have more than 2 defective batteries.
CAIE S1 2009 November Q2
6 marks Moderate -0.3
2 The probability distribution of the random variable \(X\) is shown in the following table.
\(x\)- 2- 10123
\(\mathrm { P } ( X = x )\)0.08\(p\)0.120.16\(q\)0.22
The mean of \(X\) is 1.05 .
  1. Write down two equations involving \(p\) and \(q\) and hence find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
CAIE S1 2009 November Q3
7 marks Moderate -0.3
3 The times for a certain car journey have a normal distribution with mean 100 minutes and standard deviation 7 minutes. Journey times are classified as follows: \begin{displayquote} 'short' (the shortest \(33 \%\) of times),
'long' (the longest \(33 \%\) of times),
'standard' (the remaining 34\% of times).
  1. Find the probability that a randomly chosen car journey takes between 85 and 100 minutes.
  2. Find the least and greatest times for 'standard' journeys. \end{displayquote}
CAIE S1 2009 November Q4
7 marks Easy -1.3
4 A library has many identical shelves. All the shelves are full and the numbers of books on each shelf in a certain section are summarised by the following stem-and-leaf diagram.
33699
467
50122
600112344444556667889
7113335667899
80245568
9001244445567788999
Key: 3 | 6 represents 36 books
  1. Find the number of shelves in this section of the library.
  2. Draw a box-and-whisker plot to represent the data. In another section all the shelves are full and the numbers of books on each shelf are summarised by the following stem-and-leaf diagram.
    212222334566679\(( 13 )\)
    301112334456677788\(( 15 )\)
    4223357789
    Key: 3 | 6 represents 36 books
  3. There are fewer books in this section than in the previous section. State one other difference between the books in this section and the books in the previous section.
CAIE S1 2009 November Q5
11 marks Moderate -0.8
5
  1. Find how many numbers between 5000 and 6000 can be formed from the digits 1, 2, 3, 4, 5 and 6
    1. if no digits are repeated,
    2. if repeated digits are allowed.
  2. Find the number of ways of choosing a school team of 5 pupils from 6 boys and 8 girls
    1. if there are more girls than boys in the team,
    2. if three of the boys are cousins and are either all in the team or all not in the team.
CAIE S1 2009 November Q6
14 marks Moderate -0.3
6 A box contains 4 pears and 7 oranges. Three fruits are taken out at random and eaten. Find the probability that
  1. 2 pears and 1 orange are eaten, in any order,
  2. the third fruit eaten is an orange,
  3. the first fruit eaten was a pear, given that the third fruit eaten is an orange. There are 121 similar boxes in a warehouse. One fruit is taken at random from each box.
  4. Using a suitable approximation, find the probability that fewer than 39 are pears.
CAIE S1 2009 November Q1
4 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{6f677fc6-3ca2-4a0d-82a2-69a7cbb8574d-2_211_1169_267_488} Measurements of wind speed on a certain island were taken over a period of one year. A box-andwhisker plot of the data obtained is displayed above, and the values of the quartiles are as shown. It is suggested that wind speed can be modelled approximately by a normal distribution with mean \(\mu \mathrm { km } \mathrm { h } ^ { - 1 }\) and standard deviation \(\sigma \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Estimate the value of \(\mu\).
  2. Estimate the value of \(\sigma\).
CAIE S1 2009 November Q2
4 marks Moderate -0.8
2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.
CAIE S1 2009 November Q3
6 marks Moderate -0.8
3 Maria chooses toast for her breakfast with probability 0.85 . If she does not choose toast then she has a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8 . If she has a bread roll then the probability that she will have jam on it is 0.4 .
  1. Draw a fully labelled tree diagram to show this information.
  2. Given that Maria did not have jam for breakfast, find the probability that she had toast.
CAIE S1 2009 November Q4
8 marks Moderate -0.8
4
    1. Find how many different four-digit numbers can be made using only the digits 1, 3, 5 and 6 with no digit being repeated.
    2. Find how many different odd numbers greater than 500 can be made using some or all of the digits \(1,3,5\) and 6 with no digit being repeated.
  2. Six cards numbered 1,2,3,4,5,6 are arranged randomly in a line. Find the probability that the cards numbered 4 and 5 are not next to each other.
CAIE S1 2009 November Q5
8 marks Moderate -0.3
5 In a particular discrete probability distribution the random variable \(X\) takes the value \(\frac { 120 } { r }\) with probability \(\frac { r } { 45 }\), where \(r\) takes all integer values from 1 to 9 inclusive.
  1. Show that \(\mathrm { P } ( X = 40 ) = \frac { 1 } { 15 }\).
  2. Construct the probability distribution table for \(X\).
  3. Which is the modal value of \(X\) ?
  4. Find the probability that \(X\) lies between 18 and 100 .
CAIE S1 2009 November Q6
9 marks Moderate -0.8
6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.
Marks\(1 - 20\)\(21 - 30\)\(31 - 40\)\(41 - 50\)\(51 - 60\)\(61 - 75\)
Frequency403456542921
  1. Draw a histogram on graph paper to represent these results.
  2. Calculate estimates of the mean mark and the standard deviation.
CAIE S1 2009 November Q7
11 marks Standard +0.3
7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.
  1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
  2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
  3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.
CAIE S1 2010 November Q1
3 marks Easy -1.2
1 Anita made observations of the maximum temperature, \(t ^ { \circ } \mathrm { C }\), on 50 days. Her results are summarised by \(\Sigma t = 910\) and \(\Sigma ( t - \bar { t } ) ^ { 2 } = 876\), where \(\bar { t }\) denotes the mean of the 50 observations. Calculate \(\bar { t }\) and the standard deviation of the observations.
CAIE S1 2010 November Q2
5 marks Moderate -0.3
2 On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a random sample of 200 apples, the number of apples which are underweight is more than 21 and less than 35.
CAIE S1 2010 November Q3
7 marks Moderate -0.3
3 The times taken by students to get up in the morning can be modelled by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.
  1. For a random sample of 350 students, find the number who would be expected to take longer than 20 minutes to get up in the morning.
  2. 'Very slow' students are students whose time to get up is more than 1.645 standard deviations above the mean. Find the probability that fewer than 3 students from a random sample of 8 students are 'very slow'.
CAIE S1 2010 November Q4
7 marks Moderate -0.8
4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table.
Weight (grams)\(1 - 10\)\(11 - 20\)\(21 - 25\)\(26 - 30\)\(31 - 50\)\(51 - 70\)
Frequency\(2 x\)\(4 x\)\(3 x\)\(5 x\)\(4 x\)\(x\)
A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .
  1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
  2. Calculate an estimate of the mean weight of the stones.
CAIE S1 2010 November Q5
8 marks Standard +0.3
5 Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, \(A , B , C\) and \(D\). Each friend chooses an entrance independently.
  • The probability that Rick chooses entrance \(A\) is \(\frac { 1 } { 3 }\). The probabilities that he chooses entrances \(B , C\) or \(D\) are all equal.
  • Brenda is equally likely to choose any of the four entrances.
  • The probability that Ali chooses entrance \(C\) is \(\frac { 2 } { 7 }\) and the probability that he chooses entrance \(D\) is \(\frac { 3 } { 5 }\). The probabilities that he chooses the other two entrances are equal.
    1. Find the probability that at least 2 friends will choose entrance \(B\).
    2. Find the probability that the three friends will all choose the same entrance.