Questions S1 - A-Level Maths

1. Calculate the equation of the least squares regression line of $y$ on $x$.
2. Interpret in context the value for the gradient of your line.
3. Comment on the value for the intercept with the $y$-axis of your line.
The tyre manufacturer states that, when one of these new tyres is fitted to the wheel of a car and then inflated to 265 kPa , a suitable regression equation is of the form $$y = 265 + b x$$ The manufacturer also states that, as the car is used, the tyre pressure will decrease at twice the rate of that found in part (a).
1. Suggest a suitable value for $b$.
2. One of these new tyres is fitted to the wheel of a car and inflated to 265 kPa . The car is then used for 8 months, after which the tyre pressure is checked for the first time. Show that, accepting the manufacturer's statements, the tyre pressure can be expected to have fallen below its minimum safety value of 220 kPa .
  (2 marks)

AQA S1 2006 June Q4

7 marks Moderate -0.3

4 The weights of packets of sultanas may be assumed to be normally distributed with a standard deviation of 6 grams. The weights of a random sample of 10 packets were as follows: $\begin{array} { l l l l l l l l l l } 498 & 496 & 499 & 511 & 503 & 505 & 510 & 509 & 513 & 508 \end{array}$

1. Construct a $99 \%$ confidence interval for the mean weight of packets of sultanas, giving the limits to one decimal place.
2. State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.
3. On each packet it states 'Contents 500 grams'. Comment on this statement using both the given sample and your confidence interval.
Given that the mean weight of all packets of sultanas is 500 grams, state the probability that a 99\% confidence interval for the mean, calculated from a random sample of packets, will not contain 500 grams.

AQA S1 2006 June Q5

17 marks Standard +0.3

5 Kirk and Les regularly play each other at darts.

The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game. Find the probability that, in a match of 15 games, Kirk wins:
1. exactly 5 games;
2. fewer than half of the games;
3. more than 2 but fewer than 7 games.
Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
1. Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
2. To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results: $$\begin{array} { l l l l l l l l l l } 8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5 \end{array}$$ Calculate the mean and standard deviation of these values.
3. Hence comment on the validity of Kirk's claim.

AQA S1 2006 June Q6

Easy -1.3

6 A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of children living in these houses are shown in the table.

\multirow{2}{*}{}	Number of children
	None	One	Two	At least three	Total
Detached house	24	32	41	23	120
Semi-detached house	40	37	88	35	200
Total	64	69	129	58	320

A house on the estate is selected at random. $D$ denotes the event 'the house is detached'. $R$ denotes the event 'no children live in the house'. $S$ denotes the event 'one child lives in the house'. $T$ denotes the event 'two children live in the house'.
( $D ^ { \prime }$ denotes the event 'not $D$ '.)

Find:
1. $\mathrm { P } ( D )$;
2. $\quad \mathrm { P } ( D \cap R )$;
3. $\quad \mathrm { P } ( D \cup T )$;
4. $\mathrm { P } ( D \mid R )$;
5. $\mathrm { P } \left( R \mid D ^ { \prime } \right)$.
1. Name two of the events $D , R , S$ and $T$ that are mutually exclusive.
2. Determine whether the events $D$ and $R$ are independent. Justify your answer.
Define, in the context of this question, the event:
1. $D ^ { \prime } \cup T$;
2. $D \cap ( R \cup S )$.

AQA S1 2015 June Q1

6 marks Easy -1.2

1 The number of passengers getting off the 11.45 am train at a railway station on each of 35 days is summarised as follows.

AQA S1 2015 June Q2

10 marks Moderate -0.8

2 The length of aluminium baking foil on a roll may be modelled by a normal distribution with mean 91 metres and standard deviation 0.8 metres.

Determine the probability that the length of foil on a particular roll is:
1. less than 90 metres;
2. not exactly 90 metres;
3. between 91 metres and 92.5 metres.
The length of cling film on a roll may also be modelled by a normal distribution but with mean 153 metres and standard deviation $\sigma$ metres. It is required that $1 \%$ of rolls of cling film should have a length less than 150 metres.
Find the value of $\sigma$ that is needed to satisfy this requirement.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-04_1526_1714_1181_153}

AQA S1 2015 June Q3

11 marks Moderate -0.5

3 Fourteen candidates each sat two test papers, Paper 1 and Paper 2, on the same day. The marks, out of a total of 50, achieved by the students on each paper are shown in the table.

