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CAIE S1 2021 June Q6
10 marks Standard +0.8
6
  1. How many different arrangements are there of the 11 letters in the word REQUIREMENT? [2]
  2. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which the two Rs are together and the three Es are together?
  3. How many different arrangements are there of the 11 letters in the word REQUIREMENT in which there are exactly three letters between the two Rs?
    Five of the 11 letters in the word REQUIREMENT are selected.
  4. How many possible selections contain at least two Es and at least one R?
CAIE S1 2021 June Q7
11 marks Moderate -0.3
7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Quality of broadband service
\cline { 3 - 5 } \multicolumn{2}{c|}{}ExcellentGoodPoor
\multirow{3}{*}{Village}Reeta7511832
\cline { 2 - 5 }Shan22317740
\cline { 2 - 5 }Teber126063
    1. Find the probability that a randomly chosen household is in Shan and has poor broadband service.
    2. Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.
      In the whole of Arka there are a large number of households. A survey showed that \(35 \%\) of households in Arka have no broadband service.
    1. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.
    2. 120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.
      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 June Q1
6 marks Moderate -0.3
1
  1. Find the number of different arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.
  2. Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.
CAIE S1 2022 June Q2
6 marks Moderate -0.3
2 There are 6 men and 8 women in a Book Club. The committee of the club consists of five of its members. Mr Lan and Mrs Lan are members of the club.
  1. In how many different ways can the committee be selected if exactly one of Mr Lan and Mrs Lan must be on the committee?
  2. In how many different ways can the committee be selected if Mrs Lan must be on the committee and there must be more women than men on the committee?
CAIE S1 2022 June Q3
9 marks Moderate -0.8
3 The times taken to travel to college by 2500 students are summarised in the table.
Time taken \(( t\) minutes \()\)\(0 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 60\)\(60 \leqslant t < 90\)
Frequency440720920300120
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{d69f6a47-7c88-46b3-9e8f-07727106e987-04_1201_1198_1050_516} From the data, the estimate of the mean value of \(t\) is 31.44 .
  2. Calculate an estimate of the standard deviation of the times taken to travel to college.
  3. In which class interval does the upper quartile lie?
    It was later discovered that the times taken to travel to college by two students were incorrectly recorded. One student's time was recorded as 15 instead of 5 and the other's time was recorded as 65 instead of 75 .
  4. Without doing any further calculations, state with a reason whether the estimate of the standard deviation in part (b) would be increased, decreased or stay the same.
CAIE S1 2022 June Q4
10 marks Standard +0.3
4 Jacob has four coins. One of the coins is biased such that when it is thrown the probability of obtaining a head is \(\frac { 7 } { 10 }\). The other three coins are fair. Jacob throws all four coins once. The number of heads that he obtains is denoted by the random variable \(X\). The probability distribution table for \(X\) is as follows.
\(x\)01234
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 80 }\)\(a\)\(b\)\(c\)\(\frac { 7 } { 80 }\)
  1. Show that \(a = \frac { 1 } { 5 }\) and find the values of \(b\) and \(c\).
  2. Find \(\mathrm { E } ( X )\).
    Jacob throws all four coins together 10 times.
  3. Find the probability that he obtains exactly one head on fewer than 3 occasions.
  4. Find the probability that Jacob obtains exactly one head for the first time on the 7th or 8th time that he throws the 4 coins.
CAIE S1 2022 June Q5
11 marks Standard +0.3
5 The lengths, in cm, of the leaves of a particular type are modelled by the distribution \(\mathrm { N } \left( 5.2,1.5 ^ { 2 } \right)\).
  1. Find the probability that a randomly chosen leaf of this type has length less than 6 cm .
    The lengths of the leaves of another type are also modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves of this type and finds that 46 are less than 3 cm long and 95 are more than 8 cm long.
  2. Find estimates for the mean and standard deviation of the lengths of leaves of this type.
  3. In a random sample of 2000 leaves of this second type, how many would the scientist expect to find with lengths more than 1 standard deviation from the mean?
CAIE S1 2022 June Q6
8 marks Standard +0.3
6 Janice is playing a computer game. She has to complete level 1 and level 2 to finish the game. She is allowed at most two attempts at any level.
  • For level 1 , the probability that Janice completes it at the first attempt is 0.6 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.3 .
  • If Janice completes level 1, she immediately moves on to level 2.
  • For level 2, the probability that Janice completes it at the first attempt is 0.4 . If she fails at her first attempt, the probability that she completes it at the second attempt is 0.2 .
