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CAIE S1 2024 June Q2
8 marks Standard +0.3
2 In a certain country, the heights of the adult population are normally distributed with mean 1.64 m and standard deviation 0.25 m .
  1. Find the probability that an adult chosen at random from this country will have height greater than 1.93 m . \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-04_2716_35_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-05_2724_35_136_20} In another country, the heights of the adult population are also normally distributed. \(33 \%\) of the adult population have height less than \(1.56 \mathrm {~m} .25 \%\) of the adult population have height greater than 1.86 m .
  2. Find the mean and the standard deviation of this distribution.
CAIE S1 2024 June Q3
7 marks Moderate -0.8
3 Box \(A\) contains 6 green balls and 3 yellow balls.
Box \(B\) contains 4 green balls and \(x\) yellow balls.
A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\).
  1. Draw a tree diagram to represent this information, showing the probability on each of the branches.
    [0pt] [4] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-06_2727_38_132_2010}
    The probability that both the balls chosen are the same colour is \(\frac { 8 } { 15 }\).
  2. Find the value of \(x\).
CAIE S1 2024 June Q4
8 marks Easy -1.3
4 The times taken, in seconds, by 15 members of each of two swimming clubs, the Penguins and the Dolphins, to swim 50 metres are shown in the following table.
Penguins353942444545485056585961666872
Dolphins364143484949505154565660616471
  1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Penguins on the left-hand side. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_2720_33_141_20} The diagram shows a box-and-whisker plot representing the times for the Penguins.
  2. On the same diagram, draw a box-and-whisker plot to represent the times for the Dolphins. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_719_1219_424_424}
  3. Hence state one difference between the distributions of the times for the Penguins and the Dolphins.
CAIE S1 2024 June Q5
11 marks Standard +0.3
5 Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.
[0pt]
  1. Find the probability that Salah completes the puzzle for the first time on the 5th day. [1]
  2. Find the probability that Salah completes the puzzle for the second time on the 5th day.
  3. Find the probability that Salah completes the puzzle fewer than 5 times in a week (7 days). [3] \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-10_2713_31_145_2014}
  4. Use a suitable approximation to find the probability that Salah completes the puzzle more than 50 times in a period of 84 days.
CAIE S1 2024 June Q6
8 marks Standard +0.3
6
  1. How many different arrangements are there of the 9 letters in the word RECORDERS?
  2. How many different arrangements are there of the 9 letters in the word RECORDERS in which there is an E at the beginning, an E at the end and the three Rs are not all together? \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-12_2725_40_136_2007}
    The 9 letters of the word RECORDERS are divided at random into two groups: a group of 5 letters and a group of 4 letters.
  3. Find the probability that the three Rs are in the same group.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-14_2715_35_143_2012}
CAIE S1 2020 March Q1
3 marks Easy -1.2
1 The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car. In how many ways can the members who will travel in the coach be chosen?
CAIE S1 2020 March Q2
8 marks Moderate -0.8
2 An ordinary fair die is thrown repeatedly until a 1 or a 6 is obtained.
  1. Find the probability that it takes at least 3 throws but no more than 5 throws to obtain a 1 or a 6 .
    On another occasion, this die is thrown 3 times. The random variable \(X\) is the number of times that a 1 or a 6 is obtained.
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\).
CAIE S1 2020 March Q3
7 marks Moderate -0.3
3 The weights of apples of a certain variety are normally distributed with mean 82 grams. \(22 \%\) of these apples have a weight greater than 87 grams.
  1. Find the standard deviation of the weights of these apples.
  2. Find the probability that the weight of a randomly chosen apple of this variety differs from the mean weight by less than 4 grams.
CAIE S1 2020 March Q4
6 marks Standard +0.3
4 Richard has 3 blue candles, 2 red candles and 6 green candles. The candles are identical apart from their colours. He arranges the 11 candles in a line.