AQA S1 2015 June Q4

15 marks Moderate -0.8

Chris shops at his local store on his way to and from work every Friday.
The event that he buys a morning newspaper is denoted by $M$, and the event that he buys an evening newspaper is denoted by $E$. On any one Friday, Chris may buy neither, exactly one or both of these newspapers.
1. Complete the table of probabilities, printed on the opposite page, where $M ^ { \prime }$ and $E ^ { \prime }$ denote the events 'not $M$ ' and 'not $E$ ' respectively.
2. Hence, or otherwise, find the probability that, on any given Friday, Chris buys exactly one newspaper.
3. Give a numerical justification for the following statement.
  'The events $M$ and $E$ are not mutually exclusive.'
The event that Chris buys a morning newspaper on Saturday is denoted by $S$, and the event that he buys a morning newspaper on the following day, Sunday, is denoted by $T$. The event that he buys a morning newspaper on both Saturday and Sunday is denoted by $S \cap T$. Each combination of the events $S$ and $T$ is independent of any combination of the events $M$ and $E$. However, the events $S$ and $T$ are not independent, with $$\mathrm { P } ( S ) = 0.85 , \quad \mathrm { P } ( T \mid S ) = 0.20 \quad \text { and } \quad \mathrm { P } \left( T \mid S ^ { \prime } \right) = 0.75$$ Find the probability that, on a particular Friday, Saturday and Sunday, Chris buys:
1. all four newspapers;
2. none of the four newspapers.
1. State, as briefly as possible, in the context of the question, the event that is denoted by $M \cap E ^ { \prime } \cap S \cap T ^ { \prime }$.
2. Calculate the value of $\mathrm { P } \left( M \cap E ^ { \prime } \cap S \cap T ^ { \prime } \right)$. \section*{Answer space for question 4}
  1. (i)
    \cline { 2 - 4 } \multicolumn{1}{c|}{} $\boldsymbol { M }$ $\boldsymbol { M } ^ { \prime }$ Total
    $\boldsymbol { E }$ 0.16 0.28
    $\boldsymbol { E } ^ { \prime }$
    Total 0.60 1.00
    
    \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-11_2050_1707_687_153}

AQA S1 2015 June Q5

11 marks Moderate -0.8

5 The table shows the number of customers, $x$, and the takings, $\pounds y$, recorded to the nearest $\pounds 10$, at a local butcher's shop on each of 10 randomly selected weekdays.

$\boldsymbol { x }$	86	60	65	46	71	93	56	81	75	57
$\boldsymbol { y }$	940	790	620	530	770	1050	690	780	860	550

The first 6 pairs of data values in this table are plotted on the scatter diagram shown on the opposite page. Plot the final 4 pairs of data values on the scatter diagram.
1. Calculate the equation of the least squares regression line in the form $y = a + b x$ and draw your line on the scatter diagram.
2. Interpret your value for $b$ in the context of the question.
3. State why your value for $a$ has no practical interpretation.
Estimate, to the nearest $\pounds 10$, the shop's takings when the number of customers is 50 .
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-14_1255_1705_1448_155}
Butcher's shop \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Answer space for question 5} \includegraphics[alt={},max width=\textwidth]{4c679380-894f-4d36-aec8-296b662058e2-15_2335_1760_372_100}
\end{figure}

AQA S1 2015 June Q6

12 marks Moderate -0.8

6 Customers at a supermarket can pay at a checkout either by cash, debit card or credit card.

The probability that a customer pays by cash is 0.22 . Calculate the probability that exactly 2 customers from a random sample of 24 customers pay by cash.
The probability that a customer pays by debit card is 0.45 . Determine the probability that the number of customers who pay by debit card from a random sample of $\mathbf { 4 0 }$ customers is:
1. fewer than 20 ;
2. more than 15 ;
3. at least 12 but at most 24 .
The random variable $W$ denotes the number of customers who pay by credit card from a random sample of $\mathbf { 2 0 0 }$ customers. Calculate values for the mean and the variance of $W$.
[0pt] [3 marks]

AQA S1 2015 June Q7

10 marks Moderate -0.3

A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
1. Construct a $\mathbf { 9 6 \% }$ confidence interval for the mean weight of a bag of pears.
2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
  [0pt] [2 marks]

AQA S1 2015 June Q1

4 marks Moderate -0.8

1
The table shows the annual gas consumption, $x \mathrm { kWh }$, and the annual electricity consumption, $y \mathrm { kWh }$, for a sample of 10 bungalows of similar size and occupancy.

