Questions — CAIE (7646 questions)

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CAIE S1 2006 November Q7
11 marks Standard +0.3
7 A manufacturer makes two sizes of elastic bands: large and small. \(40 \%\) of the bands produced are large bands and \(60 \%\) are small bands. Assuming that each pack of these elastic bands contains a random selection, calculate the probability that, in a pack containing 20 bands, there are
  1. equal numbers of large and small bands,
  2. more than 17 small bands. An office pack contains 150 elastic bands.
  3. Using a suitable approximation, calculate the probability that the number of small bands in the office pack is between 88 and 97 inclusive.
CAIE S1 2007 November Q1
4 marks Moderate -0.8
1 A summary of 24 observations of \(x\) gave the following information: $$\Sigma ( x - a ) = - 73.2 \quad \text { and } \quad \Sigma ( x - a ) ^ { 2 } = 2115 .$$ The mean of these values of \(x\) is 8.95 .
  1. Find the value of the constant \(a\).
  2. Find the standard deviation of these values of \(x\).
CAIE S1 2007 November Q2
6 marks Easy -1.3
2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2007 November Q3
6 marks Moderate -0.8
3 The six digits 4, 5, 6, 7, 7, 7 can be arranged to give many different 6-digit numbers.
  1. How many different 6-digit numbers can be made?
  2. How many of these 6-digit numbers start with an odd digit and end with an odd digit?
CAIE S1 2007 November Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{1a10471c-5810-44ca-9353-c2c76e190a2b-2_542_876_1425_632} The random variable \(X\) has a normal distribution with mean 4.5. It is given that \(\mathrm { P } ( X > 5.5 ) = 0.0465\) (see diagram).
  1. Find the standard deviation of \(X\).
  2. Find the probability that a random observation of \(X\) lies between 3.8 and 4.8.
CAIE S1 2007 November Q5
8 marks Easy -1.8
5 The arrival times of 204 trains were noted and the number of minutes, \(t\), that each train was late was recorded. The results are summarised in the table.
Number of minutes late \(( t )\)\(- 2 \leqslant t < 0\)\(0 \leqslant t < 2\)\(2 \leqslant t < 4\)\(4 \leqslant t < 6\)\(6 \leqslant t < 10\)
Number of trains4351692219
  1. Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
  2. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.
CAIE S1 2007 November Q6
9 marks Moderate -0.8
6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.
  1. Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .
  2. On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
  3. On another day she performs the routine \(n\) times. Find the smallest value of \(n\) for which the expected number of correct performances is at least 8 .
CAIE S1 2007 November Q7
10 marks Moderate -0.3
7 Box \(A\) contains 5 red paper clips and 1 white paper clip. Box \(B\) contains 7 red paper clips and 2 white paper clips. One paper clip is taken at random from box \(A\) and transferred to box \(B\). One paper clip is then taken at random from box \(B\).
  1. Find the probability of taking both a white paper clip from box \(A\) and a red paper clip from box \(B\).
  2. Find the probability that the paper clip taken from box \(B\) is red.
  3. Find the probability that the paper clip taken from box \(A\) was red, given that the paper clip taken from box \(B\) is red.
  4. The random variable \(X\) denotes the number of times that a red paper clip is taken. Draw up a table to show the probability distribution of \(X\).
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.