Questions S1 (1967 questions)

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OCR MEI S1 Q5
3 marks Easy -1.2
5 My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.
  1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
  2. If I now guess my number, find the probability that I guess it correctly.
OCR MEI S1 Q6
6 marks Moderate -0.8
6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems.
\includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-3_654_804_1262_699}
  1. One of the 100 plants is selected at random. Find the probability that this plant has
    (A) at most one of the problems,
    (B) exactly two of the problems.
  2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems.
OCR MEI S1 Q1
18 marks Easy -1.2
1 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information.
\includegraphics[max width=\textwidth, alt={}, center]{10679ff3-494d-4f4e-a38a-0832faa91690-1_605_1650_534_284}
  1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
  2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
  3. Find the probability that Laura's flight is delayed due to just one of the three problems.
  4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
  5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
  6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 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 \(\mathrm { P } ( R \cap L ) = 0.2\).
  1. Determine whether the events \(R\) and \(L\) are independent.
  2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.
OCR MEI S1 Q3
8 marks Moderate -0.8
3 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults. [1]
OCR MEI S1 Q1
16 marks Moderate -0.3
1 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining \(1 \%\) overseas. Two people are selected at random. You may use the tree diagram below in answering this question.
\includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-1_944_1118_626_547}
  1. Find the probability that
    (A) both of these people were born in the rest of the UK,
    (B) at least one of these people was born in England,
    (C) neither of these people was born overseas.
  2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
  3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
    (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.
  1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
  2. Find the probability that he is delayed on at least one of the two flights.
  3. Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.
OCR MEI S1 Q3
8 marks Standard +0.8
3 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. ( \(\mathbf { S }\) represents a match won by Sophie; \(\mathbf { J }\) represents a match won by James.) \section*{SJSS SJSJS}
  1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
  2. Write down the other three possible sequences in which Sophie wins the competition.
  3. The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.
OCR MEI S1 Q4
8 marks Easy -1.2
4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-3_803_804_520_717} One of the residents is selected at random.
  1. Find the probability that this resident recycles
    (A) at least one of the materials,
    (B) exactly one of the materials.
  2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
  3. Find the probability that exactly one of them recycles kitchen waste.
OCR MEI S1 Q1
18 marks Standard +0.3
1 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.
  1. Copy and complete the tree diagram to show this information.
    \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-1_833_1156_851_573}
  2. Find the probability that a randomly selected person tests negative and is clear.
  3. Find the probability that a randomly selected person has the disease.
  4. Find the probability that a randomly selected person tests negative given that the person has the disease.
  5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
  6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .
  1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
  2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
  3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
    (A) wears either a jacket or a tie (or both),
    (B) wears no tie or no jacket (or wears neither).
OCR MEI S1 Q3
8 marks Standard +0.3
3 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.
  • \(A\) is the event that Isobel's parents watch a match.
  • \(\quad B\) is the event that Isobel scores in a match.
You are given that \(\frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).
  1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
  2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
  3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
  4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.
OCR MEI S1 Q4
18 marks Moderate -0.5
4 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .
  1. Copy and complete the probability tree diagram below, to illustrate the information above.
    \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-3_466_667_834_737} Calculate the probabilities of the following events.
  2. The test gives a positive result.
  3. The test gives a correct result.
  4. The painting is genuine, given a positive result.
  5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
  6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
  7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.
OCR MEI S1 Q1
5 marks Moderate -0.8
1 A school athletics team has 10 members. The table shows which competitions each of the members can take part in.
Competiton
100 m200 m110 m hurdles400 mLong jump
\multirow{10}{*}{Athlete}Abel✓✓✓
Bernoulli✓✓
Cauchy✓✓✓
Descartes✓✓
Einstein✓✓
Fermat✓✓
Galois✓✓
Hardy✓✓✓
Iwasawa✓✓
Jacobi✓
An athlete is selected at random. Events \(A , B , C , D\) are defined as follows.
\(A\) : the athlete can take part in exactly 2 competitions.
\(B\) : the athlete can take part in the 200 m .
\(C\) : the athlete can take part in the 110 m hurdles.
\(D\) : the athlete can take part in the long jump.
  1. Write down the value of \(\mathrm { P } ( A \cap B )\).
  2. Write down the value of \(\mathrm { P } ( C \cup D )\).
  3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
  4. Show that events \(B\) and \(D\) are not independent.
OCR MEI S1 Q2
18 marks Standard +0.3
2 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.
  1. All 3 doughnuts eaten contain jam.
  2. All 3 doughnuts are of the same kind.
  3. The 3 doughnuts are all of a different kind.
  4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
  5. exactly 2 Saturdays out of the 5 ,
  6. at least 1 Saturday out of the 5 .
OCR MEI S1 Q5
3 marks Standard +0.3
5 The Venn diagram illustrates the occurrence of two events \(A\) and \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{64f25a40-d3bf-4212-b92e-655f980c702b-5_480_771_452_655} You are given that \(\mathrm { P } ( A \cap B ) = 0.3\) and that the probability that neither \(A\) nor \(B\) occurs is 0.1 . You are also given that \(\mathrm { P } ( A ) = 2 \mathrm { P } ( B )\). Find \(\mathrm { P } ( B )\).
OCR MEI S1 Q1
8 marks Easy -1.2
1 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table,
\multirow{2}{*}{}Weather Forecast\multirow{2}{*}{Total}
SunnyCloudyWet
\multirow{3}{*}{Actual Weather}Sunny5512774
Cloudy1712829174
Wet33381117
Total75173117365
A day is selected at random from that year.
  1. Show that the probability that the forecast is correct is \(\frac { 264 } { 365 }\). Find the probability that
  2. the forecast is correct, given that the forecast is sunny,
  3. the forecast is correct, given that the weather is wet,
  4. the weather is cloudy, given that the forecast is correct.
OCR MEI S1 Q2
17 marks Standard +0.3
2 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.
  1. (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.
  2. 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.
  3. 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 Q3
18 marks Standard +0.3
3 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
  1. 10 customers are selected at random.
    (A) Find the probability that exactly 5 of them are accessing the internet.
    (B) Find the probability that at least 5 of them are accessing the internet.
    (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35. Give a reason for your choice of alternative hypothesis.
  3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.
OCR MEI S1 Q1
18 marks Standard +0.3
1 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.
  1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
    (B) Find the probability that there are more than 2 faulty tiles in the sample.
    (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 Q2
5 marks Moderate -0.8
2 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.
OCR MEI S1 Q3
16 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{dfb0acd8-d84b-4291-a811-a68f4942794b-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected.
    \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q4
18 marks Moderate -0.3
4 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
  1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
    (A) both seeds germinate,
    (B) just one seed germinates,
    (C) neither seed germinates.
  2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
  3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
  4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
  5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.
OCR MEI S1 Q3
18 marks Moderate -0.3
3 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{79f1015b-7c3d-4576-8d5b-e9fc89d8a49e-2_1266_1546_487_335}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected.
    \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 Q1
8 marks Easy -1.2
1 The stem and leaf diagram illustrates the heights in metres of 25 young oak trees.
3467899
402234689
501358
6245
746
81
Key: 4 |2 represents 4.2
  1. State the type of skewness of the distribution.
  2. Use your calculator to find the mean and standard deviation of these data.
  3. Determine whether there are any outliers.