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CAIE S1 2005 June Q3
7 marks Moderate -0.8
3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable \(X\) is the number of dice that fall on a green face.
  1. Show that the probability of 4 dice landing on a green face is 0.0146 , correct to 4 decimal places.
  2. Draw up a table for the probability distribution of \(X\), giving your answers correct to 4 decimal places.
CAIE S1 2005 June Q4
8 marks Easy -1.2
4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves.
People who exercisePeople who do not exercise
(9)98764322131577(4)
(12)9888766533224234458(6)
(9)87776533151222344567889(13)
(7)6666432612333455577899(14)
(3)8417245566788(9)
(4)95528133467999(9)
(1)4914558(5)
(0)10336(3)
Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.
  1. Give one useful feature of a stem-and-leaf diagram.
  2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
  3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.
CAIE S1 2005 June Q5
8 marks Easy -1.3
5 Data about employment for males and females in a small rural area are shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}UnemployedEmployed
Male206412
Female358305
A person from this area is chosen at random. Let \(M\) be the event that the person is male and let \(E\) be the event that the person is employed.
  1. Find \(\mathrm { P } ( M )\).
  2. Find \(\mathrm { P } ( M\) and \(E )\).
  3. Are \(M\) and \(E\) independent events? Justify your answer.
  4. Given that the person chosen is unemployed, find the probability that the person is female.
CAIE S1 2005 June Q6
8 marks Moderate -0.3
6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.
  1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
  2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.
CAIE S1 2005 June Q7
8 marks Moderate -0.8
7
  1. A football team consists of 3 players who play in a defence position, 3 players who play in a midfield position and 5 players who play in a forward position. Three players are chosen to collect a gold medal for the team. Find in how many ways this can be done
    1. if the captain, who is a midfield player, must be included, together with one defence and one forward player,
    2. if exactly one forward player must be included, together with any two others.
  2. Find how many different arrangements there are of the nine letters in the words GOLD MEDAL
    1. if there are no restrictions on the order of the letters,
    2. if the two letters D come first and the two letters L come last.
CAIE S1 2006 June Q1
3 marks Easy -1.2
1 The salaries, in thousands of dollars, of 11 people, chosen at random in a certain office, were found to be: $$40 , \quad 42 , \quad 45 , \quad 41 , \quad 352 , \quad 40 , \quad 50 , \quad 48 , \quad 51 , \quad 49 , \quad 47 .$$ Choose and calculate an appropriate measure of central tendency (mean, mode or median) to summarise these salaries. Explain briefly why the other measures are not suitable.
CAIE S1 2006 June Q2
6 marks Moderate -0.8
2 The probability that Henk goes swimming on any day is 0.2 . On a day when he goes swimming, the probability that Henk has burgers for supper is 0.75 . On a day when he does not go swimming the probability that he has burgers for supper is \(x\). This information is shown on the following tree diagram. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-2_693_1038_845_555} The probability that Henk has burgers for supper on any day is 0.5 .
  1. Find \(x\).
  2. Given that Henk has burgers for supper, find the probability that he went swimming that day.
CAIE S1 2006 June Q3
8 marks Standard +0.3
3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that \(5 \%\) of the fish are longer than 50 cm .
  1. Find the standard deviation.
  2. When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.
  3. 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .
CAIE S1 2006 June Q4
8 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733} The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.
  1. How many possible seating arrangements are there for the 11 passengers?
  2. How many possible seating arrangements are there if 5 particular people sit in the back row? Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.
  3. In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?
CAIE S1 2006 June Q5
7 marks Moderate -0.8
5 Each father in a random sample of fathers was asked how old he was when his first child was born. The following histogram represents the information. \includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_789_1627_1468_260}
  1. What is the modal age group?
  2. How many fathers were between 25 and 30 years old when their first child was born?
  3. How many fathers were in the sample?
  4. Find the probability that a father, chosen at random from the group, was between 25 and 30 years old when his first child was born, given that he was older than 25 years. 632 teams enter for a knockout competition, in which each match results in one team winning and the other team losing. After each match the winning team goes on to the next round, and the losing team takes no further part in the competition. Thus 16 teams play in the second round, 8 teams play in the third round, and so on, until 2 teams play in the final round.
CAIE S1 2006 June Q7
9 marks Standard +0.3
7 A survey of adults in a certain large town found that \(76 \%\) of people wore a watch on their left wrist, \(15 \%\) wore a watch on their right wrist and \(9 \%\) did not wear a watch.
  1. A random sample of 14 adults was taken. Find the probability that more than 2 adults did not wear a watch.
  2. A random sample of 200 adults was taken. Using a suitable approximation, find the probability that more than 155 wore a watch on their left wrist.
CAIE S1 2007 June Q1
4 marks Easy -1.2
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
6 marks Moderate -0.8
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
7 marks Standard +0.3
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
7 marks Easy -1.3
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
7 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
4 marks Easy -1.2
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
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
8 marks Moderate -0.8
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
9 marks Moderate -0.8
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
11 marks Standard +0.3
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.