Questions — CAIE (7646 questions)

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CAIE S1 2015 November Q5
9 marks Easy -1.3
5 The weights, in kilograms, of the 15 rugby players in each of two teams, \(A\) and \(B\), are shown below.
Team \(A\)9798104841001091159912282116968410791
Team \(B\)75799410196771111088384861158211395
  1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team \(A\) on the lefthand side of the diagram and team \(B\) on the right-hand side.
  2. Find the interquartile range of the weights of the players in team \(A\).
  3. A new player joins team \(B\) as a substitute. The mean weight of the 16 players in team \(B\) is now 93.9 kg . Find the weight of the new player.
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.
  1. Copy and complete the table showing the possible values of \(X\).
    Spinner \(A\)
    \cline { 2 - 6 }1233
    Spinner \(B\)- 2
    - 21
    - 1
    1
  2. Draw up a table showing the probability distribution of \(X\).
  3. Find \(\operatorname { Var } ( X )\).
  4. Find the probability that \(X\) is even, given that \(X\) is positive.
CAIE S1 2015 November Q7
13 marks Standard +0.3
7
  1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
    1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
    2. Find the value of \(m\).
    3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
  2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).
CAIE S1 2015 November Q1
3 marks Easy -1.2
1 The time taken, \(t\) hours, to deliver letters on a particular route each day is measured on 250 working days. The mean time taken is 2.8 hours. Given that \(\Sigma ( t - 2.5 ) ^ { 2 } = 96.1\), find the standard deviation of the times taken.
CAIE S1 2015 November Q2
5 marks Moderate -0.8
2 In country \(X , 25 \%\) of people have fair hair. In country \(Y , 60 \%\) of people have fair hair. There are 20 million people in country \(X\) and 8 million people in country \(Y\). A person is chosen at random from these 28 million people.
  1. Find the probability that the person chosen is from country \(X\).
  2. Find the probability that the person chosen has fair hair.
  3. Find the probability that the person chosen is from country \(X\), given that the person has fair hair.
CAIE S1 2015 November Q3
6 marks Moderate -0.3
3 Ellie throws two fair tetrahedral dice, each with faces numbered 1, 2, 3 and 4. She notes the numbers on the faces that the dice land on. Event \(S\) is 'the sum of the two numbers is 4 '. Event \(T\) is 'the product of the two numbers is an odd number'.
  1. Determine whether events \(S\) and \(T\) are independent, showing your working.
  2. Are events \(S\) and \(T\) exclusive? Justify your answer.
CAIE S1 2015 November Q4
7 marks Standard +0.3
4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation \(\sigma\) hours.
  1. It is found that \(13 \%\) of seeds take longer than 136 hours to germinate. Find the value of \(\sigma\).
  2. 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.
CAIE S1 2015 November Q5
8 marks Moderate -0.3
5
  1. Find the number of different ways that the 13 letters of the word ACCOMMODATION can be arranged in a line if all the vowels (A, I, O) are next to each other.
  2. There are 7 Chinese, 6 European and 4 American students at an international conference. Four of the students are to be chosen to take part in a television broadcast. Find the number of different ways the students can be chosen if at least one Chinese and at least one European student are included.
CAIE S1 2015 November Q6
9 marks Moderate -0.8
6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.
Height (m)\(21 - 40\)\(41 - 45\)\(46 - 50\)\(51 - 60\)\(61 - 80\)
Frequency1815215228
  1. Draw a histogram on graph paper to illustrate the data.
  2. Calculate estimates of the mean and standard deviation of these heights.
CAIE S1 2015 November Q7
12 marks Standard +0.3
7 A factory makes water pistols, \(8 \%\) of which do not work properly.
  1. A random sample of 19 water pistols is taken. Find the probability that at most 2 do not work properly.
  2. In a random sample of \(n\) water pistols, the probability that at least one does not work properly is greater than 0.9 . Find the smallest possible value of \(n\).
  3. A random sample of 1800 water pistols is taken. Use an approximation to find the probability that there are at least 152 that do not work properly.
  4. Justify the use of your approximation in part (iii).
CAIE S1 2016 November Q1
3 marks Moderate -0.8
1 The random variable \(X\) is such that \(X \sim \mathrm {~N} ( 20,49 )\). Given that \(\mathrm { P } ( X > k ) = 0.25\), find the value of \(k\).
CAIE S1 2016 November Q2
5 marks Moderate -0.8
2 Two fair six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 are thrown and the two scores are noted. The difference between the two scores is defined as follows.
  • If the scores are equal the difference is zero.
  • If the scores are not equal the difference is the larger score minus the smaller score.
Find the expectation of the difference between the two scores.
CAIE S1 2016 November Q3
6 marks Moderate -0.8
3 Visitors to a Wildlife Park in Africa have independent probabilities of 0.9 of seeing giraffes, 0.95 of seeing elephants, 0.85 of seeing zebras and 0.1 of seeing lions.
  1. Find the probability that a visitor to the Wildlife Park sees all these animals.
  2. Find the probability that, out of 12 randomly chosen visitors, fewer than 3 see lions.
  3. 50 people independently visit the Wildlife Park. Find the mean and variance of the number of these people who see zebras.
