Questions - CAIE S1 - A-Level Maths

CAIE S1 2021 November Q4

6 marks Standard +0.3

4

In how many different ways can the 9 letters of the word TELESCOPE be arranged?
In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?

CAIE S1 2021 November Q5

7 marks Moderate -0.8

5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.

Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
Find the probability that the first wet day in October is 8 October.
For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.

CAIE S1 2021 November Q6

10 marks Moderate -0.8

6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.

Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
$20 \%$ of employees take longer than $t$ minutes to complete the task. Find the value of $t$.
Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.

CAIE S1 2021 November Q8

Easy -1.8

8	MATHEMATICS	9709/52
0	Paper 5 Probability \	Statistics 1	October/November 2021
$\infty$		1 hour 15 minutes
	You must answer on the question paper.
	You will need: List of formulae (MF19)

\section*{INSTRUCTIONS}

Answer all questions.
Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
Write your name, centre number and candidate number in the boxes at the top of the page.
Write your answer to each question in the space provided.
Do not use an erasable pen or correction fluid.
Do not write on any bar codes.
If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown.
You should use a calculator where appropriate.
You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

\section*{INFORMATION}

The total mark for this paper is 50.
The number of marks for each question or part question is shown in brackets [ ].

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.

	Piano	Guitar	Drums
Male	25	44	11
Female	42	38	20

A student at the college is chosen at random.

Find the probability that the student plays the guitar.
Find the probability that the student is male given that the student plays the drums.
Determine whether the events 'the student plays the guitar' and 'the student is female' are independent, justifying your answer.
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?
  3 A bag contains 5 yellow and 4 green marbles. Three marbles are selected at random from the bag, without replacement.
3. 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.
4. Draw up the probability distribution table for $X$.
5. Find $\mathrm { E } ( X )$.
  4
6. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
7. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
  5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
8. Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
9. Find the probability that the first wet day in October is 8 October.
10. For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
  6 The times taken, in minutes, to complete a particular task by employees at a large company are normally distributed with mean 32.2 and standard deviation 9.6.
11. Find the probability that a randomly chosen employee takes more than 28.6 minutes to complete the task.
12. $20 \%$ of employees take longer than $t$ minutes to complete the task. Find the value of $t$.
13. Find the probability that the time taken to complete the task by a randomly chosen employee differs from the mean by less than 15.0 minutes.
  7 The distances, $x \mathrm {~m}$, travelled to school by 140 children were recorded. The results are summarised in the table below.
  Distance, $x \mathrm {~m}$ $x \leqslant 200$ $x \leqslant 300$ $x \leqslant 500$ $x \leqslant 900$ $x \leqslant 1200$ $x \leqslant 1600$
  Cumulative frequency 16 46 88 122 134 140
14. On the grid, draw a cumulative frequency graph to represent these results. \includegraphics[max width=\textwidth, alt={}, center]{93ff111b-0267-4b4b-a41c-64c3307115af-10_1593_1593_701_306}
15. Use your graph to estimate the interquartile range of the distances.
16. Calculate estimates of the mean and standard deviation of the distances.
  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 March Q1

8 marks Moderate -0.8

Each year the total number of hours, $x$, of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

$x$	$30 \leqslant x < 60$	$60 \leqslant x < 90$	$90 \leqslant x < 110$	$110 \leqslant x < 140$	$140 \leqslant x < 180$	$180 \leqslant x \leqslant 240$
Number of years	4	8	14	25	7	2

Draw a cumulative frequency graph to illustrate the data. [3]
Use your graph to estimate the 70th percentile of the data. [2]
Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years. [3]

CAIE S1 2023 March Q2

7 marks Moderate -0.3

Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6. The other three coins are fair. Alisha throws the four coins at the same time. The random variable $X$ denotes the number of heads obtained.

Show that the probability of obtaining exactly one head is 0.225. [3]
Complete the following probability distribution table for $X$. [2]
$x$ 0 1 2 3 4
P($X = x$) 0.05 0.225 0.075
Given that E($X$) = 2.1, find the value of Var($X$). [2]

CAIE S1 2023 March Q3

6 marks Moderate -0.8

80\% of the residents of Kinwawa are in favour of a leisure centre being built in the town. 20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.

