Questions — CAIE S1 (789 questions)

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CAIE S1 2002 November Q2
6 marks Easy -1.2
2 Ivan throws three fair dice.
  1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
  2. Find the probability of Ivan obtaining a total score of 7.
CAIE S1 2002 November Q3
6 marks Moderate -0.8
3 The distance in metres that a ball can be thrown by pupils at a particular school follows a normal distribution with mean 35.0 m and standard deviation 11.6 m .
  1. Find the probability that a randomly chosen pupil can throw a ball between 30 and 40 m .
  2. The school gives a certificate to the \(10 \%\) of pupils who throw further than a certain distance. Find the least distance that must be thrown to qualify for a certificate.
CAIE S1 2002 November Q4
7 marks Moderate -0.3
4 In a certain hotel, the lock on the door to each room can be opened by inserting a key card. The key card can be inserted only one way round. The card has a pattern of holes punched in it. The card has 4 columns, and each column can have either 1 hole, 2 holes, 3 holes or 4 holes punched in it. Each column has 8 different positions for the holes. The diagram illustrates one particular key card with 3 holes punched in the first column, 3 in the second, 1 in the third and 2 in the fourth. \includegraphics[max width=\textwidth, alt={}, center]{2bcbd4d3-0d41-48fa-8f70-192b158c0bbe-2_410_214_1811_968}
  1. Show that the number of different ways in which a column could have exactly 2 holes is 28 .
  2. Find how many different patterns of holes can be punched in a column.
  3. How many different possible key cards are there?
CAIE S1 2002 November Q5
9 marks Standard +0.3
5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,
  1. find the conditional probability that Rachel wins the first game, given that she loses the second,
  2. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.
CAIE S1 2002 November Q6
9 marks Standard +0.3
6
  1. A manufacturer of biscuits produces 3 times as many cream ones as chocolate ones. Biscuits are chosen randomly and packed into boxes of 10 . Find the probability that a box contains equal numbers of cream biscuits and chocolate biscuits.
  2. A random sample of 8 boxes is taken. Find the probability that exactly 1 of them contains equal numbers of cream biscuits and chocolate biscuits.
  3. A large box of randomly chosen biscuits contains 120 biscuits. Using a suitable approximation, find the probability that it contains fewer than 35 chocolate biscuits.
CAIE S1 2002 November Q7
9 marks Moderate -0.8
7 The weights in kilograms of two groups of 17-year-old males from country \(P\) and country \(Q\) are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, ... \(4 | 7 | 1 \ldots\) denotes weights of 74 kg for a male in country \(P\) and 71 kg for a male in country \(Q\).
Country \(P\)Country \(Q\)
515
62348
9876471345677889
88665382367788
97765554290224
544311045
  1. Find the median and quartile weights for country \(Q\).
  2. You are given that the lower quartile, median and upper quartile for country \(P\) are 84,94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data.
  3. Make two comments on the weights of the two groups.
CAIE S1 2003 November Q1
4 marks Easy -1.8
1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.
CAIE S1 2003 November Q2
4 marks Easy -1.3
2 The floor areas, \(x \mathrm {~m} ^ { 2 }\), of 20 factories are as follows.
150350450578595644722798802904
1000133015331561177819602167233024333231
Represent these data by a histogram on graph paper, using intervals $$0 \leqslant x < 500,500 \leqslant x < 1000,1000 \leqslant x < 2000,2000 \leqslant x < 3000,3000 \leqslant x < 4000 .$$
CAIE S1 2003 November Q3
6 marks Standard +0.3
3 In a normal distribution, 69\% of the distribution is less than 28 and 90\% is less than 35. Find the mean and standard deviation of the distribution.
CAIE S1 2003 November Q4
6 marks Moderate -0.8
4 Single cards, chosen at random, are given away with bars of chocolate. Each card shows a picture of one of 20 different football players. Richard needs just one picture to complete his collection. He buys 5 bars of chocolate and looks at all the pictures. Find the probability that
  1. Richard does not complete his collection,
  2. he has the required picture exactly once,
  3. he completes his collection with the third picture he looks at.
CAIE S1 2003 November Q5
6 marks Moderate -0.8
5 In a certain country \(54 \%\) of the population is male. It is known that \(5 \%\) of the males are colour-blind and \(2 \%\) of the females are colour-blind. A person is chosen at random and found to be colour-blind. By drawing a tree diagram, or otherwise, find the probability that this person is male.
CAIE S1 2003 November Q6
8 marks Easy -1.2
6
  1. A collection of 18 books contains one Harry Potter book. Linda is going to choose 6 of these books to take on holiday.
    1. In how many ways can she choose 6 books?
    2. How many of these choices will include the Harry Potter book?
  2. In how many ways can 5 boys and 3 girls stand in a straight line
    1. if there are no restrictions,
    2. if the boys stand next to each other?
CAIE S1 2003 November Q7
8 marks Standard +0.3
7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.
  1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
  2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
  3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
    \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).
CAIE S1 2003 November Q8
8 marks Easy -1.3
8 A discrete random variable \(X\) has the following probability distribution.
\(x\)1234
\(\mathrm { P } ( X = x )\)\(3 c\)\(4 c\)\(5 c\)\(6 c\)
  1. Find the value of the constant \(c\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
CAIE S1 2004 November Q1
5 marks Moderate -0.3
1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.
  1. Find the number of different arrangements using all nine letters.
  2. How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?
CAIE S1 2004 November Q2
6 marks Easy -1.8
2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.
Length of car \(( x\) metres \()\)FrequencyFrequency density
\(2.80 \leqslant x < 3.00\)1785
\(3.00 \leqslant x < 3.10\)24240
\(3.10 \leqslant x < 3.20\)19190
\(3.20 \leqslant x < 3.40\)8\(a\)
  1. Find the value of \(a\).
  2. Draw a histogram on graph paper to represent the data.
  3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.
CAIE S1 2004 November Q3
6 marks Moderate -0.8
3 When Andrea needs a taxi, she rings one of three taxi companies, A, B or C. 50\% of her calls are to taxi company \(A , 30 \%\) to \(B\) and \(20 \%\) to \(C\). A taxi from company \(A\) arrives late \(4 \%\) of the time, a taxi from company \(B\) arrives late \(6 \%\) of the time and a taxi from company \(C\) arrives late \(17 \%\) of the time.
  1. Find the probability that, when Andrea rings for a taxi, it arrives late.
  2. Given that Andrea's taxi arrives late, find the conditional probability that she rang company \(B\).
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.