Questions — CAIE S1 (785 questions)

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CAIE S1 2007 June Q1
1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.
CAIE S1 2007 June Q2
2 Jamie is equally likely to attend or not to attend a training session before a football match. If he attends, he is certain to be chosen for the team which plays in the match. If he does not attend, there is a probability of 0.6 that he is chosen for the team.
  1. Find the probability that Jamie is chosen for the team.
  2. Find the conditional probability that Jamie attended the training session, given that he was chosen for the team.
CAIE S1 2007 June Q3
3
  1. The random variable \(X\) is normally distributed. The mean is twice the standard deviation. It is given that \(\mathrm { P } ( X > 5.2 ) = 0.9\). Find the standard deviation.
  2. A normal distribution has mean \(\mu\) and standard deviation \(\sigma\). If 800 observations are taken from this distribution, how many would you expect to be between \(\mu - \sigma\) and \(\mu + \sigma\) ?
CAIE S1 2007 June Q4
4 The lengths of time in minutes to swim a certain distance by the members of a class of twelve 9 -year-olds and by the members of a class of eight 16 -year-olds are shown below.
9-year-olds:13.016.116.014.415.915.114.213.716.716.415.013.2
16-year-olds:14.813.011.411.716.513.712.812.9
  1. Draw a back-to-back stem-and-leaf diagram to represent the information above.
  2. A new pupil joined the 16 -year-old class and swam the distance. The mean time for the class of nine pupils was now 13.6 minutes. Find the new pupil's time to swim the distance.
CAIE S1 2007 June Q5
5
  1. Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
    (a) if there are no restrictions,
    (b) if the Rs must all be together.
  2. How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?
CAIE S1 2007 June Q6
6 The probability that New Year's Day is on a Saturday in a randomly chosen year is \(\frac { 1 } { 7 }\).
  1. 15 years are chosen randomly. Find the probability that at least 3 of these years have New Year's Day on a Saturday.
  2. 56 years are chosen randomly. Use a suitable approximation to find the probability that more than 7 of these years have New Year's Day on a Saturday.
CAIE S1 2007 June Q7
7 A vegetable basket contains 12 peppers, of which 3 are red, 4 are green and 5 are yellow. Three peppers are taken, at random and without replacement, from the basket.
  1. Find the probability that the three peppers are all different colours.
  2. Show that the probability that exactly 2 of the peppers taken are green is \(\frac { 12 } { 55 }\).
  3. The number of green peppers taken is denoted by the discrete random variable \(X\). Draw up a probability distribution table for \(X\).
CAIE S1 2008 June Q1
1 The stem-and-leaf diagram below represents data collected for the number of hits on an internet site on each day in March 2007. There is one missing value, denoted by \(x\).
00156
1135668
2112344489
31222\(x\)89
425679
Key: 1 | 5 represents 15 hits
  1. Find the median and lower quartile for the number of hits each day.
  2. The interquartile range is 19 . Find the value of \(x\).
CAIE S1 2008 June Q2
2 In country \(A 30 \%\) of people who drink tea have sugar in it. In country \(B 65 \%\) of people who drink tea have sugar in it. There are 3 million people in country \(A\) who drink tea and 12 million people in country \(B\) who drink tea. A person is chosen at random from these 15 million people.
  1. Find the probability that the person chosen is from country \(A\).
  2. Find the probability that the person chosen does not have sugar in their tea.
  3. Given that the person chosen does not have sugar in their tea, find the probability that the person is from country \(B\).
CAIE S1 2008 June Q3
3 Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.
  1. How many different arrangements of all 11 CDs on a shelf are there if the jazz CDs are all next to each other?
  2. Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different possible selections can be made?
CAIE S1 2008 June Q4
4 In a certain country the time taken for a common infection to clear up is normally distributed with mean \(\mu\) days and standard deviation 2.6 days. \(25 \%\) of these infections clear up in less than 7 days.
  1. Find the value of \(\mu\). In another country the standard deviation of the time taken for the infection to clear up is the same as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
  2. Find the probability that, in a randomly chosen case from this country, the infection takes longer than 6.2 days to clear up.
CAIE S1 2008 June Q5
5 As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0.1 hour. The results are displayed in the following table.
Time spent \(( t\) hours \()\)\(0.1 \leqslant t \leqslant 0.5\)\(0.6 \leqslant t \leqslant 1.0\)\(1.1 \leqslant t \leqslant 2.0\)\(2.1 \leqslant t \leqslant 3.0\)\(3.1 \leqslant t \leqslant 4.5\)
Frequency1115183021
  1. Draw, on graph paper, a histogram to illustrate this information.
  2. Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.
CAIE S1 2008 June Q6
6 Every day Eduardo tries to phone his friend. Every time he phones there is a \(50 \%\) chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.