Find the probability that more than 17 of these residents are in favour of the leisure centre. [3]
Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre. [1]
Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre. [2]

CAIE S1 2023 March Q4

3 marks Standard +0.3

The probability that it will rain on any given day is $x$. If it is raining, the probability that Aran wears a hat is 0.8 and if it is not raining, the probability that he wears a hat is 0.3. Whether it is raining or not, if Aran wears a hat, the probability that he wears a scarf is 0.4. If he does not wear a hat, the probability that he wears a scarf is 0.1. The probability that on a randomly chosen day it is not raining and Aran is not wearing a hat or a scarf is 0.36. Find the value of $x$. [3]

CAIE S1 2023 March Q5

3 marks Standard +0.8

Marco has four boxes labelled $K$, $L$, $M$ and $N$. He places them in a straight line in the order $K$, $L$, $M$, $N$ with $K$ on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events $A$ and $B$ are defined as follows. $A$: The white marble is in either box $L$ or box $M$. $B$: The red marble is to the left of both the green marble and the yellow marble. Determine whether or not events $A$ and $B$ are independent. [3]

CAIE S1 2023 March Q6

11 marks Standard +0.3

In a cycling event the times taken to complete a course are modelled by a normal distribution with mean 62.3 minutes and standard deviation 8.4 minutes.

Find the probability that a randomly chosen cyclist has a time less than 74 minutes. [2]
Find the probability that 4 randomly chosen cyclists all have times between 50 and 74 minutes. [4]

In a different cycling event, the times can also be modelled by a normal distribution. 23\% of the cyclists have times less than 36 minutes and 10\% of the cyclists have times greater than 54 minutes.

Find estimates for the mean and standard deviation of this distribution. [5]

CAIE S1 2023 March Q7

12 marks Standard +0.3

Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]

Five letters are selected from the 9 letters in the word DELIVERED.

Find the number of different selections if the 5 letters include at least one D and at least one E. [3]

CAIE S1 2002 June Q1

4 marks Easy -1.2

Events $A$ and $B$ are such that $\text{P}(A) = 0.3$, $\text{P}(B) = 0.8$ and $\text{P}(A \text{ and } B) = 0.4$. State, giving a reason in each case, whether events $A$ and $B$ are

independent, [2]
mutually exclusive. [2]

CAIE S1 2002 June Q2

6 marks Easy -1.2

The manager of a company noted the times spent in 80 meetings. The results were as follows.

Time ($t$ minutes)	$0 < t \leq 15$	$15 < t \leq 30$	$30 < t \leq 60$	$60 < t \leq 90$	$90 < t \leq 120$
Number of meetings	4	7	24	38	7

Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]

CAIE S1 2002 June Q3

7 marks Moderate -0.8

A fair cubical die with faces numbered 1, 1, 1, 2, 3, 4 is thrown and the score noted. The area $A$ of a square of side equal to the score is calculated, so, for example, when the score on the die is 3, the value of $A$ is 9.

Draw up a table to show the probability distribution of $A$. [3]
Find $\text{E}(A)$ and $\text{Var}(A)$. [4]

CAIE S1 2002 June Q4

7 marks Moderate -0.8

In a spot check of the speeds $x \text{ km h}^{-1}$ of 30 cars on a motorway, the data were summarised by $\Sigma(x - 110) = -47.2$ and $\Sigma(x - 110)^2 = 5460$. Calculate the mean and standard deviation of these speeds. [4]
On another day the mean speed of cars on the motorway was found to be $107.6 \text{ km h}^{-1}$ and the standard deviation was $13.8 \text{ km h}^{-1}$. Assuming these speeds follow a normal distribution and that the speed limit is $110 \text{ km h}^{-1}$, find what proportion of cars exceed the speed limit. [3]

CAIE S1 2002 June Q5

8 marks Moderate -0.3

The digits of the number 1223678 can be rearranged to give many different 7-digit numbers. Find how many different 7-digit numbers can be made if

there are no restrictions on the order of the digits, [2]
the digits 1, 3, 7 (in any order) are next to each other, [3]
these 7-digit numbers are even. [3]

CAIE S1 2002 June Q6

8 marks Standard +0.3

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, $\text{P}(X > 3.6) = 0.5$ and $\text{P}(X > 2.8) = 0.6554$. Write down the value of $\mu$, and calculate the value of $\sigma$. [4]
If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]

CAIE S1 2002 June Q7

10 marks Moderate -0.3

A garden shop sells polyanthus plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95.
1. Find the number of plants per box. [4]
2. Find the probability that a box contains exactly 12 plants which produce yellow flowers. [2]
Another garden shop sells polyanthus plants in boxes of 100. The shop's advertisement states that the probability of any polyanthus plant producing a pink flower is 0.3. Use a suitable approximation to find the probability that a box contains fewer than 35 plants which produce pink flowers. [4]

