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CAIE S1 2004 November Q4
7 marks Moderate -0.3
4 The ages, \(x\) years, of 18 people attending an evening class are summarised by the following totals: \(\Sigma x = 745 , \Sigma x ^ { 2 } = 33951\).
  1. Calculate the mean and standard deviation of the ages of this group of people.
  2. One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find the age of the person who left and the standard deviation of the ages of the remaining 17 people.
CAIE S1 2004 November Q5
7 marks Standard +0.8
5 The length of Paulo's lunch break follows a normal distribution with mean \(\mu\) minutes and standard deviation 5 minutes. On one day in four, on average, his lunch break lasts for more than 52 minutes.
  1. Find the value of \(\mu\).
  2. Find the probability that Paulo's lunch break lasts for between 40 and 46 minutes on every one of the next four days.
CAIE S1 2004 November Q6
9 marks Standard +0.3
6 A box contains five balls numbered \(1,2,3,4,5\). Three balls are drawn randomly at the same time from the box.
  1. By listing all possible outcomes (123, 124, etc.), find the probability that the sum of the three numbers drawn is an odd number. The random variable \(L\) denotes the largest of the three numbers drawn.
  2. Find the probability that \(L\) is 4 .
  3. Draw up a table to show the probability distribution of \(L\).
  4. Calculate the expectation and variance of \(L\).
CAIE S1 2004 November Q7
10 marks Moderate -0.8
7
  1. State two conditions which must be satisfied for a situation to be modelled by a binomial distribution. In a certain village 28\% of all cars are made by Ford.
  2. 14 cars are chosen randomly in this village. Find the probability that fewer than 4 of these cars are made by Ford.
  3. A random sample of 50 cars in the village is taken. Estimate, using a normal approximation, the probability that more than 18 cars are made by Ford.
CAIE S1 2005 November Q1
4 marks Easy -1.8
1 A study of the ages of car drivers in a certain country produced the results shown in the table. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Percentage of drivers in each age group}
YoungMiddle-agedElderly
Males403525
Females207010
\end{table} Illustrate these results diagrammatically.
CAIE S1 2005 November Q2
6 marks Standard +0.3
2 Boxes of sweets contain toffees and chocolates. Box \(A\) contains 6 toffees and 4 chocolates, box \(B\) contains 5 toffees and 3 chocolates, and box \(C\) contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are taken out, one after the other, and eaten.
  1. Find the probability that they are both toffees.
  2. Given that they are both toffees, find the probability that they both came from box \(A\).
CAIE S1 2005 November Q3
7 marks Moderate -0.8
3 A staff car park at a school has 13 parking spaces in a row. There are 9 cars to be parked.
  1. How many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?
  2. How many different arrangements are there if the 4 empty spaces are next to each other?
  3. If the parking is random, find the probability that there will not be 4 empty spaces next to each other.
CAIE S1 2005 November Q4
7 marks Moderate -0.3
4 A group of 10 married couples and 3 single men found that the mean age \(\bar { x } _ { w }\) of the 10 women was 41.2 years and the standard deviation of the women's ages was 15.1 years. For the 13 men, the mean age \(\bar { x } _ { m }\) was 46.3 years and the standard deviation was 12.7 years.
  1. Find the mean age of the whole group of 23 people.
  2. The individual women's ages are denoted by \(x _ { w }\) and the individual men's ages by \(x _ { m }\). By first finding \(\Sigma x _ { w } ^ { 2 }\) and \(\Sigma x _ { m } ^ { 2 }\), find the standard deviation for the whole group.
CAIE S1 2005 November Q5
8 marks Moderate -0.3
5 A box contains 300 discs of different colours. There are 100 pink discs, 100 blue discs and 100 orange discs. The discs of each colour are numbered from 0 to 99 . Five discs are selected at random, one at a time, with replacement. Find
  1. the probability that no orange discs are selected,
  2. the probability that exactly 2 discs with numbers ending in a 6 are selected,
  3. the probability that exactly 2 orange discs with numbers ending in a 6 are selected,
  4. the mean and variance of the number of pink discs selected.
CAIE S1 2005 November Q6
8 marks Standard +0.3
6 In a competition, people pay \(\\) 1\( to throw a ball at a target. If they hit the target on the first throw they receive \)\\( 5\). If they hit it on the second or third throw they receive \(\\) 3\(, and if they hit it on the fourth or fifth throw they receive \)\\( 1\). People stop throwing after the first hit, or after 5 throws if no hit is made. Mario has a constant probability of \(\frac { 1 } { 5 }\) of hitting the target on any throw, independently of the results of other throws.
  1. Mario misses with his first and second throws and hits the target with his third throw. State how much profit he has made.
  2. Show that the probability that Mario's profit is \(\\) 0$ is 0.184 , correct to 3 significant figures.
  3. Draw up a probability distribution table for Mario's profit.
  4. Calculate his expected profit.
CAIE S1 2005 November Q7
10 marks Standard +0.3
7 In tests on a new type of light bulb it was found that the time they lasted followed a normal distribution with standard deviation 40.6 hours. 10\% lasted longer than 5130 hours.
  1. Find the mean lifetime, giving your answer to the nearest hour.
  2. Find the probability that a light bulb fails to last for 5000 hours.
  3. A hospital buys 600 of these light bulbs. Using a suitable approximation, find the probability that fewer than 65 light bulbs will last longer than 5130 hours.
CAIE S1 2006 November Q1
4 marks Easy -1.8
1 The weights of 30 children in a class, to the nearest kilogram, were as follows.
50456153554752494651
60525447575942465153
56485051445249585545
Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 41.5 kg and the fifth class having an upper boundary of 61.5 kg .
CAIE S1 2006 November Q2
5 marks Easy -1.3
2 The discrete random variable \(X\) has the following probability distribution.
\(x\)01234
\(\mathrm { P } ( X = x )\)0.26\(q\)\(3 q\)0.050.09
  1. Find the value of \(q\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2006 November Q3
6 marks Easy -1.3
3 In a survey, people were asked how long they took to travel to and from work, on average. The median time was 3 hours 36 minutes, the upper quartile was 4 hours 42 minutes and the interquartile range was 3 hours 48 minutes. The longest time taken was 5 hours 12 minutes and the shortest time was 30 minutes.
  1. Find the lower quartile.
  2. Represent the information by a box-and-whisker plot, using a scale of 2 cm to represent 60 minutes.
CAIE S1 2006 November Q4
7 marks Moderate -0.8
4 Two fair dice are thrown.
  1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
  2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
  3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.
CAIE S1 2006 November Q5
8 marks Moderate -0.3
5
  1. Give an example of a variable in real life which could be modelled by a normal distribution.
  2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 21.0. Given that \(\mathrm { P } ( X > 10.0 ) = 0.7389\), find the value of \(\mu\).
  3. If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.
CAIE S1 2006 November Q6
9 marks Moderate -0.8
6 Six men and three women are standing in a supermarket queue.
  1. How many possible arrangements are there if there are no restrictions on order?
  2. How many possible arrangements are there if no two of the women are standing next to each other?
  3. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?
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 \(X\).