CAIE S1 2016 November Q4
8 marks Standard +0.3
4 Packets of rice are filled by a machine and have weights which are normally distributed with mean 1.04 kg and standard deviation 0.017 kg .
  1. Find the probability that a randomly chosen packet weighs less than 1 kg .
  2. How many packets of rice, on average, would the machine fill from 1000 kg of rice? The factory manager wants to produce more packets of rice. He changes the settings on the machine so that the standard deviation is the same but the mean is reduced to \(\mu \mathrm { kg }\). With this mean the probability that a packet weighs less than 1 kg is 0.0388 .
  3. Find the value of \(\mu\).
  4. How many packets of rice, on average, would the machine now fill from 1000 kg of rice?
CAIE S1 2016 November Q5
9 marks Standard +0.8
5
  1. Find the number of different ways of arranging all nine letters of the word PINEAPPLE if no vowel (A, E, I) is next to another vowel.
  2. A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 allrounders and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicket-keeper.
    1. Find the number of different teams the manager can choose.
    2. Find the number of different teams the manager can choose if one particular batsman refuses to be in the team when one particular bowler is in the team.
CAIE S1 2016 November Q6
9 marks Standard +0.3
6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.
  1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
  2. Find \(\mathrm { P } ( X = 1 )\).
  3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.
CAIE S1 2016 November Q7
10 marks Easy -1.3
7 The masses, in grams, of components made in factory \(A\) and components made in factory \(B\) are shown below.
Factory \(A\)0.0490.0500.0530.0540.0570.0580.058
0.0590.0610.0610.0610.0630.065
Factory \(B\)0.0310.0560.0490.0440.0380.0480.051
0.0640.0350.0420.0470.0540.058
  1. Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the two factories.
  2. Find the median and the interquartile range for the masses of components made in factory \(B\).
  3. Make two comparisons between the masses of components made in factory \(A\) and the masses of those made in factory \(B\).
CAIE S1 2016 November Q1
5 marks Moderate -0.8
1 When Anya goes to school, the probability that she walks is 0.3 and the probability that she cycles is 0.65 ; if she does not walk or cycle she takes the bus. When Anya walks the probability that she is late is 0.15 . When she cycles the probability that she is late is 0.1 and when she takes the bus the probability that she is late is 0.6 . Given that Anya is late, find the probability that she cycles.
CAIE S1 2016 November Q2
7 marks Moderate -0.8
2 Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses 3 items at random. The random variable \(X\) is the number of T-shirts chosen.
  1. Show that the probability that Noor chooses exactly one T-shirt is \(\frac { 27 } { 55 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2016 November Q3
8 marks Standard +0.3
3 On any day at noon, the probabilities that Kersley is asleep or studying are 0.2 and 0.6 respectively.
  1. Find the probability that, in any 7-day period, Kersley is either asleep or studying at noon on at least 6 days.
  2. Use an approximation to find the probability that, in any period of 100 days, Kersley is asleep at noon on at most 30 days.
CAIE S1 2016 November Q4
9 marks Standard +0.3
4 The time taken to cook an egg by people living in a certain town has a normal distribution with mean 4.2 minutes and standard deviation 0.6 minutes.
  1. Find the probability that a person chosen at random takes between 3.5 and 4.5 minutes to cook an egg. \(12 \%\) of people take more than \(t\) minutes to cook an egg.
  2. Find the value of \(t\).
  3. A random sample of \(n\) people is taken. Find the smallest possible value of \(n\) if the probability that none of these people takes more than \(t\) minutes to cook an egg is less than 0.003 .
CAIE S1 2016 November Q5
9 marks Moderate -0.8
5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.
Capacity\(3000 - 7000\)\(8000 - 12000\)\(13000 - 22000\)\(23000 - 42000\)\(43000 - 82000\)
Number of stadiums403018348
  1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.
  2. Calculate an estimate of the mean capacity of these 130 stadiums.
  3. Find which class in the table contains the median and which contains the lower quartile.
CAIE S1 2016 November Q6
12 marks Standard +0.3
6 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that
  1. the vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { O }\) ) are together and the consonants ( \(\mathrm { C } , \mathrm { G } , \mathrm { H } , \mathrm { N } , \mathrm { P }\) ) are together, [3]
  2. the Es are not next to each other. Four letters are selected from the 10 letters of the word COPENHAGEN.
  3. Find the number of different selections if the four letters must contain the same number of Es and Ns with at least one of each.
CAIE S1 2016 November Q1
3 marks Moderate -0.3
1 A committee of 5 people is to be chosen from 4 men and 6 women. William is one of the 4 men and Mary is one of the 6 women. Find the number of different committees that can be chosen if William and Mary refuse to be on the committee together.
CAIE S1 2016 November Q2
6 marks Standard +0.3
2 A fair triangular spinner has three sides numbered 1, 2, 3. When the spinner is spun, the score is the number of the side on which it lands. The spinner is spun four times.
  1. Find the probability that at least two of the scores are 3 .
  2. Find the probability that the sum of the four scores is 5 .