CAIE S1 2010 June Q1

5 marks Moderate -0.8

The times in minutes for seven students to become proficient at a new computer game were measured. The results are shown below. $$15 \quad 10 \quad 48 \quad 10 \quad 19 \quad 14 \quad 16$$

Find the mean and standard deviation of these times. [2]
State which of the mean, median or mode you consider would be most appropriate to use as a measure of central tendency to represent the data in this case. [1]
For each of the two measures of average you did not choose in part (ii), give a reason why you consider it inappropriate. [2]

CAIE S1 2010 June Q2

5 marks Moderate -0.8

The lengths of new pencils are normally distributed with mean 11 cm and standard deviation 0.095 cm.

Find the probability that a pencil chosen at random has a length greater than 10.9 cm. [2]
Find the probability that, in a random sample of 6 pencils, at least two have lengths less than 10.9 cm. [3]

CAIE S1 2010 June Q3

6 marks Moderate -0.8

\includegraphics{figure_3} The birth weights of random samples of 900 babies born in country $A$ and 900 babies born in country $B$ are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies. [6]

CAIE S1 2010 June Q4

6 marks Standard +0.3

The random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$.

Given that $5\sigma = 3\mu$, find $\mathrm{P}(X < 2\mu)$. [3]
With a different relationship between $\mu$ and $\sigma$, it is given that $\mathrm{P}(X < \frac{4\mu}{3}) = 0.8524$. Express $\mu$ in terms of $\sigma$. [3]

CAIE S1 2010 June Q5

8 marks Moderate -0.8

Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the numbers on the sides which land face down are noted. Events $Q$ and $R$ are defined as follows. $Q$: the product of the two numbers is 24. $R$: both of the numbers are greater than 8.

Find $\mathrm{P}(Q)$. [2]
Find $\mathrm{P}(R)$. [2]
Are events $Q$ and $R$ exclusive? Justify your answer. [2]
Are events $Q$ and $R$ independent? Justify your answer. [2]

CAIE S1 2010 June Q6

10 marks Moderate -0.3

A small farm has 5 ducks and 2 geese. Four of these birds are to be chosen at random. The random variable $X$ represents the number of geese chosen.

Draw up the probability distribution of $X$. [3]
Show that $\mathrm{E}(X) = \frac{8}{7}$ and calculate $\mathrm{Var}(X)$. [3]
When the farmer's dog is let loose, it chases either the ducks with probability $\frac{3}{5}$ or the geese with probability $\frac{2}{5}$. If the dog chases the ducks there is a probability of $\frac{1}{10}$ that they will attack the dog. If the dog chases the geese there is a probability of $\frac{1}{4}$ that they will attack the dog. Given that the dog is not attacked, find the probability that it was chasing the geese. [4]

CAIE S1 2010 June Q7

10 marks Moderate -0.8

Nine cards, each of a different colour, are to be arranged in a line.

How many different arrangements of the 9 cards are possible? [1]

The 9 cards include a pink card and a green card.

How many different arrangements do not have the pink card next to the green card? [3]

Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.

How many different arrangements in total of 3 cards are possible? [2]
How many of the arrangements of 3 cards in part (iii) contain the pink card? [2]
How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card? [2]

\(x\)	\(30 \leqslant x < 60\)	\(60 \leqslant x < 90\)	\(90 \leqslant x < 110\)	\(110 \leqslant x < 140\)	\(140 \leqslant x < 180\)	\(180 \leqslant x \leqslant 240\)
Number of years	4	8	14	25	7	2

Time (\(t\) minutes)	\(0 < t \leq 15\)	\(15 < t \leq 30\)	\(30 < t \leq 60\)	\(60 < t \leq 90\)	\(90 < t \leq 120\)
Number of meetings	4	7	24	38	7

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CAIE S1 2010 June Q7

Distance, \(x \mathrm {~m}\)	\(x \leqslant 200\)	\(x \leqslant 300\)	\(x \leqslant 500\)	\(x \leqslant 900\)	\(x \leqslant 1200\)	\(x \leqslant 1600\)
Cumulative frequency	16	46	88	122	134	140

\(x\)	0	1	2	3	4
P(\(X = x\))	0.05	0.225			0.075