Questions — CAIE S1 (785 questions)

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CAIE S1 2002 November Q5
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.