Questions — CAIE (7659 questions)

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CAIE S1 2019 November Q1
4 marks Easy -1.2
1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$
  1. Find the median and the interquartile range for the data.
  2. Give a disadvantage of using the mean as a measure of the central tendency in this case.
CAIE S1 2019 November Q2
5 marks Moderate -0.3
2 Benju cycles to work each morning and he has two possible routes. He chooses the hilly route with probability 0.4 and the busy route with probability 0.6 . If he chooses the hilly route, the probability that he will be late for work is \(x\) and if he chooses the busy route the probability that he will be late for work is \(2 x\). The probability that Benju is late for work on any day is 0.36 .
  1. Show that \(x = 0.225\).
  2. Given that Benju is not late for work, find the probability that he chooses the hilly route.
CAIE S1 2019 November Q3
6 marks Moderate -0.8
3 The speeds, in \(\mathrm { km } \mathrm { h } ^ { - 1 }\), of 90 cars as they passed a certain marker on a road were recorded, correct to the nearest \(\mathrm { km } \mathrm { h } ^ { - 1 }\). The results are summarised in the following table.
Speed \(\left( \mathrm { km } \mathrm { h } ^ { - 1 } \right)\)\(10 - 29\)\(30 - 39\)\(40 - 49\)\(50 - 59\)\(60 - 89\)
Frequency1024301412
  1. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{5307cf3d-3d3a-441a-83d7-4adad917e168-04_1594_1198_657_516}
  2. Calculate an estimate for the mean speed of these 90 cars as they pass the marker.
CAIE S1 2019 November Q4
8 marks Moderate -0.3
4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.
  1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
  2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]
CAIE S1 2019 November Q5
7 marks Moderate -0.3
5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.
  1. Draw up the probability distribution table for \(X\).
  2. Find \(\operatorname { Var } ( X )\).
CAIE S1 2019 November Q6
9 marks Standard +0.8
6 The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8 .
  1. Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.
  2. Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.
    In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high.
  3. Find the mean and standard deviation of the heights of trees of this type.
CAIE S1 2019 November Q7
11 marks Standard +0.3
7
  1. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that all three Os are together and both Ts are together.
  2. Find the number of different ways in which the 9 letters of the word TOADSTOOL can be arranged so that the Ts are not together.
  3. Find the probability that a randomly chosen arrangement of the 9 letters of the word TOADSTOOL has a T at the beginning and a T at the end.
  4. Five letters are selected from the 9 letters of the word TOADSTOOL. Find the number of different selections if the five letters include at least 2 Os and at least 1 T .
    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 2019 November Q1
3 marks Moderate -0.8
1 There are 300 students at a music college. All students play exactly one of the guitar, the piano or the flute. The numbers of male and female students that play each of the instruments are given in the following table.
GuitarPianoFlute
Female students623543
Male students784042
  1. Find the probability that a randomly chosen student at the college is a male who does not play the piano.
  2. Determine whether the events 'a randomly chosen student is male' and 'a randomly chosen student does not play the piano' are independent, justifying your answer.
CAIE S1 2019 November Q2
5 marks Moderate -0.8
2
  1. How many different arrangements are there of the 9 letters in the word CORRIDORS?
  2. How many different arrangements are there of the 9 letters in the word CORRIDORS in which the first letter is D and the last letter is R or O ?
CAIE S1 2019 November Q3
6 marks Standard +0.3
3 A sports team of 7 people is to be chosen from 6 attackers, 5 defenders and 4 midfielders. The team must include at least 3 attackers, at least 2 defenders and at least 1 midfielder.
  1. In how many different ways can the team of 7 people be chosen?
    The team of 7 that is chosen travels to a match in two cars. A group of 4 travel in one car and a group of 3 travel in the other car.
  2. In how many different ways can the team of 7 be divided into a group of 4 and a group of 3 ?
CAIE S1 2019 November Q4
7 marks Moderate -0.8
4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .
  1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
    120 Mainland students are chosen at random.
  2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.
CAIE S1 2019 November Q5
9 marks Easy -1.8
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
10 marks Moderate -0.3
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
10 marks Moderate -0.3
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
4 marks Moderate -0.5
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
4 marks Moderate -0.3
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
6 marks Easy -1.8
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
7 marks Moderate -0.8
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
9 marks Moderate -0.3
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
9 marks Moderate -0.8
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
11 marks Standard +0.3
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 S2 2023 March Q1
4 marks Moderate -0.8
1 Anita carried out a survey of 140 randomly selected students at her college. She found that 49 of these students watched a TV programme called Bunch.
  1. Calculate an approximate \(98 \%\) confidence interval for the proportion, \(p\), of students at Anita's college who watch Bunch.
    Carlos says that the confidence interval found in (a) is not useful because it is too wide.
  2. Without calculation, explain briefly how Carlos can use the results of Anita's survey to find a narrower confidence interval for \(p\).
CAIE S2 2023 March Q2
13 marks Standard +0.3
2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).
  1. State two assumptions that are required for the Poisson model to be valid in this context.
    1. Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
    2. Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
  3. The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.
CAIE S2 2023 March Q3
8 marks
3 \includegraphics[max width=\textwidth, alt={}, center]{823cc2e5-e408-4b81-ac4d-7e9f584107cc-06_558_1077_260_523} The diagram shows the graph of the probability density function, f, of a random variable \(X\) that takes values between \(x = 0\) and \(x = 3\) only. The graph is symmetrical about the line \(x = 1.5\).
  1. It is given that \(\mathrm { P } ( X < 0.6 ) = a\) and \(\mathrm { P } ( 0.6 < X < 1.2 ) = b\). Find \(\mathrm { P } ( 0.6 < X < 1.8 )\) in terms of \(a\) and \(b\).
  2. It is now given that the equation of the probability density function of \(X\) is $$f ( x ) = \begin{cases} k x ^ { 2 } ( 3 - x ) ^ { 2 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 10 } { 81 }\).
    2. Find \(\operatorname { Var } ( X )\).
CAIE S2 2023 March Q4
5 marks Standard +0.3
4 The number of accidents per 3-month period on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) has been 5.7. Following some changes to the road, the council carries out a hypothesis test to determine whether the value of \(\lambda\) has decreased. If there are fewer than 3 accidents in a randomly chosen 3 -month period, the council will conclude that the value of \(\lambda\) has decreased.
  1. Find the probability of a Type I error.
  2. Find the probability of a Type II error if the mean number of accidents per 3-month period is now actually 0.9 .