$\boldsymbol { x }$	21371	18521	15222	17312	19854	23561	20738	22111	17897	24523
$\boldsymbol { y }$	2281	2327	2221	2378	2787	2856	3078	2647	2566	2559

$$S _ { x x } = 76581640 \quad S _ { y y } = 694250 \quad S _ { x y } = 3629670$$

Calculate the value of $r _ { x y }$, the product moment correlation coefficient between $x$ and $y$.
Interpret your value of $r _ { x y }$ in the context of this question.

AQA S1 2015 June Q2

6 marks Easy -1.2

2 The table summarises the diameters, $d$ millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.

AQA S1 2015 June Q3

13 marks Moderate -0.8

3 A ferry sails once each day from port D to port A. The ferry departs from D on time or late but never early. However, the ferry can arrive at A early, on time or late. The probabilities for some combined events of departing from $D$ and arriving at $A$ are shown in the table below.

Complete the table.
Write down the probability that, on a particular day, the ferry:
1. both departs and arrives on time;
2. departs late.
Find the probability that, on a particular day, the ferry:
1. arrives late, given that it departed late;
2. does not arrive late, given that it departed on time.
On three particular days, the ferry departs from port D on time. Find the probability that, on these three days, the ferry arrives at port A early once, on time once and late once. Give your answer to three decimal places.
[0pt] [4 marks]
1. \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Answer space for question 3}
  \multirow{2}{*}{} Arrive at A
  Early On time Late Total
  \multirow{2}{*}{Depart from D} On time 0.16 0.56 0.08
  Late
  Total 0.22 0.65 1.00
  \end{table}

AQA S1 2015 June Q4

15 marks Moderate -0.3

4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, $x _ { i }$, and the time taken, $y _ { i }$ hours. His results are shown in the table.

Roof $( \boldsymbol { i } )$	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$	$\mathbf { 9 }$	$\mathbf { 1 0 }$	$\mathbf { 1 1 }$
$\boldsymbol { x } _ { \boldsymbol { i } }$	8	11	14	14	16	20	22	22	25	27	30
$\boldsymbol { y } _ { \boldsymbol { i } }$	5.0	5.2	6.3	7.2	8.0	8.8	10.6	11.0	11.8	12.1	13.0

The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page. Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
1. Calculate the equation of the least squares regression line of $y _ { i }$ on $x _ { i }$, and draw your line on the scatter diagram.
2. Interpret your values for the gradient and for the intercept of this regression line.
Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
Given that $r _ { i }$ denotes the residual for the point representing roof $i$ :
1. calculate the value of $r _ { 6 }$;
2. state why the value of $\sum _ { i = 1 } ^ { 11 } r _ { i }$ gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
  [0pt] [1 mark]
  \section*{Answer space for question 4}
  \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}

AQA S1 2015 June Q5

12 marks Moderate -0.3

Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, $X$ metres, of a roll may be modelled by a normal distribution with mean 1.81 and standard deviation 0.08 . Determine the probability that a randomly selected roll has length:
1. less than 1.90 m ;
2. greater than 1.85 m ;
3. between 1.81 m and 1.85 m .
Plastic lawn edging is supplied in 9 m length rolls. The actual length, $Y$ metres, of a roll may be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. An analysis of a batch of rolls, selected at random, showed that $$\mathrm { P } ( Y < 9.25 ) = 0.88$$
1. Use this probability to find the value of $z$ such that $$9.25 - \mu = z \times \sigma$$ where $z$ is a value of $Z \sim \mathrm {~N} ( 0,1 )$.
2. Given also that $$\mathrm { P } ( Y > 8.75 ) = 0.975$$ find values for $\mu$ and $\sigma$.

AQA S1 2015 June Q6

13 marks Moderate -0.8

In a particular country, 35 per cent of the population is estimated to have at least one mobile phone. A sample of 40 people is selected from the population.
Use the distribution $\mathrm { B } ( 40,0.35 )$ to estimate the probability that the number of people in the sample that have at least one mobile phone is:
1. at most 15 ;
2. more than 10 ;
3. more than 12 but fewer than 18 ;
4. exactly equal to the mean of the distribution.
In the same country, 70 per cent of households have a landline telephone connection. A sample of 50 households is selected from all households in the country.
Stating a necessary condition regarding this selection, estimate the probability that fewer than 30 households have a landline telephone connection.
[0pt] [4 marks]

AQA S1 2015 June Q7

12 marks Moderate -0.3

The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, $x$ grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
1. Construct a $99 \%$ confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
  Claim 2: All pouches contain more than 300 grams of dry cat food.
  [0pt] [4 marks]
  \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}

OCR S1 Q3

8 marks Moderate -0.8

3 In a supermarket the proportion of shoppers who buy washing powder is denoted by $p$. 16 shoppers are selected at random.