    1. Show that the probability that Janice moves on to level 2 is 0.72 .
    2. Find the probability that Janice finishes the game.
    3. Find the probability that Janice fails exactly one attempt, given that she finishes the game.
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 June Q1
3 marks Easy -1.8
1 The time taken, \(t\) minutes, to complete a puzzle was recorded for each of 150 students. These times are summarised in the table.
Time taken \(( t\) minutes \()\)\(t \leqslant 25\)\(t \leqslant 50\)\(t \leqslant 75\)\(t \leqslant 100\)\(t \leqslant 150\)\(t \leqslant 200\)
Cumulative frequency164486104132150
  1. Draw a cumulative frequency graph to illustrate the data.
    \multirow{2}{*}{}
    \multirow{3}{*}}{
  2. Use your graph to estimate the 20th percentile of the data.
CAIE S1 2022 June Q2
3 marks Easy -1.3
2 Twenty children were asked to estimate the height of a particular tree. Their estimates, in metres, were as follows.
4.14.24.44.54.64.85.05.25.35.4
5.55.86.06.26.36.46.66.86.919.4
  1. Find the mean of the estimated heights.
  2. Find the median of the estimated heights.
  3. Give a reason why the median is likely to be more suitable than the mean as a measure of the central tendency for this information.
CAIE S1 2022 June Q3
6 marks Easy -1.2
3 The random variable \(X\) takes the values \(- 2,1,2,3\). It is given that \(\mathrm { P } ( X = x ) = k x ^ { 2 }\), where \(k\) is a constant.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2022 June Q4
7 marks Moderate -0.3
4 Ramesh throws an ordinary fair 6-sided die.
  1. Find the probability that he obtains a 4 for the first time on his 8th throw.
  2. Find the probability that it takes no more than 5 throws for Ramesh to obtain a 4 .
    Ramesh now repeatedly throws two ordinary fair 6-sided dice at the same time. Each time he adds the two numbers that he obtains.
  3. For 10 randomly chosen throws of the two dice, find the probability that Ramesh obtains a total of less than 4 on at least three throws.
CAIE S1 2022 June Q5
10 marks Moderate -0.3
5 Farmer Jones grows apples. The weights, in grams, of the apples grown this year are normally distributed with mean 170 and standard deviation 25. Apples that weigh between 142 grams and 205 grams are sold to a supermarket.
  1. Find the probability that a randomly chosen apple grown by Farmer Jones this year is sold to the supermarket.
    Farmer Jones sells the apples to the supermarket at \(\\) 0.24\( each. He sells apples that weigh more than 205 grams to a local shop at \)\\( 0.30\) each. He does not sell apples that weigh less than 142 grams. The total number of apples grown by Farmer Jones this year is 20000.
  2. Calculate an estimate for his total income from this year's apples.
    Farmer Tan also grows apples. The weights, in grams, of the apples grown this year follow the distribution \(\mathrm { N } \left( 182,20 ^ { 2 } \right) .72 \%\) of these apples have a weight more than \(w\) grams.
  3. Find the value of \(w\).
CAIE S1 2022 June Q6
10 marks Standard +0.8
6 Sajid is practising for a long jump competition. He counts any jump that is longer than 6 m as a success. On any day, the probability that he has a success with his first jump is 0.2 . For any subsequent jump, the probability of a success is 0.3 if the previous jump was a success and 0.1 otherwise. Sajid makes three jumps.
  1. Draw a tree diagram to illustrate this information, showing all the probabilities.
  2. Find the probability that Sajid has exactly one success given that he has at least one success.
    On another day, Sajid makes six jumps.
  3. Find the probability that only his first three jumps are successes or only his last three jumps are successes.
CAIE S1 2022 June Q7
11 marks Standard +0.8
7 A group of 15 friends visit an adventure park. The group consists of four families.
  • Mr and Mrs Kenny and their four children
  • Mr and Mrs Lizo and their three children
  • Mrs Martin and her child
  • Mr and Mrs Nantes
The group travel to the park in three cars, one containing 6 people, one containing 5 people and one containing 4 people. The cars are driven by Mr Lizo, Mrs Martin and Mr Nantes respectively.
  1. In how many different ways can the remaining 12 members of the group be divided between the three cars?
    The group enter the park by walking through a gate one at a time.
  2. In how many different orders can the 15 friends go through the gate if Mr Lizo goes first and each family stays together?
    In the park, the group enter a competition which requires a team of 4 adults and 3 children.