  1. Find the number of different arrangements of the 11 candles if there is a red candle at each end.
  2. Find the number of different arrangements of the 11 candles if all the blue candles are together and the red candles are not together.
CAIE S1 2020 March Q5
8 marks Moderate -0.3
5 In Greenton, 70\% of the adults own a car. A random sample of 8 adults from Greenton is chosen.
[0pt]
  1. Find the probability that the number of adults in this sample who own a car is less than 6 . [3]
    A random sample of 120 adults from Greenton is now chosen.
  2. Use an approximation to find the probability that more than 75 of them own a car.
CAIE S1 2020 March Q6
9 marks Moderate -0.8
6 Box \(A\) contains 7 red balls and 1 blue ball. Box \(B\) contains 9 red balls and 5 blue balls. A ball is chosen at random from box \(A\) and placed in box \(B\). A ball is then chosen at random from box \(B\). The tree diagram below shows the possibilities for the colours of the balls chosen.
  1. Complete the tree diagram to show the probabilities. Box \(A\) \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-08_624_428_667_621} \section*{Box \(B\)} Red Blue Red Blue
  2. Find the probability that the two balls chosen are not the same colour.
  3. Find the probability that the ball chosen from box \(A\) is blue given that the ball chosen from box \(B\) is blue.
CAIE S1 2020 March Q7
9 marks Moderate -0.8
7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.
Length (cm)\(0 - 9\)\(10 - 14\)\(15 - 19\)\(20 - 30\)
Frequency15486621
  1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
  2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
    The mean length of these 150 fish is 15.295 cm .
  3. Calculate an estimate for the variance of the lengths of the fish.
    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 2021 March Q1
3 marks Moderate -0.8
1 A fair spinner with 5 sides numbered 1,2,3,4,5 is spun repeatedly. The score on each spin is the number on the side on which the spinner lands.
  1. Find the probability that a score of 3 is obtained for the first time on the 8th spin.
  2. Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.
CAIE S1 2021 March Q2
5 marks Moderate -0.8
2 Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these scarves. The probabilities for the three colours are \(0.2,0.45\) and 0.35 respectively. When she wears a red scarf, she always wears a hat. When she wears a blue scarf, she wears a hat with probability 0.4 . When she wears a yellow scarf, she wears a hat with probability 0.3 .
  1. Find the probability that on a randomly chosen day Georgie wears a hat.
  2. Find the probability that on a randomly chosen day Georgie wears a yellow scarf given that she does not wear a hat.
CAIE S1 2021 March Q3
6 marks Moderate -0.3
3 The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes and standard deviation 18 minutes.
  1. Find the probability that a shopper chosen at random spends between 85 and 100 minutes in the shopping centre. \(88 \%\) of shoppers spend more than \(t\) minutes in the shopping centre.
  2. Find the value of \(t\).
CAIE S1 2021 March Q4
6 marks Moderate -0.5
4 The random variable \(X\) takes the values \(1,2,3,4\) only. The probability that \(X\) takes the value \(x\) is \(k x ( 5 - x )\), where \(k\) is a constant.
  1. Draw up the probability distribution table for \(X\), in terms of \(k\).
  2. Show that \(\operatorname { Var } ( X ) = 1.05\).
CAIE S1 2021 March Q5
9 marks Easy -1.3
5 A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km , are summarised in the following table.
Distance \(( \mathrm { km } )\)\(0 - 4\)\(5 - 10\)\(11 - 20\)\(21 - 30\)\(31 - 40\)\(41 - 60\)
Frequency12163266204
  1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{3f05dc2a-b466-40bc-9f5f-0fd2bff120c8-06_1593_1397_852_415}
  2. For 30\% of these journeys the distance travelled is \(d \mathrm {~km}\) or more. Use your graph to estimate the value of \(d\).
  3. Calculate an estimate of the mean distance travelled for the 150 journeys.
CAIE S1 2021 March Q6
10 marks Standard +0.8
6
  1. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
  2. Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
  3. Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.