Given that $p = 0.35$, use tables to find the probability that the number of shoppers who buy washing powder is
1. at least 8,
2. between 4 and 9 inclusive.
3. Given instead that $p = 0.38$, find the probability that the number of shoppers who buy washing powder is exactly 6 . \section*{June 2005}

OCR S1 Q4

8 marks Moderate -0.3

4 The table shows the latitude, $x$ (in degrees correct to 3 significant figures), and the average rainfall $y$ (in cm correct to 3 significant figures) of five European cities.

City	$x$	$y$
Berlin	52.5	58.2
Bucharest	44.4	58.7
Moscow	55.8	53.3
St Petersburg	60.0	47.8
Warsaw	52.3	56.6

$$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

Calculate the product moment correlation coefficient.
The values of $y$ in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54. State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
It is required to estimate the annual rainfall at Bergen, where $x = 60.4$. Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate. \section*{June 2005}

OCR S1 Q5

13 marks Moderate -0.8

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-04_1335_1319_404_413} Use the curve to estimate

the interquartile range of the marks,
$x$, if $40 \%$ of the candidates scored more than $x$ marks,
the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored $35,36,37 , \ldots , 55$.
What does this information suggest about the estimate of the interquartile range found in part (i)? \section*{June 2005}

OCR S1 Q6

13 marks Standard +0.3

6 Two bags contain coloured discs. At first, bag $P$ contains 2 red discs and 2 green discs, and bag $Q$ contains 3 red discs and 1 green disc. A disc is chosen at random from bag $P$, its colour is noted and it is placed in bag $Q$. A disc is then chosen at random from bag $Q$, its colour is noted and it is placed in bag $P$. A disc is then chosen at random from bag $P$. The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-05_858_980_573_585}

Write down the values of $a , b , c , d , e$ and $f$. The total number of red discs chosen, out of 3, is denoted by $R$. The table shows the probability distribution of $R$.
$r$ 0 1 2 3
$\mathrm { P } ( R = r )$ $\frac { 1 } { 10 }$ $k$ $\frac { 9 } { 20 }$ $\frac { 1 } { 5 }$
Show how to obtain the value $\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }$.
Find the value of $k$.
Calculate the mean and variance of $R$.

OCR S1 Q7

14 marks Moderate -0.3

7 A committee of 7 people is to be chosen at random from 18 volunteers.

In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
include exactly 5 people from Worcester,
include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
Jenny estimates that her chances of success are as follows.
- The probability that she will pass on her first attempt is $\frac { 2 } { 3 }$.
- If she fails on her first attempt, the probability that she will pass on her second attempt is $\frac { 3 } { 4 }$. Calculate the probability that Jenny will pass.
- John estimates that his chances of success are as follows.
- The probability that he will pass on his first attempt is $\frac { 2 } { 3 }$.
- Overall, the probability that he will pass is $\frac { 5 } { 6 }$.
Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, $h \mathrm {~cm}$, was measured and the value of ( $h - 100$ ) was recorded. The mean and standard deviation of $( h - 100 )$ were found to be 24.5 and 4.8 respectively.
Write down the mean and standard deviation of $h$. The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
Describe briefly how the heights of the second group of plants compare with the first.
Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let $X$ be the number of tins that Mr Kendall opens.
Show that $\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }$.
The probability distribution of $X$ is given in the table below.
$x$ 2 3 4 5
$\mathrm { P } ( X = x )$ $\frac { 1 } { 10 }$ $\frac { 1 } { 5 }$ $\frac { 3 } { 10 }$ $\frac { 2 } { 5 }$
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR S1 Q8

13 marks Moderate -0.3

8 The table shows the population, $x$ million, of each of nine countries in Western Europe together with the population, $y$ million, of its capital city.

