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240 people were surveyed about which of three countries, A, B, or C, they had been to. The Venn diagram below shows the number of people who had been to each combination of these countries, as well as those who had been to none of the three.
In this question, the event A is the event that a person picked at random from the 240 people surveyed had been to country A, and so on.
(i)Show that P(A) = 14.
(a)(i)
(ii)Verify that, for the values in this diagram: P(A ∪ C) = P(A) + P(C) − P(A ∩ C).
(a)(ii)
(iii)Are A and B independent events? Use calculations to justify your answer.
(a)(iii)
(b)Two of the 240 people are picked at random. Find the probability that one of them had been to all three countries, and the other had been to none of the three countries. Give your answer as a fraction in its simplest form.
(b)
Lee has a solar farm. There are 128 solar panels in Lee’s solar farm. Each week, he chooses 2 of the 128 solar panels at random to test.
(c)Find the probability that, in a given week, he chooses at least one of the same solar panels as he chose the week before. Give your answer correct to 3 decimal places.
(c)
In a game, each player wants to collect as many tokens as possible. When each player takes their turn, they roll a standard unbiased 6-sided die once. Each time they roll the die: if they roll a 1, 2, 3, or 4, they get nothing; if they roll a 5 or a 6, they get 1 token.
(i)Write down the probability that a player gets 1 token on their first turn.
(b)(i)
Amelie takes her turn 5 times in total.
(ii)Work out the probability that she gets 1 token for the first time on her 5th turn. Give your answer as a fraction.
(b)(ii)
(iii)Work out the probability that she gets 4 or more tokens in her 5 turns. Give your answer as a fraction.
(b)(iii)
The rules of the game are changed. Each time a player takes their turn, if they roll a 1, 2, 3, or 4, they still get nothing. If a player rolls a 5 or a 6, they have a choice. They can either: take 1 token and finish their turn; or roll the die again, as part of the same turn. If they roll a 1, 2, 3, or 4 this time, they get nothing at all for this turn. If they roll a 5 or a 6, they get a total of 4 tokens for this turn.
p is the probability that a player who gets a 5 or a 6 when they roll the die the first time will take 1 token and finish their turn, where 0 ≤ p ≤ 1.
(i)Work out the expected value (in tokens) of taking one turn using these rules. Remember that a turn can involve one or two rolls of the die. Give your answer in the form a + bp9 where a, b are constants. It may be helpful to draw a tree diagram.
(c)(i) — tree diagram & working
(ii)The expected value from part (c)(i) is biggest when p = 0. Explain what this means each player should do, in order to maximise the expected value (in tokens) of each turn.
(c)(ii)
In parts (a) and (b) of this question, give answers correct to 4 decimal places, where relevant.
(a)Assume that 6·7% of people in Ireland have diabetes. For a particular test for diabetes, each person tests either positive or negative. The probability that someone who has diabetes gets a correct positive result is 99%. However, the probability that someone who does not have diabetes gets an incorrect positive result is 7·8%. One person is picked at random and tested. Use the information above to complete the tree diagram below, by (i) writing the proportion on each branch in the appropriate box, and (ii) working out the probability for each outcome in the final column. Some values are already filled in.
(a)(i) & (ii) — write the branch proportions and outcome probabilities directly in the boxes
(iii)Find the probability that the person picked at random tests positive for diabetes.
(a)(iii)
(iv)The person picked at random tests positive for diabetes, using this test. Find the probability that they actually have diabetes.
(a)(iv)
(b)5 people are picked at random from the people in Ireland. Assuming that 6·7% of people in Ireland have diabetes, work out the probability that 2 or more of these 5 people have diabetes.
(b)
(c)20 people take part in a clinical trial. 10 of them will be picked to be in group A. The remaining 10 people will be in group B.
(i)How many different combinations of 10 people can be picked to be in group A?
(c)(i)
10 people are picked and are put in group A.
(ii)Each person in group A is now paired with a person in group B, making 10 pairs. How many different sets of 10 pairs can be made?
(c)(ii)
(d)In a different clinical trial, 24 people are split into two groups, X and Y. 8 people are in group X and 16 people are in group Y. Each person in group X is paired with two people in group Y, so that everyone in group Y is in exactly one of these pairings. How many different sets of such pairings can be made? Give your answer in the form a × 10n where 1 ≤ a < 10, n ∈ ℕ, and a is correct to 3 decimal places.
