Questions S1 (1967 questions)

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CAIE S1 2016 November Q5
Standard +0.8
5
  1. Find the number of different ways of arranging all nine letters of the word PINEAPPLE if no vowel (A, E, I) is next to another vowel.
  2. A certain country has a cricket squad of 16 people, consisting of 7 batsmen, 5 bowlers, 2 allrounders and 2 wicket-keepers. The manager chooses a team of 11 players consisting of 5 batsmen, 4 bowlers, 1 all-rounder and 1 wicket-keeper.
    1. Find the number of different teams the manager can choose.
    2. Find the number of different teams the manager can choose if one particular batsman refuses to be in the team when one particular bowler is in the team.
CAIE S1 2016 November Q6
Standard +0.3
6 Deeti has 3 red pens and 1 blue pen in her left pocket and 3 red pens and 1 blue pen in her right pocket. 'Operation \(T\) ' consists of Deeti taking one pen at random from her left pocket and placing it in her right pocket, then taking one pen at random from her right pocket and placing it in her left pocket.
  1. Find the probability that, when Deeti carries out operation \(T\), she takes a blue pen from her left pocket and then a blue pen from her right pocket. The random variable \(X\) is the number of blue pens in Deeti's left pocket after carrying out operation \(T\).
  2. Find \(\mathrm { P } ( X = 1 )\).
  3. Given that the pen taken from Deeti's right pocket is blue, find the probability that the pen taken from Deeti's left pocket is blue.
CAIE S1 2016 November Q7
Easy -1.3
7 The masses, in grams, of components made in factory \(A\) and components made in factory \(B\) are shown below.
Factory \(A\)0.0490.0500.0530.0540.0570.0580.058
0.0590.0610.0610.0610.0630.065
Factory \(B\)0.0310.0560.0490.0440.0380.0480.051
0.0640.0350.0420.0470.0540.058
  1. Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the two factories.
  2. Find the median and the interquartile range for the masses of components made in factory \(B\).
  3. Make two comparisons between the masses of components made in factory \(A\) and the masses of those made in factory \(B\).
CAIE S1 2016 November Q1
Moderate -0.8
1 When Anya goes to school, the probability that she walks is 0.3 and the probability that she cycles is 0.65 ; if she does not walk or cycle she takes the bus. When Anya walks the probability that she is late is 0.15 . When she cycles the probability that she is late is 0.1 and when she takes the bus the probability that she is late is 0.6 . Given that Anya is late, find the probability that she cycles.
CAIE S1 2016 November Q2
Moderate -0.8
2 Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses 3 items at random. The random variable \(X\) is the number of T-shirts chosen.
  1. Show that the probability that Noor chooses exactly one T-shirt is \(\frac { 27 } { 55 }\).
  2. Draw up the probability distribution table for \(X\).
CAIE S1 2016 November Q3
Standard +0.3
3 On any day at noon, the probabilities that Kersley is asleep or studying are 0.2 and 0.6 respectively.
  1. Find the probability that, in any 7-day period, Kersley is either asleep or studying at noon on at least 6 days.
  2. Use an approximation to find the probability that, in any period of 100 days, Kersley is asleep at noon on at most 30 days.
CAIE S1 2016 November Q4
Standard +0.3
4 The time taken to cook an egg by people living in a certain town has a normal distribution with mean 4.2 minutes and standard deviation 0.6 minutes.
  1. Find the probability that a person chosen at random takes between 3.5 and 4.5 minutes to cook an egg.
    \(12 \%\) of people take more than \(t\) minutes to cook an egg.
  2. Find the value of \(t\).
  3. A random sample of \(n\) people is taken. Find the smallest possible value of \(n\) if the probability that none of these people takes more than \(t\) minutes to cook an egg is less than 0.003 .
CAIE S1 2016 November Q5
Moderate -0.8
5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.
Capacity\(3000 - 7000\)\(8000 - 12000\)\(13000 - 22000\)\(23000 - 42000\)\(43000 - 82000\)
Number of stadiums403018348
  1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.
