Questions S1 (1967 questions)

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CAIE S1 2024 November Q2
2
  1. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
  2. Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As.
    \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}
CAIE S1 2024 November Q3
3 A fair coin and an ordinary fair six-sided dice are thrown at the same time.The random variable \(X\) is defined as follows.
-If the coin shows a tail,\(X\) is twice the score on the dice.
-If the coin shows a head,\(X\) is the score on the dice if the score is even and \(X\) is 0 otherwise.
  1. Draw up the probability distribution table for \(X\) .
  2. Find \(\operatorname { Var } ( X )\) .
CAIE S1 2024 November Q4
4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.
  1. How many of these trees would you expect to have height less than 18.2 metres?
    The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
  2. Find the value of \(\sigma\).
CAIE S1 2024 November Q5
5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.
  1. In how many ways can the group of 7 be chosen?
    On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
  2. In how many ways can the group of 5 be chosen?
CAIE S1 2024 November Q6
6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.
Falcons383942444648505152565859646976
Kites324040454748525458595960616365
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
  2. Find the median and the interquartile range of the times for the Falcons.
    Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
  3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.
CAIE S1 2024 November Q7
7 In a game,players attempt to score a goal by kicking a ball into a net.The probability that Leno scores a goal is 0.4 on any attempt,independently of all other attempts.The random variable \(X\) denotes the number of attempts that it takes Leno to score a goal.
  1. Find \(\mathrm { P } ( X = 5 )\) .
    ............................................................................................................................................
  2. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\) .
  3. Find the probability that Leno scores his second goal on or before his 5th attempt.
    \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-10_2715_33_106_2017}
    \includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-11_2723_33_99_22} Leno has 75 attempts to score a goal.
  4. Use a suitable approximation to find the probability that Leno scores more than 28 goals but fewer than 35 goals.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S1 2024 November Q2
2 A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled \(1,1,2,2,3,3\). Both dice are thrown. The random variable \(X\) is the product of the scores on the two dice.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2024 November Q3
3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .
  1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
    60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
  2. Find the value of \(k\), giving your answer correct to 1 decimal place.
CAIE S1 2024 November Q4
4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.
Height
\(( \mathrm { cm } )\)
\(10 - 19\)\(20 - 29\)\(30 - 39\)\(40 - 44\)\(45 - 49\)\(50 - 54\)\(55 - 59\)
Frequency1018324228146
  1. Draw a cumulative frequency graph to illustrate the data.
    \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
  2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants.
    \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
  3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.
CAIE S1 2024 November Q5
5 A factory produces chocolates. 30\% of the chocolates are wrapped in gold foil, 25\% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.
  1. Find the probability that she obtains no more than 2 chocolates that are wrapped in red foil.
    Jake chooses chocolates one at a time at random from the production line.
  2. Find the probability that the first time he obtains a chocolate that is wrapped in red foil is before the 7th choice.
    \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-08_2720_35_106_2015}
    \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-09_2717_29_105_22} Keifa chooses chocolates one at a time at random from the production line.
  3. Find the probability that the second chocolate chosen is the first one wrapped in gold foil given that the fifth chocolate chosen is the first unwrapped chocolate.
CAIE S1 2024 November Q6
6
  1. Find the number of different arrangements of the 9 letters in the word HAPPINESS.
  2. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the first and last letters are not the same as each other.
    \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-10_2715_35_110_2012}
  3. Find the number of different arrangements of the 9 letters in the word HAPPINESS in which the two Ps are together and there are exactly two letters between the two Ss.
    The 9 letters in the word HAPPINESS are divided at random into a group of 5 and a group of 4 .
  4. Find the probability that both Ps are in one group and both Ss are in the other group.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE S1 2002 June Q1
1 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.8\) and \(\mathrm { P } ( A\) and \(B\) )=0.4. State, giving a reason in each case, whether events \(A\) and \(B\) are
  1. independent,
  2. mutually exclusive.
CAIE S1 2002 June Q2
2 The manager of a company noted the times spent in 80 meetings. The results were as follows.
Time ( \(t\) minutes)\(0 < t \leqslant 15\)\(15 < t \leqslant 30\)\(30 < t \leqslant 60\)\(60 < t \leqslant 90\)\(90 < t \leqslant 120\)
Number of meetings4724387
Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range.
CAIE S1 2002 June Q3
3 A fair cubical die with faces numbered \(1,1,1,2,3,4\) is thrown and the score noted. The area \(A\) of a square of side equal to the score is calculated, so, for example, when the score on the die is 3 , the value of \(A\) is 9 .
  1. Draw up a table to show the probability distribution of \(A\).
  2. Find \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
  3. In a spot check of the speeds \(x \mathrm {~km} \mathrm {~h} ^ { - 1 }\) of 30 cars on a motorway, the data were summarised by \(\Sigma ( x - 110 ) = - 47.2\) and \(\Sigma ( x - 110 ) ^ { 2 } = 5460\). Calculate the mean and standard deviation of these speeds.