(d)
(a)The table below shows the prizes, in euro, that a player can win in a game, as well as the probability of winning each prize. The player wins at most one prize each time that she plays. Some of the prizes are given in terms of x ∈ ℝ.
| Prize (€) | None | 2 | x − 10 | x |
|---|---|---|---|---|
| Probability | 30% | 40% | 28% | 2% |
It costs €10 to play the game once. The game is fair – that is, the expected value of the winnings, taking the cost into account, is €0. Work out the value of x.
(a)
(b)A and B are mutually exclusive events. P(A) = 0·1 and P(B) = 0·4. Write down the value of each of the following: P(A ∩ B) and P(A ∪ B).
(b)
(c)C and D are two other events, with universal set U. P(C) = 0·5 and P(D) = 0·7. Find the maximum value of P[(C ∪ D)′]. Note: (C ∪ D)′ is the complement of the set C ∪ D in the set U.
(c)
(d)E is the event that it will be raining tomorrow morning. F is the event that I will wear a coat going outside tomorrow morning. Explain why it would not be reasonable to assume that E and F are independent events.
(d)
(c)In the co-ordinate diagram below, 16 points are marked with a dot (●). These are all of the points of the form (m, n), where m, n ∈ ℕ and m, n ≤ 4.
A pair of these points is picked at random.
(i)How many different pairs of points can be picked from these 16 points?
(c)(i)
(ii)The two points that are picked are joined with a straight line. Find the probability that this line is horizontal.
(c)(ii)
PK Hotels is a hotel chain in Europe.
(b)During their most recent stay, 15 of PK Hotel customers used the pool.
(i)6 of the PK Hotel customers are picked at random. Find the probability that exactly 2 of them used the pool.
(b)(i)
(ii)n of the PK Hotel customers are picked at random, where n ∈ ℕ. The probability that none of them used the pool, correct to 4 decimal places, is 0·0047. Work out the value of n.
(b)(ii)
(c)PK Hotels are testing a new booking system. 45% of people who log on to the PK Hotels website are shown the old booking system; the other 55% are shown the new booking system. People are assigned the booking system (old or new) at random. One third of people who see the old booking system end up booking a room. Two fifths of people who see the new booking system end up booking a room. One person is selected at random from those who booked a room through the PK Hotels website. Find the probability that this person used the new booking system. Give your answer as a percentage, correct to the nearest percent.
(c)
(c)Mattie is driving home. On the way, she passes five traffic lights. Each traffic light is either red (R), green (G), or orange (O) when she arrives at it. One day, she notes the pattern made by the colour of each traffic light when she arrives at it. For example, this pattern could be R R O G R.
(i)How many different patterns could the five traffic lights make?
(c)(i)
(ii)How many different patterns could the five traffic lights make, if the first light is red and the fifth light is not red?
(c)(ii)
(iii)How many different patterns could the five traffic lights make if no two consecutive lights are the same colour?
(c)(iii)
A circular spinner has 12 sectors, as follows: 5 sectors are labelled €6, 3 sectors are labelled €9, and the rest are labelled €0. In a game, the spinner is spun once. The spinner is equally likely to land on each sector. The player gets the amount of money shown on the sector that the spinner lands on.
(a)Fiona plays the game a number of times. Work out the probability that Fiona gets €6, then €9, then €6 the first three times she plays. Give your answer correct to 4 decimal places.
(a)
(b)Rohan also plays the game a number of times. Find the probability that Rohan gets €9 for the 3rd time, on the 8th time that he plays the game. Give your answer correct to 4 decimal places.
(b)
(c)Olga plays the game 2 times. Find the probability that Olga gets less than €16 in total from playing the game. Give your answer correct to 4 decimal places.
(c)
(d)A player wins the game if they guess the word with 6 guesses or less. The organisers record the following statistics for players in Europe: 80% of players win the game each day; 90% of players who win will play again the following day; 25% of players who do not win will play again the following day. Some of this is shown in the tree diagram below, based on the 360 000 people who played on 1st June. The squares represent proportions; the rectangles (PLAY, WIN, LOSE) represent numbers of people. Assume winning the game on one day is independent of winning it on another day.
(i)Complete the tree diagram, by writing the proportion associated with each branch into the appropriate square and writing the number of people who play, win, and lose the game into the appropriate rectangle.
(d)(i)
(ii)One person is picked at random from those who played the game on all three days (1st, 2nd, and 3rd June). Find the probability that this person lost on 1st June or 2nd June (or on both).
(d)(ii)
(b)Séamus needs to pick a new PIN code. It must be a 4-digit code. It must use exactly 4 of the digits from 1 to 9, so no digit can be used more than once.