  1. Draw a tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)
  3. Calculate the expected number of unanswered phone calls on a day.
CAIE S1 2008 June Q7
7 A die is biased so that the probability of throwing a 5 is 0.75 and the probabilities of throwing a 1,2 , 3 , 4 or 6 are all equal.
  1. The die is thrown three times. Find the probability that the result is a 1 followed by a 5 followed by any even number.
  2. Find the probability that, out of 10 throws of this die, at least 8 throws result in a 5 .
  3. The die is thrown 90 times. Using an appropriate approximation, find the probability that a 5 is thrown more than 60 times.
CAIE S1 2009 June Q1
1 The volume of milk in millilitres in cartons is normally distributed with mean \(\mu\) and standard deviation 8. Measurements were taken of the volume in 900 of these cartons and it was found that 225 of them contained more than 1002 millilitres.
  1. Calculate the value of \(\mu\).
  2. Three of these 900 cartons are chosen at random. Calculate the probability that exactly 2 of them contain more than 1002 millilitres.
CAIE S1 2009 June Q2
2 Gohan throws a fair tetrahedral die with faces numbered \(1,2,3,4\). If she throws an even number then her score is the number thrown. If she throws an odd number then she throws again and her score is the sum of both numbers thrown. Let the random variable \(X\) denote Gohan's score.
  1. Show that \(\mathrm { P } ( X = 2 ) = \frac { 5 } { 16 }\).
  2. The table below shows the probability distribution of \(X\).
    \(x\)234567
    \(\mathrm { P } ( X = x )\)\(\frac { 5 } { 16 }\)\(\frac { 1 } { 16 }\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 16 }\)
    Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2009 June Q3
3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.
  1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
  2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.
CAIE S1 2009 June Q4
4 A choir consists of 13 sopranos, 12 altos, 6 tenors and 7 basses. A group consisting of 10 sopranos, 9 altos, 4 tenors and 4 basses is to be chosen from the choir.
  1. In how many different ways can the group be chosen?
  2. In how many ways can the 10 chosen sopranos be arranged in a line if the 6 tallest stand next to each other?
  3. The 4 tenors and 4 basses in the group stand in a single line with all the tenors next to each other and all the basses next to each other. How many possible arrangements are there if three of the tenors refuse to stand next to any of the basses?
CAIE S1 2009 June Q5
5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.
  1. Find the probabilities that he goes on each of the three rides. The probabilities that Ravi is frightened on each of the rides are as follows: $$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
  2. Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened. Ravi goes on a ride.
  3. Find the probability that he is frightened.
  4. Given that Ravi is not frightened, find the probability that he went on the camel ride.
CAIE S1 2009 June Q6
6 During January the numbers of people entering a store during the first hour after opening were as follows.
Time after opening,
\(x\) minutes
Frequency
Cumulative
frequency
\(0 < x \leqslant 10\)210210
\(10 < x \leqslant 20\)134344
\(20 < x \leqslant 30\)78422
\(30 < x \leqslant 40\)72\(a\)
\(40 < x \leqslant 60\)\(b\)540
  1. Find the values of \(a\) and \(b\).
  2. Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.
  3. Use your graph to estimate the median time after opening that people entered the store.
  4. Calculate estimates of the mean, \(m\) minutes, and standard deviation, \(s\) minutes, of the time after opening that people entered the store.
  5. Use your graph to estimate the number of people entering the store between ( \(m - \frac { 1 } { 2 } s\) ) and \(\left( m + \frac { 1 } { 2 } s \right)\) minutes after opening.
CAIE S1 2010 June Q1
1 The probability distribution of the discrete random variable \(X\) is shown in the table below.
\(x\)- 3- 104
\(\mathrm { P } ( X = x )\)\(a\)\(b\)0.150.4
Given that \(\mathrm { E } ( X ) = 0.75\), find the values of \(a\) and \(b\).
CAIE S1 2010 June Q2
2 The numbers of people travelling on a certain bus at different times of the day are as follows.
17522316318
22142535172712
623192123826
  1. Draw a stem-and-leaf diagram to illustrate the information given above.
  2. Find the median, the lower quartile, the upper quartile and the interquartile range.
  3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.
CAIE S1 2010 June Q3
3 The random variable \(X\) is the length of time in minutes that Jannon takes to mend a bicycle puncture. \(X\) has a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is given that \(\mathrm { P } ( X > 30.0 ) = 0.1480\) and \(\mathrm { P } ( X > 20.9 ) = 0.6228\). Find \(\mu\) and \(\sigma\).
CAIE S1 2010 June Q4
2 marks
4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.
Roller
coaster
Water
slide
Revolving
drum
Fei420
Graeme136
  1. The mean cost of Fei's rides is \(
    ) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(
    ) 2.50\( per ride. [2]
  2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.
CAIE S1 2010 June Q5
5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.
  1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
  2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
  3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.