CAIE S1 2021 March Q7
11 marks Moderate -0.3
7 There are 400 students at a school in a certain country. Each student was asked whether they preferred swimming, cycling or running and the results are given in the following table.
SwimmingCyclingRunning
Female1045066
Male315792
A student is chosen at random.
    1. Find the probability that the student prefers swimming.
    2. Determine whether the events 'the student is male' and 'the student prefers swimming' are independent, justifying your answer.
      On average at all the schools in this country \(30 \%\) of the students do not like any sports.
    1. 10 of the students from this country are chosen at random. Find the probability that at least 3 of these students do not like any sports.
    2. 90 students from this country are now chosen at random. Use an approximation to find the probability that fewer than 32 of them do not like any sports.
      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 2022 March Q1
5 marks Moderate -0.8
1 A fair red spinner has edges numbered \(1,2,2,3\). A fair blue spinner has edges numbered \(- 3 , - 2 , - 1 , - 1\). Each spinner is spun once and the number on the edge on which each spinner lands is noted. The random variable \(X\) denotes the sum of the resulting two numbers.
  1. Draw up the probability distribution table for \(X\).
  2. Given that \(\mathrm { E } ( X ) = 0.25\), find the value of \(\operatorname { Var } ( X )\).
CAIE S1 2022 March Q2
6 marks Standard +0.3
2 In a certain country, the probability of more than 10 cm of rain on any particular day is 0.18 , independently of the weather on any other day.
  1. Find the probability that in any randomly chosen 7-day period, more than 2 days have more than 10 cm of rain.
  2. For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than 10 cm of rain.
CAIE S1 2022 March Q3
6 marks Moderate -0.8
3 At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.
Time taken, in minutes\(1 - 30\)\(31 - 45\)\(46 - 65\)\(66 - 75\)\(76 - 100\)
Frequency2130688645
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{c1bc5ac2-6b0e-48c7-92e9-9b8b56b57d90-05_1000_1198_785_516}
  2. State which class interval contains the median.
  3. Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.
CAIE S1 2022 March Q4
11 marks Standard +0.3
4 The weights of male leopards in a particular region are normally distributed with mean 55 kg and standard deviation 6 kg .
  1. Find the probability that a randomly chosen male leopard from this region weighs between 46 and 62 kg .
    The weights of female leopards in this region are normally distributed with mean 42 kg and standard deviation \(\sigma \mathrm { kg }\). It is known that \(25 \%\) of female leopards in the region weigh less than 36 kg .
  2. Find the value of \(\sigma\).
    The distributions of the weights of male and female leopards are independent of each other. A male leopard and a female leopard are each chosen at random.
  3. Find the probability that both the weights of these leopards are less than 46 kg .
CAIE S1 2022 March Q5
10 marks Standard +0.3
5 A group of 12 people consists of 3 boys, 4 girls and 5 adults.
  1. In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
  2. In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?
    The same group of 12 people stand in a line.
  3. How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?
CAIE S1 2022 March Q6
12 marks Moderate -0.3
6 A factory produces chocolates in three flavours: lemon, orange and strawberry in the ratio \(3 : 5 : 7\) respectively. Nell checks the chocolates on the production line by choosing chocolates randomly one at a time.
  1. Find the probability that the first chocolate with lemon flavour that Nell chooses is the 7th chocolate that she checks.
  2. Find the probability that the first chocolate with lemon flavour that Nell chooses is after she has checked at least 6 chocolates.
    'Surprise' boxes of chocolates each contain 15 chocolates: 3 are lemon, 5 are orange and 7 are strawberry. Petra has a box of Surprise chocolates. She chooses 3 chocolates at random from the box. She eats each chocolate before choosing the next one.
  3. Find the probability that none of Petra's 3 chocolates has orange flavour.
  4. Find the probability that each of Petra's 3 chocolates has a different flavour.
  5. Find the probability that at least 2 of Petra's 3 chocolates have strawberry flavour given that none of them has orange flavour.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.