(i)Work out how many such codes are possible.
(b)(i)
(ii)Work out how many of these codes contain the digit 2.
(b)(ii)
(iii)Work out how many of the codes in part (b)(i) have the sum of their first three digits equal to their fourth digit. (The codes can contain the digit 2, although they do not have to.)
(b)(iii)
(a)The table below gives details on the number of different types of student in a university. There are 22 714 students in total.
| 23 or younger | 24 or older | Total | |
| Undergraduate | 12 785 | 2922 | 15 707 |
| Postgraduate | 1353 | ||
| Total | 8576 | 22 714 |
(i)Fill in the three missing values to complete the table above.
(a)(i)
(ii)One student is picked at random. Let O be the event that the student is 24 or older. Let U be the event that the student is an undergraduate. Are the events O and U independent? Justify your answer.
(a)(ii)
(b)Three people are picked at random from a class. Find the probability that all three were born on the same day of the week. Assume that the probability of being born on each day is the same.
(b)
(c)There are b boys and g girls in a class, where b, g ∈ ℕ. 35 of the students are girls. 4 boys and 4 girls join the class. One student is then picked at random from the whole class. The probability that this student is a girl is now 47. Find the value of b and the value of g.
(c)
(d)John bought a car a number of years ago. The table gives an estimate of the probability that each of three events happens to John's car in the next year.
| Event | Head gasket blows | Timing belt goes | Air filters break |
|---|---|---|---|
| Probability | 0·095 | 0·041 | 0·073 |
(i)If the head gasket blows, John will have to replace his car, at an estimated cost of €20 000. If the head gasket is replaced now, it will cost €1450, and the probability that it blows in the next year will be reduced to 0·005. Based on these figures, use expected values to work out if it is worth replacing the head gasket now, or not.
(d)(i)
(ii)Work out the probability that at least one of the events in the table happens to John's car this year, taking these events to be independent. Give your answer correct to 3 decimal places.
(d)(ii)
In a particular population 15% of the people are left footed. A soccer team of 11 players, including 1 goalkeeper, is picked at random from the population.
(a)Find the probability that there is exactly one left footed player on the team. Give your answer correct to three decimal places.
(a)
(b)Find the probability that less than three players on the team are left footed. Give your answer correct to two decimal places.
(b)
(c)The goalkeeper is left footed. Find the probability that at least eight of the remainder of the team are right footed. Give your answer correct to two decimal places.
(c)
(c)The school caretaker has a box with 23 room keys: 12 for general classrooms, 6 for science labs, and 5 for offices.
(i)Four keys are drawn at random from the box. What is the probability that the 4th key drawn is the first office key drawn? Give your answer correct to 4 decimal places.
(c)(i)
(ii)All the keys are returned. Then 3 keys are drawn at random, one after the other, without replacement. What is the probability that one is for a general classroom, one for a science lab, and one for an office? Give your answer correct to 4 decimal places.
(c)(ii)
(a)Two events A and B are such that P(A) = 34 and P(A ∩ B) = 12.
(i)Find P(B|A) (the conditional probability). Give your answer as a fraction in its simplest form.
(a)(i)
(ii)P(A ∪ B) = 1112. Investigate if the events A and B are independent.
(a)(ii)
(b)A spinner consists of 4 segments (labelled 1, 1, 2, 3), as shown. Each segment is equally likely to be landed on. Liam, Sorcha and Lee play a game in which the spinner is spun twice and the numbers landed on are added together. The result is divided by 3 and the remainder is recorded. If the remainder is 0 then Liam wins; if 1 then Sorcha wins; if 2 then Lee wins. Is this a fair game? (i.e. Are all 3 participants equally likely to win?) Justify your answer by relevant calculations.
(b)
(a)A class group carried out a study of the makes and fuel types of cars in a large carpark. It found that 30% of the cars ran on diesel and 70% of these diesel cars were Volkswagen. It found that 60% of the cars ran on petrol and 25% of these petrol cars were Volkswagen. It found that 10% of the cars were hybrid/electric and 9% of these cars were Volkswagen. One car is selected at random from the car park. Find the probability that it is a Volkswagen car.
(a)
(b)The Road Safety Authority has data on driving test pass rates at all its test centres.
(i)In a particular Driving Test Centre the probability that a person taking the test for the first time will pass is 14. All of the test results are independent. In this centre on a particular day Joe, along with 5 others, takes the test. All six are taking the test for the first time. Find the probability that Joe passes the test along with exactly 2 others.
