Questions - OCR MEI S1 - A-Level Maths

Calculate the mean and standard deviation of $x$.
The weights, $y$ grams, of the potatoes after they have been peeled are given by the formula $y = 0.9 x - 15$. Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

OCR MEI S1 2013 June Q2

2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.

Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.

OCR MEI S1 2013 June Q3

3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average $10 \%$ of bags are underweight.

Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.

OCR MEI S1 2013 June Q4

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain $\pounds 5$ and the other three each contain $\pounds 1000$. Since the envelopes are identical on the outside, he chooses three of them at random. Let $\pounds X$ be the total amount of money that he receives in prize money.

Show that $\mathrm { P } ( X = 15 ) = 0.05$. The probability distribution of $X$ is given in the table below.
$r$ 15 1010 2005 3000
$\mathrm { P } ( X = r )$ 0.05 0.45 0.45 0.05
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2013 June Q5

5 A researcher is investigating whether people can identify whether a glass of water they are given is bottled water or tap water. She suspects that people do no better than they would by guessing. Twenty people are selected at random; thirteen make a correct identification. She carries out a hypothesis test.

Explain why the null hypothesis should be $p = 0.5$, where $p$ represents the probability that a randomly selected person makes a correct identification.
Briefly explain why she uses an alternative hypothesis of $p > 0.5$.
Complete the test at the $5 \%$ significance level.

OCR MEI S1 2013 June Q6

6 The birth weights in kilograms of 25 female babies are shown below, in ascending order.

1.39	2.50	2.68	2.76	2.82	2.82	2.84	3.03	3.06	3.16	3.16	3.24	3.32
3.36	3.40	3.54	3.56	3.56	3.70	3.72	3.72	3.84	4.02	4.24	4.34

Find the median and interquartile range of these data.
Draw a box and whisker plot to illustrate the data.
Show that there is exactly one outlier. Discuss whether this outlier should be removed from the data. The cumulative frequency curve below illustrates the birth weights of 200 male babies.
\includegraphics[max width=\textwidth, alt={}, center]{6b886da6-3fb8-4b4c-b572-f4b770ae5a4c-3_929_1569_1450_248}
Find the median and interquartile range of the birth weights of the male babies.
Compare the weights of the female and male babies.
Two of these male babies are chosen at random. Calculate an estimate of the probability that both of these babies weigh more than any of the female babies.

OCR MEI S1 2013 June Q7

7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.

Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
Find the probability that
(A) she hits the bull's-eye with at least one of her first three darts,
(B) she hits the bull's-eye with exactly one of her first three darts.
Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
Find the probability that she hits the bull's-eye three times in total.

OCR MEI S1 2014 June Q1

1 The ages, $x$ years, of the senior members of a running club are summarised in the table below.

Age $( x )$	$20 \leqslant x < 30$	$30 \leqslant x < 40$	$40 \leqslant x < 50$	$50 \leqslant x < 60$	$60 \leqslant x < 70$	$70 \leqslant x < 80$	$80 \leqslant x < 90$
Frequency	10	30	42	23	9	5	1

Draw a cumulative frequency diagram to illustrate the data.
Use your diagram to estimate the median and interquartile range of the data.

OCR MEI S1 2014 June Q2

2 Candidates applying for jobs in a large company take an aptitude test, as a result of which they are either accepted, rejected or retested, with probabilities $0.2,0.5$ and 0.3 respectively. When a candidate is retested for the first time, the three possible outcomes and their probabilities remain the same as for the original test. When a candidate is retested for the second time there are just two possible outcomes, accepted or rejected, with probabilities 0.4 and 0.6 respectively.

Draw a probability tree diagram to illustrate the outcomes.
Find the probability that a randomly selected candidate is accepted.
Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.

OCR MEI S1 2014 June Q3

3 Each weekday, Marta travels to school by bus. Sometimes she arrives late.

$L$ is the event that Marta arrives late.
$R$ is the event that it is raining.

You are given that $\mathrm { P } ( L ) = 0.15 , \mathrm { P } ( R ) = 0.22$ and $\mathrm { P } ( L \mid R ) = 0.45$.

Use this information to show that the events $L$ and $R$ are not independent.
Find $\mathrm { P } ( L \cap R )$.
Draw a Venn diagram showing the events $L$ and $R$, and fill in the probability corresponding to each of the four regions of your diagram.

OCR MEI S1 2014 June Q4

4 There are 16 girls and 14 boys in a class. Four of them are to be selected to form a quiz team. The team is to be selected at random.

Find the probability that all 4 members of the team will be girls.
Find the probability that the team will contain at least one girl and at least one boy.

OCR MEI S1 2014 June Q5

5 The probability distribution of the random variable $X$ is given by the formula $$\mathrm { P } ( X = r ) = k + 0.01 r ^ { 2 } \text { for } r = 1,2,3,4,5 \text {. }$$

Show that $k = 0.09$. Using this value of $k$, display the probability distribution of $X$ in a table.
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2014 June Q7

7 It is known that on average $85 \%$ of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.

(A) Find the probability that exactly 12 germinate.
(B) Find the probability that fewer than 12 germinate. The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting $n$ of these seeds and sowing them. He then carries out a hypothesis test at the $1 \%$ significance level to investigate whether he is correct.
Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
In a trial with $n = 20$, Ramesh finds that 13 seeds germinate. Carry out the test.
Suppose instead that Ramesh conducts the trial with $n = 50$, and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
If $n$ is small, there is no point in carrying out the test at the $1 \%$ significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of $n$ for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.