  2. Calculate an estimate of the mean capacity of these 130 stadiums.
  3. Find which class in the table contains the median and which contains the lower quartile.
CAIE S1 2016 November Q6
3 marks Standard +0.3
6 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that
  1. the vowels ( \(\mathrm { A } , \mathrm { E } , \mathrm { O }\) ) are together and the consonants ( \(\mathrm { C } , \mathrm { G } , \mathrm { H } , \mathrm { N } , \mathrm { P }\) ) are together, [3]
  2. the Es are not next to each other. Four letters are selected from the 10 letters of the word COPENHAGEN.
  3. Find the number of different selections if the four letters must contain the same number of Es and Ns with at least one of each.
CAIE S1 2016 November Q1
Moderate -0.3
1 A committee of 5 people is to be chosen from 4 men and 6 women. William is one of the 4 men and Mary is one of the 6 women. Find the number of different committees that can be chosen if William and Mary refuse to be on the committee together.
CAIE S1 2016 November Q2
Standard +0.3
2 A fair triangular spinner has three sides numbered 1, 2, 3. When the spinner is spun, the score is the number of the side on which it lands. The spinner is spun four times.
  1. Find the probability that at least two of the scores are 3 .
  2. Find the probability that the sum of the four scores is 5 .
CAIE S1 2016 November Q3
Moderate -0.8
3 Numbers are formed using some or all of the digits 4, 5, 6, 7 with no digit being used more than once.
  1. Show that, using exactly 3 of the digits, there are 12 different odd numbers that can be formed.
  2. Find how many odd numbers altogether can be formed.
CAIE S1 2016 November Q4
Easy -1.3
4 For a group of 250 cars the numbers, classified by colour and country of manufacture, are shown in the table.
GermanyJapanKorea
Silver402634
White322226
Red281230
One car is selected at random from this group. Find the probability that the selected car is
  1. a red or silver car manufactured in Korea,
  2. not manufactured in Japan.
    \(X\) is the event that the selected car is white. \(Y\) is the event that the selected car is manufactured in Germany.
  3. By using appropriate probabilities, determine whether events \(X\) and \(Y\) are independent.
CAIE S1 2016 November Q5
1 marks Easy -1.3
5 The tables summarise the heights, \(h \mathrm {~cm}\), of 60 girls and 60 boys.
Height of girls (cm)\(140 < h \leqslant 150\)\(150 < h \leqslant 160\)\(160 < h \leqslant 170\)\(170 < h \leqslant 180\)\(180 < h \leqslant 190\)
Frequency122117100
Height of boys \(( \mathrm { cm } )\)\(140 < h \leqslant 150\)\(150 < h \leqslant 160\)\(160 < h \leqslant 170\)\(170 < h \leqslant 180\)\(180 < h \leqslant 190\)
Frequency02023125
  1. On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
  2. On a school trip the students have to enter a cave which is 165 cm high. Use your graph to estimate the percentage of the girls who will be unable to stand upright.
    [0pt]
  3. The students are asked to compare the heights of the girls and the boys. State one advantage of using a pair of box-and-whisker plots instead of the cumulative frequency graphs to do this. [1]
CAIE S1 2016 November Q6
Moderate -0.3
6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams
  1. Find the proportion of bananas that are small.
  2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
  3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
    (b) Calculate the expected total cost of 100 randomly chosen bananas.
CAIE S1 2016 November Q7
Standard +0.3
7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .
  1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
  2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
  3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.
CAIE S1 2017 November Q1
Moderate -0.8
1 The discrete random variable \(X\) has the following probability distribution.
\(x\)1236
\(\mathrm { P } ( X = x )\)0.15\(p\)0.4\(q\)
Given that \(\mathrm { E } ( X ) = 3.05\), find the values of \(p\) and \(q\).
CAIE S1 2017 November Q2
Easy -1.8
2 The time taken by a car to accelerate from 0 to 30 metres per second was measured correct to the nearest second. The results from 48 cars are summarised in the following table.
