Questions — OCR MEI S1 (292 questions)

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OCR MEI S1 Q3
3 The table shows the probability distribution of the random variable \(X\).
\(r\)10203040
\(\mathrm { P } ( X = r )\)0.20.30.30.2
  1. Explain why \(\mathrm { E } ( X ) = 25\).
  2. Calculate \(\operatorname { Var } ( X )\).
OCR MEI S1 Q4
4 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.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. 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 Q5
5 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.
  1. Find the probability that in a batch of 50 there is
    (A) exactly one faulty teapot,
    (B) more than one faulty teapot.
  2. 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 Q6
6 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.
Area \(( x )\)\(0 < x \leqslant 3\)\(3 < x \leqslant 5\)\(5 < x \leqslant 7\)\(7 < x \leqslant 10\)\(10 < x \leqslant 20\)
Frequency3813146
  1. Calculate an estimate of the sample mean and the sample standard deviation.
  2. Determine whether there could be any outliers at the upper end of the distribution.
OCR MEI S1 Q7
7 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.
\(r\)1234
\(\mathrm { P } ( X = r )\)0.20.160.1280.512
  1. Find the expectation and variance of \(X\).
  2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
  3. Draw a suitable diagram to illustrate the distribution of \(X\).
OCR MEI S1 Q2
2 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$
  1. Calculate the mean and standard deviation of the marks.
  2. The highest mark scored by any of the 56 students in the examination was 93. Show that this result may be considered to be an outlier.
  3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.
OCR MEI S1 Q3
3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.
  1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.
    \(r\)01234
    \(\mathrm { P } ( X = r )\)0\(\frac { 4 } { 35 }\)\(\frac { 18 } { 35 }\)\(\frac { 12 } { 35 }\)\(\frac { 1 } { 35 }\)
  2. Find the expectation and variance of \(X\).
OCR MEI S1 Q4
4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6$$
  1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(k\)\(11 k\)
  2. Find the mean of \(X\). A fair six-sided die is rolled three times.
  3. Find the probability that the total score is 16 .
OCR MEI S1 Q5
5 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
  1. Show that \(k = \frac { 1 } { 50 }\).
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q1
1 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.
\(r\)01
\(\mathrm { P } ( X = r )\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 4 }\)0\(\frac { 1 } { 24 }\)
  1. Explain why the case \(X = 3\) is impossible.
  2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 Q2
2 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.
  1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
  2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.
OCR MEI S1 Q4
4 A sprinter runs many 100 -metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$
  1. Calculate the sample mean, \(\bar { x }\), and sample standard deviation, \(s\), of these times.
  2. Show that the time of 10.04 seconds may be regarded as an outlier.
  3. Discuss briefly whether or not the time of 10.04 seconds should be discarded.
OCR MEI S1 Q5
5 The number, \(X\), of children per family in a certain city is modelled by the probability distribution \(\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )\) for \(r = 0,1,2,3,4\).
  1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac { 1 } { 50 }\).
    \(r\)01234
    \(\mathrm { P } ( X = r )\)\(6 k\)\(10 k\)
  2. Calculate \(\mathrm { E } ( X )\).
  3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children.
OCR MEI S1 Q6
6 The weights, \(w\) grams, of a random sample of 60 carrots of variety A are summarised in the table below.
Weight\(30 \leqslant w < 50\)\(50 \leqslant w < 60\)\(60 \leqslant w < 70\)\(70 \leqslant w < 80\)\(80 \leqslant w < 90\)
Frequency111018147
  1. Draw a histogram to illustrate these data.
  2. Calculate estimates of the mean and standard deviation of \(w\).
  3. Use your answers to part (ii) to investigate whether there are any outliers. The weights, \(x\) grams, of a random sample of 50 carrots of variety B are summarised as follows. $$n = 50 \quad \sum x = 3624.5 \quad \sum x ^ { 2 } = 265416$$
  4. Calculate the mean and standard deviation of \(x\).
  5. Compare the central tendency and variation of the weights of varieties A and B .
OCR MEI S1 Q7
7 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\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. 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 Q1
1 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]{870b6ef1-60f7-42e3-95f8-0544a2a07b15-1_725_921_728_622} One of the people is selected at random.
  1. 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.
  2. Given that this person has watched cycling, find the probability that this person has not watched football.
OCR MEI S1 Q2
2 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.
  1. Find the probability that neither card is an ace.
  2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.
OCR MEI S1 Q3
3 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.
  1. Draw a probability tree diagram to illustrate the outcomes.
  2. Find the probability that a randomly selected candidate is accepted.
  3. Find the probability that a randomly selected candidate is retested at least once, given that this candidate is accepted.
OCR MEI S1 Q4
4 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 \(\mathrm { P } ( L \mid R ) = 0.45\).
  1. Use this information to show that the events \(L\) and \(R\) are not independent.
  2. Find \(\mathrm { P } ( L \cap R )\).
  3. 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 Q5
5 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.
  • \(W\) is the event that Alan has to wait at the level crossing.
  • \(L\) is the event that Alan is late for work.
You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).
  1. Calculate \(\mathrm { P } ( L \cap W )\).
  2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.
OCR MEI S1 Q6
6 Malik is playing a game in which he has to throw a 6 on a fair six-sided die to start the game. Find the probability that
  1. Malik throws a 6 for the first time on his third attempt,
  2. Malik needs at most ten attempts to throw a 6.
OCR MEI S1 Q7
7 At a garden centre there is a box containing 50 hyacinth bulbs. Of these, 30 will produce a blue flower and the remaining 20 will produce a red flower. Unfortunately they have become mixed together so that it is not known which of the bulbs will produce a blue flower and which will produce a red flower. Karen buys 3 of these bulbs.
  1. Find the probability that all 3 of these bulbs will produce blue flowers.
  2. Find the probability that Karen will have at least one flower of each colour from her 3 bulbs.
OCR MEI S1 Q8
8 At a call centre, 85\% of callers are put on hold before being connected to an operator. A random sample of 30 callers is selected.
  1. Find the probability that exactly 29 of these callers are put on hold.
  2. Find the probability that at least 29 of these callers are put on hold.
  3. If 10 random samples, each of 30 callers, are selected, find the expected number of samples in which at least 29 callers are put on hold.
OCR MEI S1 Q1
1 It is known that \(8 \%\) of the population of a large city use a particular web browser. A researcher wishes to interview some people from the city who use this browser. He selects people at random, one at a time.
  1. Find the probability that the first person that he finds who uses this browser is
    (A) the third person selected,
    (B) the second or third person selected.
  2. Find the probability that at least one of the first 20 people selected uses this browser.
OCR MEI S1 Q2
2 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is
  • 0.7 if he won the previous set,
  • 0.4 if Alan won the previous set.
It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
  2. Find the probability that Alan wins the match.
  3. Find the probability that the match ends after exactly four sets have been played.