Questions — CAIE S1 (785 questions)

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CAIE S1 2019 November Q5
5 Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.
Time \(( t\) minutes \()\)\(t \leqslant 10\)\(t \leqslant 20\)\(t \leqslant 30\)\(t \leqslant 50\)\(t \leqslant 70\)\(t \leqslant 90\)
Cumulative frequency1650106146176200
  1. On the grid, draw a cumulative frequency graph to illustrate the data.
    \includegraphics[max width=\textwidth, alt={}, center]{06f6c8dd-170c-4e94-a960-0c649a7363a1-08_1198_1399_735_415}
  2. Use your graph to estimate the median of the data.
  3. For 80 of the drivers, the time taken was at least \(T\) minutes. Use your graph to estimate the value of \(T\).
  4. Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park.
CAIE S1 2019 November Q6
6 A box contains 3 red balls and 5 white balls. One ball is chosen at random from the box and is not returned to the box. A second ball is now chosen at random from the box.
  1. Find the probability that both balls chosen are red.
  2. Show that the probability that the balls chosen are of different colours is \(\frac { 15 } { 28 }\).
  3. Given that the second ball chosen is red, find the probability that the first ball chosen is red.
    The random variable \(X\) denotes the number of red balls chosen.
  4. Draw up the probability distribution table for \(X\).
  5. Find \(\operatorname { Var } ( X )\).
CAIE S1 2019 November Q7
7 A competition is taking place between two choirs, the Notes and the Classics. There is a large audience for the competition.
  • \(30 \%\) of the audience are Notes supporters.
  • \(45 \%\) of the audience are Classics supporters.
  • The rest of the audience are not supporters of either of these choirs.
  • No one in the audience supports both of these choirs.
    1. A random sample of 6 people is chosen from the audience.
      (a) Find the probability that no more than 2 of the 6 people are Notes supporters.
      (b) Find the probability that none of the 6 people support either of these choirs.
    2. A random sample of 240 people is chosen from the audience. Use a suitable approximation to find the probability that fewer than 50 do not support either of the choirs.
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 Specimen Q1
1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.
CAIE S1 Specimen Q2
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 54.1 ) = 0.5\) and \(\mathrm { P } ( X > 50.9 ) = 0.8665\). Find the values of \(\mu\) and \(\sigma\).
CAIE S1 Specimen Q3
3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by \(a\) and \(b\).
Time \(( t\) minutes \()\)\(60 - 62\)\(63 - 64\)\(65 - 67\)\(68 - 71\)
Frequency (number of days)396\(b\)
Frequency density1\(a\)21.5
  1. Find the values of \(a\) and \(b\).
  2. Draw a histogram to represent Robert's times.
    \includegraphics[max width=\textwidth, alt={}]{34ae4f06-d485-4138-82d8-902b70f08995-04_206_100_1516_441}"□\(\_\_\_\_\)□□ □\includegraphics[max width=\textwidth, alt={}]{34ae4f06-d485-4138-82d8-902b70f08995-04_204_28_1518_1197}\(\_\_\_\_\)□
CAIE S1 Specimen Q4
4
  1. Amy measured her pulse rate while resting, \(x\) beats per minute, at the same time each day on 30 days. The results are summarised below. $$\Sigma ( x - 80 ) = - 147 \quad \Sigma ( x - 80 ) ^ { 2 } = 952$$ Find the mean and standard deviation of Amy's pulse rate.
  2. Amy's friend Marok measured her pulse rate every day after running for half an hour. Marok's pulse rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming that pulse rates have a normal distribution, find what proportion of Marok's pulse rates, after running for half an hour, were above 160 beats per minute.
CAIE S1 Specimen Q5
5
  1. Find the number of ways in which all nine letters of the word TENNESSEE can be arranged
    1. if all the letters E are together,
    2. if the T is at one end and there is an S at the other end.
  2. Four letters are selected from the nine letters of the word VENEZUELA. Find the number of possible selections which contain exactly one E .
CAIE S1 Specimen Q6
6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.
  1. Draw a fully labelled tree diagram to illustrate this situation.
  2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Complete the following table to show the probability distribution of \(X\).
    \(x\)0123
    \(\mathrm { P } ( X = x )\)0.24
  3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.
CAIE S1 Specimen Q7
7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.
  1. Find the probability of throwing a 3 .
    \includegraphics[max width=\textwidth, alt={}, center]{34ae4f06-d485-4138-82d8-902b70f08995-10_51_1563_495_331}
  2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
  3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.
CAIE S1 2010 November Q1
1 Name the distribution and suggest suitable numerical parameters that you could use to model the weights in kilograms of female 18-year-old students.
CAIE S1 2010 November Q2
2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.
  1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
  2. Find \(\mathrm { E } ( X )\).
CAIE S1 2010 November Q3
3 It was found that \(68 \%\) of the passengers on a train used a cell phone during their train journey. Of those using a cell phone, \(70 \%\) were under 30 years old, \(25 \%\) were between 30 and 65 years old and the rest were over 65 years old. Of those not using a cell phone, \(26 \%\) were under 30 years old and \(64 \%\) were over 65 years old.