	Germany	United Kingdom	France	Italy	Spain	The Netherlands	Portugal	Austria	Switzerland
$x$	82.1	59.2	59.1	56.7	39.2	15.9	9.9	8.1	7.3
$y$	3.5	7.0	9.0	2.7	2.9	0.8	0.7	1.6	0.1

$$\left[ n = 9 , \Sigma x = 337.5 , \Sigma x ^ { 2 } = 18959.11 , \Sigma y = 28.3 , \Sigma y ^ { 2 } = 161.65 , \Sigma x y = 1533.76 . \right]$$

(a) Calculate Spearman's rank correlation coefficient, $r _ { s }$.
(b) Explain what your answer indicates about the populations of these countries and their capital cities.
Calculate the product moment correlation coefficient, $r$. The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-09_936_881_1162_632}
By considering the diagram, state the effect on the value of the product moment correlation coefficient, $r$, if the data for France and the United Kingdom were removed from the calculation.
In a certain country in Africa, most people live in remote areas and hence the population of the country is unknown. However, the population of the capital city is known to be approximately 1 million. An official suggests that the population of this country could be estimated by using a regression line drawn on the above scatter diagram.
(a) State, with a reason, whether the regression line of $y$ on $x$ or the regression line of $x$ on $y$ would need to be used.
(b) Comment on the reliability of such an estimate in this situation. 1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$
State, with a reason, whether the correlation between $x$ and $y$ is negative or positive.
Neither variable is controlled. Calculate an estimate of the value of $x$ when $y = 7.0$.
Find the values of $\bar { x }$ and $\bar { y }$. 2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that
the second disc is black, given that the first disc was black,
the second disc is black,
the two discs are of different colours. 3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.
How many different arrangements of the letters are possible?
In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
Find the probability that at least one of these 2 cards has D printed on it. 4
The random variable $X$ has the distribution $\mathrm { B } ( 25,0.2 )$. Using the tables of cumulative binomial probabilities, or otherwise, find $\mathrm { P } ( X \geqslant 5 )$.
The random variable $Y$ has the distribution $\mathrm { B } ( 10,0.27 )$. Find $\mathrm { P } ( Y = 3 )$.
The random variable $Z$ has the distribution $B ( n , 0.27 )$. Find the smallest value of $n$ such that $\mathrm { P } ( Z \geqslant 1 ) > 0.95$. 5 The probability distribution of a discrete random variable, $X$, is given in the table.
$x$ 0 1 2 3
$\mathrm { P } ( X = x )$ $\frac { 1 } { 3 }$ $\frac { 1 } { 4 }$ $p$ $q$
It is given that the expectation, $\mathrm { E } ( X )$, is $1 \frac { 1 } { 4 }$.
Calculate the values of $p$ and $q$.
Calculate the standard deviation of $X$.

OCR MEI S1 Q2

Easy -1.2

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

Number correct	1	2	3
Frequency	1	2	3

Draw a vertical line chart to illustrate the data.
State the type of skewness shown by your diagram.
Calculate the mean and the mean squared deviation of the data.
How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?

\(r\)	0	1	2	3
\(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

\(x\)	2	3	4	5
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)

\(x\)	0	1	2	3
\(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(q\)

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(\boldsymbol { M }\)	\(\boldsymbol { M } ^ { \prime }\)	Total
\(\boldsymbol { E }\)	0.16		0.28
\(\boldsymbol { E } ^ { \prime }\)
Total		0.60	1.00

\(\boldsymbol { x }\)	86	60	65	46	71	93	56	81	75	57
\(\boldsymbol { y }\)	940	790	620	530	770	1050	690	780	860	550

\multirow{2}{*}{}		Arrive at A
		Early	On time	Late	Total
\multirow{2}{*}{Depart from D}	On time	0.16	0.56	0.08
	Late
	Total	0.22	0.65		1.00

Roof \(( \boldsymbol { i } )\)	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)	\(\mathbf { 9 }\)	\(\mathbf { 1 0 }\)	\(\mathbf { 1 1 }\)
\(\boldsymbol { x } _ { \boldsymbol { i } }\)	8	11	14	14	16	20	22	22	25	27	30
\(\boldsymbol { y } _ { \boldsymbol { i } }\)	5.0	5.2	6.3	7.2	8.0	8.8	10.6	11.0	11.8	12.1	13.0

City	\(x\)	\(y\)
Berlin	52.5	58.2
Bucharest	44.4	58.7
Moscow	55.8	53.3
St Petersburg	60.0	47.8
Warsaw	52.3	56.6

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