  4. On another day the mean speed of cars on the motorway was found to be \(107.6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the standard deviation was \(13.8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Assuming these speeds follow a normal distribution and that the speed limit is \(110 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), find what proportion of cars exceed the speed limit.
CAIE S1 2002 June Q5
5 The digits of the number 1223678 can be rearranged to give many different 7 -digit numbers. Find how many different 7-digit numbers can be made if
  1. there are no restrictions on the order of the digits,
  2. the digits 1,3,7 (in any order) are next to each other,
  3. these 7 -digit numbers are even.
  4. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma , \mathrm { P } ( X > 3.6 ) = 0.5\) and \(\mathrm { P } ( X > 2.8 ) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\).
  5. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8.
CAIE S1 2002 June Q7
7
  1. A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
    (a) Find the number of plants per box.
    (b) Find the probability that a box contains exactly 12 plants which produce yellow flowers.
  2. Another garden shop sells polyanthus plants in boxes of 100 . The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3 . Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers.
CAIE S1 2003 June Q1
1
  1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Sales of Superclene Toothpaste} \includegraphics[alt={},max width=\textwidth]{df20f053-8d67-428d-bb19-9447049deed5-2_725_1073_347_497}
    \end{figure} The diagram represents the sales of Superclene toothpaste over the last few years. Give a reason why it is misleading.
  2. The following data represent the daily ticket sales at a small theatre during three weeks. $$52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86 \text {. }$$ (a) Construct a stem-and-leaf diagram to illustrate the data.
    (b) Use your diagram to find the median of the data.
CAIE S1 2003 June Q2
2 A box contains 10 pens of which 3 are new. A random sample of two pens is taken.
  1. Show that the probability of getting exactly one new pen in the sample is \(\frac { 7 } { 15 }\).
  2. Construct a probability distribution table for the number of new pens in the sample.
  3. Calculate the expected number of new pens in the sample.
CAIE S1 2003 June Q3
3
  1. The height of sunflowers follows a normal distribution with mean 112 cm and standard deviation 17.2 cm . Find the probability that the height of a randomly chosen sunflower is greater than 120 cm .
  2. When a new fertiliser is used, the height of sunflowers follows a normal distribution with mean 115 cm . Given that \(80 \%\) of the heights are now greater than 103 cm , find the standard deviation.
CAIE S1 2003 June Q4
4 Kamal has 30 hens. The probability that any hen lays an egg on any day is 0.7 . Hens do not lay more than one egg per day, and the days on which a hen lays an egg are independent.
  1. Calculate the probability that, on any particular day, Kamal's hens lay exactly 24 eggs.
  2. Use a suitable approximation to calculate the probability that Kamal's hens lay fewer than 20 eggs on any particular day.
CAIE S1 2003 June Q5
5 A committee of 5 people is to be chosen from 6 men and 4 women. In how many ways can this be done
  1. if there must be 3 men and 2 women on the committee,
  2. if there must be more men than women on the committee,
  3. if there must be 3 men and 2 women, and one particular woman refuses to be on the committee with one particular man?
CAIE S1 2003 June Q6
6 The people living in 3 houses are classified as children ( \(C\) ), parents ( \(P\) ) or grandparents ( \(G\) ). The numbers living in each house are shown in the table below.
House number 1House number 2House number 3
\(4 C , 1 P , 2 G\)\(2 C , 2 P , 3 G\)\(1 C , 1 G\)
  1. All the people in all 3 houses meet for a party. One person at the party is chosen at random. Calculate the probability of choosing a grandparent.
  2. A house is chosen at random. Then a person in that house is chosen at random. Using a tree diagram, or otherwise, calculate the probability that the person chosen is a grandparent.
  3. Given that the person chosen by the method in part (ii) is a grandparent, calculate the probability that there is also a parent living in the house.
CAIE S1 2003 June Q7
7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.
Time spent per
day \(( t\) minutes \()\)
\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 70\)
Number
of people
11203218106
  1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
  2. On graph paper, draw a fully labelled histogram to represent the data.
CAIE S1 2020 June Q1
1 The score when two fair six-sided dice are thrown is the sum of the two numbers on the upper faces.
  1. Show that the probability that the score is 4 is \(\frac { 1 } { 12 }\).
    The two dice are thrown repeatedly until a score of 4 is obtained. The number of throws taken is denoted by the random variable \(X\).
  2. Find the mean of \(X\).
  3. Find the probability that a score of 4 is first obtained on the 6th throw.
  4. Find \(\mathrm { P } ( X < 8 )\).
CAIE S1 2020 June Q2
2
  1. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.
  2. Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.