Questions — OCR MEI S1 (300 questions)

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OCR MEI S1 Q3
4 marks Easy -1.2
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
8 marks Easy -1.3
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 marks Moderate -0.8
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 marks Moderate -0.8
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
8 marks Moderate -0.8
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
8 marks Moderate -0.8
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
7 marks Moderate -0.3
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
8 marks Moderate -0.8
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
7 marks Moderate -0.8
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
5 marks Moderate -0.8
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
3 marks Easy -1.2
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
8 marks Standard +0.8
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
6 marks Moderate -0.8
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
8 marks Moderate -0.3
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
5 marks Moderate -0.8
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
6 marks Moderate -0.3
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 marks Moderate -0.3
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
8 marks Moderate -0.8
1 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-1_699_1085_849_569} One of the citizens is selected at random.
  1. Find the probability that this citizen
    (A) has avoided air travel,
    (B) has used at least two of the three methods.
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
  3. Find the probability that none of them have avoided air travel.
OCR MEI S1 Q2
6 marks Standard +0.8
2 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.
  1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
  2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
  3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.
OCR MEI S1 Q3
18 marks Moderate -0.3
3 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.
  1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
  2. Find the probability that
    (A) all three of these trains are on time,
    (B) just one of these three trains is on time,
    (C) the 1200 train is on time.
  3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time.
OCR MEI S1 Q4
8 marks Moderate -0.8
4 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.
  • \(G\) is the event that this person goes to the gym.
  • \(R\) is the event that this person goes running.
You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).
  1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
  2. Determine whether the events \(G\) and \(R\) are independent.
  3. Find \(\mathrm { P } ( R \mid G )\).
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.