  3. In how many ways can the team be chosen from the group of 15 so that the 3 children are all from different families?
  4. In how many ways can the team be chosen so that at least one of Mr Kenny or Mr Lizo is included?
    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 2023 June Q1
4 marks Moderate -0.8
1 A summary of 50 values of \(x\) gives $$\Sigma ( x - q ) = 700 , \quad \Sigma ( x - q ) ^ { 2 } = 14235$$ where \(q\) is a constant.
  1. Find the standard deviation of these values of \(x\).
  2. Given that \(\Sigma x = 2865\), find the value of \(q\).
CAIE S1 2023 June Q2
5 marks Standard +0.3
2
  1. Find the number of ways in which a committee of 6 people can be chosen from 6 men and 8 women if it must include 3 men and 3 women.
    A different committee of 6 people is to be chosen from 6 men and 8 women. Three of the 6 men are brothers.
  2. Find the number of ways in which this committee can be chosen if there are no restrictions on the numbers of men and women, but it must include no more than two of the brothers.
CAIE S1 2023 June Q3
6 marks Standard +0.8
3
  1. Find the number of different arrangements of the 8 letters in the word COCOONED.
  2. Find the number of different arrangements of the 8 letters in the word COCOONED in which the first letter is O and the last letter is N .
  3. Find the probability that a randomly chosen arrangement of the 8 letters in the word COCOONED has all three Os together given that the two Cs are next to each other.
CAIE S1 2023 June Q4
9 marks Standard +0.3
4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.
  1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
    All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
  2. Find the value of \(\mu\) and the value of \(\sigma\).
CAIE S1 2023 June Q5
7 marks Moderate -0.3
5 The populations of 150 villages in the UK, to the nearest hundred, are summarised in the table.
Population\(100 - 800\)\(900 - 1200\)\(1300 - 2000\)\(2100 - 3200\)\(3300 - 4800\)
Number of villages812504832
  1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{a7157882-d87e-4efb-abc5-9c9f58197012-08_1395_1195_1043_516}
  2. Write down the class interval which contains the median for this information.
  3. Find the greatest possible value of the interquartile range for the populations of the 150 villages.
CAIE S1 2023 June Q6
11 marks Standard +0.3
6 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.
  1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
  2. Complete the following probability distribution table for \(X\).
    \(x\)01234
    \(\mathrm { P } ( X = x )\)\(\frac { 81 } { 256 }\)\(\frac { 3 } { 64 }\)\(\frac { 1 } { 256 }\)
  3. Find \(\mathrm { E } ( X )\).
    Eli throws the four dice at the same time on 96 occasions.
  4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.
CAIE S1 2023 June Q7
8 marks Standard +0.3
7 A children's wildlife magazine is published every Monday. For the next 12 weeks it will include a model animal as a free gift. There are five different models: tiger, leopard, rhinoceros, elephant and buffalo, each with the same probability of being included in the magazine. Sahim buys one copy of the magazine every Monday.
  1. Find the probability that the first time that the free gift is an elephant is before the 6th Monday.
  2. Find the probability that Sahim will get more than two leopards in the 12 magazines.
  3. Find the probability that after 5 weeks Sahim has exactly one of each animal.
    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 2023 June Q1
6 marks Moderate -0.8
1 The random variable \(X\) takes the values \(- 2,2\) and 3. It is given that $$\mathrm { P } ( X = x ) = k \left( x ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
  1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
CAIE S1 2023 June Q2
6 marks Standard +0.3
2 A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4 . On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.
  1. Find the probability that it does not rain on any of the 4 days of the event.
  2. Find the probability that the first day on which it rains during the event is Tuesday.
  3. Find the probability that it rains on exactly one of the 4 days of the event.
CAIE S1 2023 June Q3
7 marks Moderate -0.8
3 The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, \(A\) and \(B\).
Company \(A\)Company \(B\)
\multirow{6}{*}{9}411025445667
72102601355799
4210271346688
54202801222
98529
1309
Key: 1 |27| 6 means \(\\) 2710\( for company \)A\( and \)\\( 2760\) for company \(B\)
  1. Find the median and the interquartile range of the monthly salaries of employees in company \(A\).
    The lower quartile, median and upper quartile for company \(B\) are \(\\) 2600 , \\( 2690\) and \(\\) 2780\( respectively.
  2. Draw two box-and-whisker plots in a single diagram to represent the information for the salaries of employees at companies \)A\( and \)B$. \includegraphics[max width=\textwidth, alt={}, center]{f2666d82-4711-499a-98c0-3421e4c228fb-07_810_1406_573_411}
  3. Comment on whether the mean would be a more appropriate measure than the median for comparing the given information for the two companies.