  1. Draw a tree diagram to represent this information, giving all probabilities as decimals.
  2. Given that one of the passengers is 45 years old, find the probability of this passenger using a cell phone during the journey.
CAIE S1 2010 November Q4
4 Delip measured the speeds, \(x \mathrm {~km}\) per hour, of 70 cars on a road where the speed limit is 60 km per hour. His results are summarised by \(\Sigma ( x - 60 ) = 245\).
  1. Calculate the mean speed of these 70 cars. His friend Sachim used values of \(( x - 50 )\) to calculate the mean.
  2. Find \(\Sigma ( x - 50 )\).
  3. The standard deviation of the speeds is 10.6 km per hour. Calculate \(\Sigma ( x - 50 ) ^ { 2 }\).
CAIE S1 2010 November Q5
5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower.
\includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}
  1. Copy and complete the following frequency table for the data.
    Time \(( t\) minutes \()\)\(2 < t \leqslant 4\)\(4 < t \leqslant 6\)\(6 < t \leqslant 7\)\(7 < t \leqslant 8\)\(8 < t \leqslant 10\)\(10 < t \leqslant 16\)
    Frequency
  2. Calculate an estimate of the mean time to take a shower.
  3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.
CAIE S1 2010 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-4_387_899_255_623} A small aeroplane has 14 seats for passengers. The seats are arranged in 4 rows of 3 seats and a back row of 2 seats (see diagram). 12 passengers board the aeroplane.
  1. How many possible seating arrangements are there for the 12 passengers? Give your answer correct to 3 significant figures. These 12 passengers consist of 2 married couples (Mr and Mrs Lin and Mr and Mrs Brown), 5 students and 3 business people.
  2. The 3 business people sit in the front row. The 5 students each sit at a window seat. Mr and Mrs Lin sit in the same row on the same side of the aisle. Mr and Mrs Brown sit in another row on the same side of the aisle. How many possible seating arrangements are there?
  3. If, instead, the 12 passengers are seated randomly, find the probability that Mrs Lin sits directly behind a student and Mrs Brown sits in the front row.
CAIE S1 2010 November Q7
7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.
  1. Find the standard deviation of the times spent by people visiting this dentist.
  2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
  3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
  4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE S1 2011 November Q1
1 When a butternut squash seed is sown the probability that it will germinate is 0.86 , independently of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the probability that more than 210 germinate.
CAIE S1 2011 November Q2
2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .
  1. Find the standard deviation of \(x\).
  2. Find \(\Sigma x ^ { 2 }\).
CAIE S1 2011 November Q3
3 A team of 4 is to be randomly chosen from 3 boys and 5 girls. The random variable \(X\) is the number of girls in the team.
  1. Draw up a probability distribution table for \(X\).
  2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 2 }\), calculate \(\operatorname { Var } ( X )\).
CAIE S1 2011 November Q4
4 The marks of the pupils in a certain class in a History examination are as follows. $$\begin{array} { l l l l l l l l l l l l l } 28 & 33 & 55 & 38 & 42 & 39 & 27 & 48 & 51 & 37 & 57 & 49 & 33 \end{array}$$ The marks of the pupils in a Physics examination are summarised as follows.
Lower quartile: 28 , Median: 39, Upper quartile: 67.
The lowest mark was 17 and the highest mark was 74 .
  1. Draw box-and-whisker plots in a single diagram on graph paper to illustrate the marks for History and Physics.
  2. State one difference, which can be seen from the diagram, between the marks for History and Physics.
CAIE S1 2011 November Q5
5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.
  1. Find the standard deviation of the weights of the letters.
  2. Find the probability that a randomly chosen letter weighs more than 13 g .
  3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.
CAIE S1 2011 November Q6
6
  1. Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be arranged
    1. if there are no restrictions,
    2. if the 4 vowels \(\mathrm { A } , \mathrm { E } , \mathrm { E } , \mathrm { I }\) must all be together.
    1. 4 astronauts are chosen from a certain number of candidates. If order of choosing is not taken into account, the number of ways the astronauts can be chosen is 3876 . How many ways are there if order of choosing is taken into account?
    2. 4 astronauts are chosen to go on a mission. Each of these astronauts can take 3 personal possessions with him. How many different ways can these 12 possessions be arranged in a row if each astronaut's possessions are kept together?
CAIE S1 2011 November Q7
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.
  1. Find the probability that exactly two of the selected balls have the same number.
  2. Given that exactly two of the selected balls have the same number, find the probability that they are both numbered 2 .
  3. Event \(X\) is 'exactly two of the selected balls have the same number'. Event \(Y\) is 'the ball selected from bag \(A\) has number 2'. Showing your working, determine whether events \(X\) and \(Y\) are independent or not.
CAIE S1 2011 November Q1
1 The following are the times, in minutes, taken by 11 runners to complete a 10 km run.
\(\begin{array} { l l l l l l l l l l l } 48.3 & 55.2 & 59.9 & 67.7 & 60.5 & 75.6 & 62.5 & 57.4 & 53.4 & 49.2 & 64.1 \end{array}\)
Find the mean and standard deviation of these times.