OCR MEI S1 2015 June Q2

2 A survey is being carried out into the sports viewing habits of people in a particular area. As part of the survey, 250 people are asked which of the following sports they have watched on television in the past month.

Football
Cycling
Rugby

The numbers of people who have watched these sports are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-2_723_917_1183_575} One of the people is selected at random.

Find the probability that this person has in the past month
(A) watched cycling but not football,
(B) watched either one or two of the three sports.
Given that this person has watched cycling, find the probability that this person has not watched football.

OCR MEI S1 2015 June Q3

3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

Find the probability that neither card is an ace.
This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

OCR MEI S1 2015 June Q4

4 A rugby team of 15 people is to be selected from a squad of 25 players.

How many different teams are possible?
In fact the team has to consist of 8 forwards and 7 backs. If 13 of the squad are forwards and the other 12 are backs, how many different teams are now possible?
Find the probability that, if the team is selected at random from the squad of 25 players, it contains the correct numbers of forwards and backs.

OCR MEI S1 2015 June Q5

5 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$

Construct a sorted stem and leaf diagram to represent these data.
State the type of skewness suggested by your stem and leaf diagram.
For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.

OCR MEI S1 2015 June Q6

6 Three fair six-sided dice are thrown. The random variable $X$ represents the highest of the three scores on the dice.

Show that $\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }$. The table shows the probability distribution of $X$.
$r$ 1 2 3 4 5 6
$\mathrm { P } ( X = r )$ $\frac { 1 } { 216 }$ $\frac { 7 } { 216 }$ $\frac { 19 } { 216 }$ $\frac { 37 } { 216 }$ $\frac { 61 } { 216 }$ $\frac { 91 } { 216 }$
Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.

OCR MEI S1 2015 June Q7

7 A drug for treating a particular minor illness cures, on average, $78 \%$ of patients. Twenty people with this minor illness are selected at random and treated with the drug.

$( A )$ Find the probability that exactly 19 patients are cured.
(B) Find the probability that at most 18 patients are cured.
$( C )$ Find the expected number of patients who are cured.
A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the $1 \%$ significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
If the researchers had chosen to carry out the hypothesis test at the $5 \%$ significance level, what would the result have been? Justify your answer.

OCR MEI S1 2015 June Q8

8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box.
\includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}

Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
Calculate the probabilities of the following events. A: all 3 chocolates have orange centres
$B$ : all 3 chocolates have the same centres
Find $\mathrm { P } ( A \mid B )$ and $\mathrm { P } ( B \mid A )$. The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}

OCR MEI S1 2009 January Q1

1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a $\pounds 10$ prize, 20 of them have a $\pounds 100$ prize, one of them has a $\pounds 5000$ prize and all of the rest have no prize. This information is summarised in the frequency table below.

Prize money	$\pounds 0$	$\pounds 10$	$\pounds 100$	$\pounds 5000$
Frequency	9929	50	20	1

Find the mean and standard deviation of the prize money per ticket.
I buy two of these tickets at random. Find the probability that I win either two $\pounds 10$ prizes or two $\pounds 100$ prizes.

OCR MEI S1 2009 January Q2

2 Thomas has six tiles, each with a different letter of his name on it.

Thomas arranges these letters in a random order. Find the probability that he arranges them in the correct order to spell his name.
On another occasion, Thomas picks three of the six letters at random. Find the probability that he picks the letters T, O and M (in any order).

OCR MEI S1 2009 January Q3

3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable $X$ represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of $X$ is shown in the table below.

$r$	0	1	2	3	4
$\mathrm { P } ( X = r )$	$p$	0.1	0.05	0.05	0.25

Find the value of $p$.
Find the expectation and variance of $X$.
The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.

OCR MEI S1 2009 January Q4

4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.

Find the probability that in a batch of 50 there is
(A) exactly one faulty teapot,
(B) more than one faulty teapot.
The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.

OCR MEI S1 2009 January Q5

5 Each day Anna drives to work.

$R$ is the event that it is raining.
$L$ is the event that Anna arrives at work late.

You are given that $\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25$ and $\mathrm { P } ( R \cap L ) = 0.2$.

Determine whether the events $R$ and $L$ are independent.
Draw a Venn diagram showing the events $R$ and $L$. Fill in the probability corresponding to each of the four regions of your diagram.
Find $\mathrm { P } ( L \mid R )$. State what this probability represents.

Age \(( x )\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

\(r\)	1	2	3	4	5	6
\(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 216 }\)	\(\frac { 7 } { 216 }\)	\(\frac { 19 } { 216 }\)	\(\frac { 37 } { 216 }\)	\(\frac { 61 } { 216 }\)	\(\frac { 91 } { 216 }\)

\(r\)	15	1010	2005	3000
\(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

Prize money	\(\pounds 0\)	\(\pounds 10\)	\(\pounds 100\)	\(\pounds 5000\)
Frequency	9929	50	20	1

Questions — OCR MEI S1 (292 questions)

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