Questions — CAIE S1 (789 questions)

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CAIE S1 2024 November Q2
4 marks Moderate -0.8
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
7 marks Moderate -0.8
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
11 marks Moderate -0.8
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
10 marks Standard +0.8
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
11 marks Challenging +1.2
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 2003 June Q1
5 marks Easy -1.8
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. $$52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86 \text {. }$$
    1. Construct a stem-and-leaf diagram to illustrate the data.
    2. Use your diagram to find the median of the data.
CAIE S1 2003 June Q2
6 marks Moderate -0.8
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
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
8 marks Moderate -0.8
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
9 marks Moderate -0.8
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
9 marks Easy -1.2
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
5 marks Moderate -0.8
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
6 marks Standard +0.3
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.
CAIE S1 2020 June Q3
7 marks Moderate -0.8
3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.
  1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
    The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
  2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.
CAIE S1 2020 June Q4
4 marks Standard +0.8
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
CAIE S1 2020 June Q5
8 marks Easy -1.2
5 On Mondays, Rani cooks her evening meal. She has a pizza, a burger or a curry with probabilities \(0.35,0.44,0.21\) respectively. When she cooks a pizza, Rani has some fruit with probability 0.3 . When she cooks a burger, she has some fruit with probability 0.8 . When she cooks a curry, she never has any fruit.
  1. Draw a fully labelled tree diagram to represent this information.
  2. Find the probability that Rani has some fruit.
  3. Find the probability that Rani does not have a burger given that she does not have any fruit.
CAIE S1 2020 June Q6
9 marks Standard +0.8
6 The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm .
  1. Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm .
    The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm .
  2. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.
CAIE S1 2020 June Q7
11 marks Moderate -0.8
7 The numbers of chocolate bars sold per day in a cinema over a period of 100 days are summarised in the following table.
Number of chocolate bars sold\(1 - 10\)\(11 - 15\)\(16 - 30\)\(31 - 50\)\(51 - 60\)
Number of days182430208
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{3ada18de-c4f7-4049-9032-46b796be83c3-12_1203_1399_833_415}
  2. What is the greatest possible value of the interquartile range for the data?
  3. Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2004 June Q1
4 marks Easy -1.2
1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.
Team \(A\)150220773029811816057
Team \(B\)1661421709311113014886
  1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
  2. State with a reason which team has the more consistent scores.
CAIE S1 2004 June Q2
5 marks Easy -1.8
2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.
  1. On graph paper, show these results using a fully labelled cumulative frequency graph.
  2. Use your graph to estimate how many people watched more than 50 hours of television each week.
CAIE S1 2004 June Q3
5 marks Moderate -0.8
3 Two fair dice are thrown. Let the random variable \(X\) be the smaller of the two scores if the scores are different, or the score on one of the dice if the scores are the same.
  1. Copy and complete the following table to show the probability distribution of \(X\).
    \(x\)123456
    \(\mathrm { P } ( X = x )\)
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2004 June Q4
8 marks Moderate -0.3
4 Melons are sold in three sizes: small, medium and large. The weights follow a normal distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than 350 grams are classified as small.
  1. Find the proportion of melons which are classified as small.
  2. The rest of the melons are divided in equal proportions between medium and large. Find the weight above which melons are classified as large.
CAIE S1 2004 June Q5
8 marks Moderate -0.8
5
  1. The menu for a meal in a restaurant is as follows. \begin{displayquote} Starter Course
    Melon
    or
    Soup
    or
    Smoked Salmon \end{displayquote} \begin{displayquote} Main Course
    Chicken
    or
    Steak
    or
    Lamb Cutlets
    or
    Vegetable Curry
    or
    Fish \end{displayquote} \begin{displayquote} Dessert Course
    Cheesecake
    or
    Ice Cream
    or
    Apple Pie
    All the main courses are served with salad and either
    new potatoes or french fries.
    1. How many different three-course meals are there?
    2. How many different choices are there if customers may choose only two of the three courses?
  2. In how many ways can a group of 14 people eating at the restaurant be divided between three tables seating 5, 5 and 4? \end{displayquote}
CAIE S1 2004 June Q6
9 marks Moderate -0.3
6 When Don plays tennis, \(65 \%\) of his first serves go into the correct area of the court. If the first serve goes into the correct area, his chance of winning the point is \(90 \%\). If his first serve does not go into the correct area, Don is allowed a second serve, and of these, \(80 \%\) go into the correct area. If the second serve goes into the correct area, his chance of winning the point is \(60 \%\). If neither serve goes into the correct area, Don loses the point.
  1. Draw a tree diagram to represent this information.
  2. Using your tree diagram, find the probability that Don loses the point.
  3. Find the conditional probability that Don's first serve went into the correct area, given that he loses the point.