(b)(i)
(ii)The overall pass rate for all drivers at another centre is 12 (whether it is their first attempt or a subsequent attempt). On a particular day, n people take the test in this centre. The probability that two people or less than two people pass the test can be written in the form an2 + bn + c2n+1, where a, b, c ∈ ℕ. Find the value of a, the value of b, and the value of c.
(b)(ii)
(a)A class consists of 12 boys and 8 girls.
(i)Two students are selected at random from the class. What is the probability that the two students selected will be a boy and a girl in any order?
(a)(i)
(ii)Four students are selected, one at a time, at random from the class. What is the probability that the first three students selected will be boys and the fourth will be a girl?
(a)(ii)
(b)An examination paper is made up of two sections, Section A consisting of 7 questions and Section B consisting of 8 questions. The paper contains the following instruction: "From Section A you must answer question 1 and any three other questions. From Section B you must also answer any four questions." Find how many different combinations of questions may be answered if a candidate follows this instruction.
(b)
(a)Two independent events F and S are represented in the Venn diagram. P(F \ S) = 14, P(F ∩ S) = 15, P(S \ F) = x, and P(F ∪ S)′ = y, where x, y ≠ 0. Find the value of x and the value of y.
(a)
(b)In a club there are German, Irish and Spanish children only. There are 10 Spanish children. There are twice as many Irish children as German children. They are all in a group waiting to get on a swing. One child will be selected at random to go first and will not re-join the group. Then a second child will be selected at random to go next. The probability that the first child selected will be German and that the second child selected will not be German is 16. Find how many children are in the club.
(b)
In a competition Mary has a probability of 120 of winning, a probability of 110 of finishing second, and a probability of 14 of finishing third. Winning pays €9000, second €7000, third €3000; otherwise nothing. Each participant pays €2000 to enter.
(a)Find the expected value of Mary's loss if she enters the competition.
(a)
(b)Each of the 3 prizes is increased by the same amount (€x) but the entry fee is unchanged (e.g. winning now pays €(9000 + x)). Mary now expects to break even. Find the value of x.
(b)
(a)A security code consists of six digits chosen at random from the digits 0 to 9. The code may begin with zero and digits may be repeated (e.g. 0 7 1 7 3 7 is a valid code).
(i)Find how many of the possible codes will end with a zero.
(a)(i)
(ii)Find how many of the possible codes will contain the digits 2 0 1 8 together and in this order.
(a)(ii)
(b)Find a, b, c, and d, if (n + 3)! (n + 2)!(n + 1)! (n + 1)! = an3 + bn2 + cn + d, where a, b, c, d ∈ ℕ (using factorial notation).
(b)
When Conor rings Ciara's house, the probability that Ciara answers the phone is 15.
(a)Conor rings once every day for 7 consecutive days. Find the probability that she will answer on the 2nd, 4th, and 6th days but not on the other days.
(a)
(b)Find the probability that she will answer the phone for the 4th time on the 7th day.
(b)
(c)Conor rings once every day for n days. Write, in terms of n, the probability that Ciara will answer the phone at least once.
(c)
(d)Find the minimum value of n for which the probability that Ciara will answer at least once is greater than 99%.
(d)
In Galway, rain falls in the morning on 13 of school days. When raining, P(heavy traffic) = 12; when not raining, P(heavy traffic) = 14. When raining with heavy traffic, P(late) = 12; when not raining with no heavy traffic, P(late) = 18; in any other situation P(late) = 15.
(b)(i)Complete the tree diagram by writing the probability associated with each branch and the probability of each outcome. Give each answer in the form ab, where a, b ∈ ℕ.
(b)(i)
(ii)On a random school day in Galway, find the probability of being late for school.
(b)(ii)
(iii)On a random school day in Galway, find the probability that it rained in the morning, given that you were late for school (conditional probability).
(b)(iii)
(a)(i)In an archery competition, the team of John, David, and Mike win 1st prize if at least two of them hit the bullseye with their last arrows. The probabilities of hitting the bullseye are 15, 16, and 14 respectively. Complete the table to show all the ways they could win 1st prize.
(a)(i)
(ii)Hence or otherwise find the probability that they will win the competition.
(a)(ii)
(b)Two events, A and B, are shown in the diagram. P(A ∩ B) = 0·1, P(B \ A) = 0·3 and P(A \ B) = x. Write P(A) in terms of x and hence, or otherwise, find the value of x for which A and B are independent.
