Questions — CAIE S1 (789 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE S1 2011 November Q6
9 marks Moderate -0.8
6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.
  1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
  2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
  3. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
  4. Find the proportion of winter days on which the minimum temperature is below zero.
  5. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
  6. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).
CAIE S1 2011 November Q1
6 marks Standard +0.3
1 The random variable \(X\) is normally distributed and is such that the mean \(\mu\) is three times the standard deviation \(\sigma\). It is given that \(\mathrm { P } ( X < 25 ) = 0.648\).
  1. Find the values of \(\mu\) and \(\sigma\).
  2. Find the probability that, from 6 random values of \(X\), exactly 4 are greater than 25 .
CAIE S1 2011 November Q2
6 marks Moderate -0.8
2 In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.
  1. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.
  2. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.
CAIE S1 2011 November Q3
9 marks Standard +0.3
3 A factory makes a large number of ropes with lengths either 3 m or 5 m . There are four times as many ropes of length 3 m as there are ropes of length 5 m .
  1. One rope is chosen at random. Find the expectation and variance of its length.
  2. Two ropes are chosen at random. Find the probability that they have different lengths.
  3. Three ropes are chosen at random. Find the probability that their total length is 11 m .
CAIE S1 2011 November Q4
9 marks Moderate -0.8
4 Mary saves her digital images on her computer in three separate folders named 'Family', 'Holiday' and 'Friends'. Her family folder contains 3 images, her holiday folder contains 4 images and her friends folder contains 8 images. All the images are different.
  1. Find in how many ways she can arrange these 15 images in a row across her computer screen if she keeps the images from each folder together.
  2. Find the number of different ways in which Mary can choose 6 of these images if there are 2 from each folder.
  3. Find the number of different ways in which Mary can choose 6 of these images if there are at least 3 images from the friends folder and at least 1 image from each of the other two folders.
CAIE S1 2011 November Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{b72ace6b-d3d4-401d-bffe-403c9127f2a8-3_1157_1001_258_573} The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that the lowest salary is 5000 euros and the highest salary is 80000 euros.
  1. On graph paper, draw a box-and-whisker plot to illustrate these salaries.
  2. Comment on the salaries of the people in this sample.
  3. An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
    1. How high must a salary be in order to be classified as an outlier?
    2. Show that none of the salaries is low enough to be classified as an outlier.
CAIE S1 2011 November Q6
11 marks Standard +0.3
6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).
  1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
  2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.
CAIE S1 2012 November Q1
4 marks Moderate -0.8
1 Ashok has 3 green pens and 7 red pens. His friend Rod takes 3 of these pens at random, without replacement. Draw up a probability distribution table for the number of green pens Rod takes.
CAIE S1 2012 November Q2
5 marks Moderate -0.8
2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
CAIE S1 2012 November Q3
6 marks Standard +0.3
3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .
  1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
  2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .
CAIE S1 2012 November Q4
7 marks Moderate -0.8
4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)
  1. Represent these prices by a stem-and-leaf diagram.
  2. Write down the lower quartile of the prices of the caravans in the showroom.
  3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.
CAIE S1 2012 November Q5
7 marks Standard +0.3
5 A company set up a display consisting of 20 fireworks. For each firework, the probability that it fails to work is 0.05 , independently of other fireworks.
  1. Find the probability that more than 1 firework fails to work. The 20 fireworks cost the company \(\\) 24\( each. 450 people pay the company \)\\( 10\) each to watch the display. If more than 1 firework fails to work they get their money back.
  2. Calculate the expected profit for the company.
CAIE S1 2012 November Q6
9 marks Standard +0.3
6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.
  1. Find the probability that she is on time on fewer than 20 of the next 96 days.
  2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.
CAIE S1 2012 November Q7
12 marks Standard +0.3
7
  1. In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the packets that are possible if the packets of chocolates are kept together.
  2. Jessica buys 8 different packets of biscuits. She then chooses 4 of these packets.
    1. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account? The 8 packets include 1 packet of chocolate biscuits and 1 packet of custard creams.
    2. How many different choices are possible if the order in which Jessica chooses the 4 packets is taken into account and the packet of chocolate biscuits and the packet of custard creams are both chosen?
  3. 9 different fruit pies are to be divided between 3 people so that each person gets an odd number of pies. Find the number of ways this can be done.
CAIE S1 2020 Specimen Q1
5 marks Easy -1.2
1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
FemalesMales
(4)5200203(1)
(9)98876400021007(3)
(8)8753310022004566(6)
(6)64210023002335677(9)
(6)754000240112556889(10)
(4)9500253457789(7)
(2)5026046(3)
Key: 2 | 20 | 3 means \\(20200for females and \\)20300 for males.
  1. Find the median and the quartiles of the females' salaries.
    You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
  2. Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}
CAIE S1 2020 Specimen Q2
4 marks Easy -1.2
2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \text { and } \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).
CAIE S1 2020 Specimen Q3
7 marks Moderate -0.5
3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.
  1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
  2. Draw up the probability distribution table for \(X\).
  3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).
CAIE S1 2020 Specimen Q4
10 marks Moderate -0.5
4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
  1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
    The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
  2. Find the value of \(m\).
  3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
CAIE S1 2020 Specimen Q5
7 marks Moderate -0.5
5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.
  1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
  2. Justify your use of the approximation in part (a).
    On another occasion, the same die is thrown repeatedly until a 3 is obtained.
  3. Find the probability that obtaining a 3 requires fewer than 7 throws.
CAIE S1 2020 Specimen Q6
7 marks Standard +0.3
6 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.
  1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
  2. Find the number of different seating arrangements that are now possible for the 8 friends.
CAIE S1 2020 Specimen Q7
10 marks Standard +0.3
7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.
  • Event \(X\) is 'exactly two of the selected balls have the same number'.
  • Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
    1. Find \(\mathrm { P } ( X )\).
    2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
    3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.
CAIE S1 2006 June Q6
9 marks Easy -1.2
  1. How many teams play in only 1 match?
  2. How many teams play in exactly 2 matches?
  3. Draw up a frequency table for the numbers of matches which the teams play.
  4. Calculate the mean and variance of the numbers of matches which the teams play.
CAIE S1 2021 November Q1
5 marks Easy -1.8
1 Each of the 180 students at a college plays exactly one of the piano, the guitar and the drums. The numbers of male and female students who play the piano, the guitar and the drums are given in the following table.
PianoGuitarDrums
Male254411
Female423820
A student at the college is chosen at random.
  1. Find the probability that the student plays the guitar.
  2. Find the probability that the student is male given that the student plays the drums.
  3. Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
CAIE S1 2021 November Q2
5 marks Moderate -0.3
2 A group of 6 people is to be chosen from 4 men and 11 women.
  1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
    Two of the 11 women are sisters Jane and Kate.
  2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?
CAIE S1 2021 November Q3
7 marks Moderate -0.8
3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
  1. Show that the probability that exactly one of the marbles is yellow is \(\frac { 5 } { 14 }\).
    The random variable \(X\) is the number of yellow marbles selected.
  2. Draw up the probability distribution table for \(X\).
  3. Find \(\mathrm { E } ( X )\).