Time (seconds)\(3 - 5\)\(6 - 8\)\(9 - 11\)\(12 - 16\)\(17 - 25\)
Frequency10151742
  1. On the grid, draw a cumulative frequency graph to represent this information.
    \includegraphics[max width=\textwidth, alt={}, center]{ee1e5987-315b-48eb-8dba-b9d4d34c87c9-03_1207_1406_897_411}
  2. 35 of these cars accelerated from 0 to 30 metres per second in a time more than \(t\) seconds. Estimate the value of \(t\).
CAIE S1 2017 November Q3
Moderate -0.8
3 An experiment consists of throwing a biased die 30 times and noting the number of 4 s obtained. This experiment was repeated many times and the average number of 4 s obtained in 30 throws was found to be 6.21.
  1. Estimate the probability of throwing a 4.
    ..................................................................................................................................... .
    \section*{Hence}
  2. find the variance of the number of 4 s obtained in 30 throws,
  3. find the probability that in 15 throws the number of 4 s obtained is 2 or more.
CAIE S1 2017 November Q4
Moderate -0.3
4 The ages of a group of 12 people at an Art class have mean 48.7 years and standard deviation 7.65 years. The ages of a group of 7 people at another Art class have mean 38.1 years and standard deviation 4.2 years.
  1. Find the mean age of all 19 people.
  2. The individual ages in years of people in the first Art class are denoted by \(x\) and those in the second Art class by \(y\). By first finding \(\Sigma x ^ { 2 }\) and \(\Sigma y ^ { 2 }\), find the standard deviation of the ages of all 19 people.
CAIE S1 2017 November Q5
Moderate -0.8
5 Over a period of time Julian finds that on long-distance flights he flies economy class on \(82 \%\) of flights. On the rest of the flights he flies first class. When he flies economy class, the probability that he gets a good night's sleep is \(x\). When he flies first class, the probability that he gets a good night's sleep is 0.9 .
  1. Draw a fully labelled tree diagram to illustrate this situation. The probability that Julian gets a good night's sleep on a randomly chosen flight is 0.285 .
  2. Find the value of \(x\).
  3. Given that on a particular flight Julian does not get a good night's sleep, find the probability that he is flying economy class.
CAIE S1 2017 November Q6
Moderate -0.8
6
  1. A village hall has seats for 40 people, consisting of 8 rows with 5 seats in each row. Mary, Ahmad, Wayne, Elsie and John are the first to arrive in the village hall and no seats are taken before they arrive.
    1. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John assuming there are no restrictions?
    2. How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John if Mary and Ahmad sit together in the front row and the other three sit together in one of the other rows?
  2. In how many ways can a team of 4 people be chosen from 10 people if 2 of the people, Ross and Lionel, refuse to be in the team together?
CAIE S1 2017 November Q7
Moderate -0.3
7 The weight, in grams, of pineapples is denoted by the random variable \(X\) which has a normal distribution with mean 500 and standard deviation 91.5. Pineapples weighing over 570 grams are classified as 'large'. Those weighing under 390 grams are classified as 'small' and the rest are classified as 'medium'.
  1. Find the proportions of large, small and medium pineapples.
  2. Find the weight exceeded by the heaviest \(5 \%\) of pineapples.
  3. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 610 ) = 0.3\).
CAIE S1 2017 November Q1
Easy -1.2
1 Andy counts the number of emails, \(x\), he receives each day and notes that, over a period of \(n\) days, \(\Sigma ( x - 10 ) = 27\) and the mean number of emails is 11.5 . Find the value of \(n\).
CAIE S1 2017 November Q2
Moderate -0.8
2 The circumferences, \(c \mathrm {~cm}\), of some trees in a wood were measured. The results are summarised in the table.
Circumference \(( c \mathrm {~cm} )\)\(40 < c \leqslant 50\)\(50 < c \leqslant 80\)\(80 < c \leqslant 100\)\(100 < c \leqslant 120\)
Frequency1448708
  1. On the grid, draw a cumulative frequency graph to represent the information.
    \includegraphics[max width=\textwidth, alt={}, center]{9c23b94b-e573-4e13-be90-e63a0daf18e5-03_1401_1404_854_413}
  2. Estimate the percentage of trees which have a circumference larger than 75 cm .