(b)
A club runs a weekly lotto. To win the €1000 Jackpot, contestants must match one letter (from 26) and two numbers (from 0 to 9) in the correct order; repetition of numbers is allowed (e.g. M, 3, 3).
(a)Calculate the probability that M, 3, 3 would be the winning outcome in a particular week.
(a)
(b)If a contestant matches the letter only, or the letter and one number (but not both numbers), they win €50. Using the table, or otherwise, find how much the club should expect to make or lose on each play, correct to the nearest cent, if they charge €2 per play (expected value).
(b)
(c)The club estimates 845 plays per week. If they want an average profit of €600 per week, how much should the club charge per play, correct to the nearest cent?
(c)
An experiment consists of throwing two fair, standard, six-sided dice and noting the sum of the two numbers thrown. If the sum is 9 or greater it is recorded as a "win" (W). If the sum is 8 or less it is recorded as a "loss" (L).
(a)Complete a table showing all possible outcomes of the experiment (Die 1 against Die 2), marking each as a win or a loss.
(a)
(b)(i)Find the probability of a win on one throw of the two dice.
(b)(i)
(ii)Find the probability that each of 3 successive throws of the two dice results in a loss. Give your answer correct to four decimal places.
(b)(ii)
(c)The experiment is repeated until a total of 3 wins occur. Find the probability that the third win occurs on the tenth throw of the two dice. Give your answer correct to four decimal places.
(c)
In basketball, players often take free throws. When Michael takes his first free throw in any game, the probability that he is successful is 0·7. For all subsequent free throws in the game, the probability of success is 0·8 if he was successful on the previous throw, and 0·6 if he was unsuccessful on the previous throw.
(a)Find the probability that Michael is successful (S) with all three of his first three free throws in a game, P(S, S, S).
(a)
(b)Find the probability that Michael is unsuccessful (U) with his first two free throws and successful with the third, P(U, U, S).
(b)
(c)List all the ways that Michael could be successful with his third free throw in a game and hence find the probability that Michael is successful with his third free throw.
(c)
(d)(i)Let pn be the probability that Michael is successful with his nth free throw (so 1 − pn is the probability of being unsuccessful). Show that pn+1 = 0·6 + 0·2pn.
(d)(i)
(ii)Assume that p is Michael's success rate in the long run; that is, for large n, pn+1 ≈ pn ≈ p. Using the result from (d)(i), or otherwise, show that p = 0·75.
(d)(ii)
(e)(i)For all positive integers n, let an = p − pn, where p = 0·75. Use the ratio an+1/an to show that an is a geometric sequence with common ratio 15.
(e)(i)
(ii)Find the smallest value of n for which p − pn < 0·00001.
(e)(ii)
(f)(i)You arrive at a game in which Michael is playing. You know he has already taken many free throws, but you do not know his pattern of success. Based on this, what is your estimate of the probability that Michael will be successful with his next free throw?
(f)(i)
(ii)Why would it not be appropriate to consider Michael's subsequent free throws in the game as a sequence of Bernoulli trials?
(f)(ii)
Two different games of chance can be played at a charity fundraiser. In each game, the player spins an arrow on a wheel and wins the amount shown on the sector where the arrow stops. Each game is fair in that the arrow is just as likely to stop in one sector as any other on that wheel. Game A has six equal sectors: €5, €2, €0, €0, €3, €1. Game B has six equal sectors: €5, €4, €0, €6, €0, €3.
(a)John played Game A four times and tells us that he has won a total of €8. In how many different ways could he have done this?
(a)
(b)To spin either arrow once, the player pays €3. Which game of chance would you expect to be more successful in raising funds for the charity? Give a reason for your answer.
(b)
(c)Mary plays Game B six times. Find the probability that the arrow stops in the €4 sector exactly twice.
(c)
Blood tests are sometimes used to indicate if a person has a particular disease. A test can give a false positive (indicating the person has the disease when they do not) or a false negative (indicating they do not have it when they do). It is estimated that 0·3% of a large population have a particular disease. A test gives a false positive in 4% of tests and a false negative in 1% of tests. A person picked at random is tested.
(i)Write the probability associated with each branch of the tree diagram in the blank boxes provided.
(i)
(ii)Hence, or otherwise, calculate the probability that a person selected at random from the population tests positive for the disease.
(ii)
(iii)A person tests positive for the disease. What is the probability that the person actually has the disease? Give your answer correct to three significant figures.
(iii)
(iv)The health authority is considering using the test on the general population with a view to treatment of the disease. Based on your results, do you think the above test would be an effective way to do this? Give a reason for your answer.
(iv)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.