Questions — OCR MEI (4301 questions)

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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.
OCR MEI S1 Q3
3 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.
  • \(T\) is the event that this person likes tomato soup.
  • \(M\) is the event that this person likes mushroom soup.
You are given that \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent.
  2. Find \(\mathrm { P } ( T \cap M )\).
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram.
    \(425 \%\) of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
  4. Find the probability that there are no plants with red flowers in the sample.
  5. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers.
OCR MEI S1 Q5
5 In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random.
  • \(W\) is the event that this person works part-time.
  • \(F\) is the event that this person is female.
You are given that \(\mathrm { P } ( W ) = 0.14 , \mathrm { P } ( F ) = 0.41\) and \(\mathrm { P } ( W \cap F ) = 0.11\).
  1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram.
  2. Determine whether the events \(W\) and \(F\) are independent.
  3. Find \(\mathrm { P } ( W \mid F )\) and explain what this probability represents.
OCR MEI S1 Q6
6 The table shows all the possible products of the scores on two fair four-sided dice.
\multirow{2}{*}{}Score on second die
1234
\multirow{4}{*}{Score on first die}11234
22468
336912
4481216
  1. Find the probability that the product of the two scores is less than 10 .
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than \(10 ^ { \prime }\) are not independent.
OCR MEI S1 Q7
7 Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel.
\includegraphics[max width=\textwidth, alt={}, center]{971a3594-3906-4868-b57c-e4667d42e5c8-3_712_1126_553_556} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus,
  2. Andy walks or travels by bike,
  3. the weather is dry given that Andy walks or travels by bike.
OCR MEI S1 Q1
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
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
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
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
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 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
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
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
1 marks
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
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
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
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.