Probability Exam Questions with Solutions by Henk Tijms

Probability Exam Questions with Solutions by Henk Tijms1

December 15, 2013 This note gives a large number of exam problems for a first course in probability. Fully worked-out solutions of these problems are also given, but of course you should first try to solve the problems on your own!

c 2013 by Henk Tijms, Vrije University, Amsterdam. All rights reserved. Permission is hereby given to freely print and circulate copies of these notes so long as the notes are not reproduced for commercial purposes. Comments are welcome on tijms@quicknet.nl.

1The exam questions accompany the basic probability material in the chapters 7-14 of my textbook Understanding Probability, third edition, Cambridge University Press, 2012. The simulation-based chapters 1-6 of the book present many probability applications from everyday life to help the beginning student develop a feel for probabilities. Moreover, the book gives an introduction to discrete-time Markov chains and continuous-time Markov chains in the chapters 15 and 16.

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Chapter 7

7E-1 A bridge hand in which there is no card higher than a nine is called a Yarborough. Specify an appropriate sample space and determine the probability of Yarborough when you are randomly dealt 13 cards out of a wellshuffled deck of 52 cards.

7E-2 Five dots are placed at random on a 8 ? 8 grid in such a way that no cell contains more than one dot. Specify an appropriate sample space and determine the probability that no row or column contains more than one dot.

7E-3 Five people are sitting at a table in a restaurant. Two of them order coffee and the other three order tea. The waiter forgot who ordered what and puts the drinks in a random order for the five persons. Specify an appropriate sample space and determine the probability that each person gets the correct drink

7E-4 A parking lot has 10 parking spaces arranged in a row. There are 7 cars parked. Assume that each car owner has picked at a random a parking place among the spaces available. Specify an appropriate sample space and determine the probability that the three empty places are adjacent to each other.

7E-5 Somebody is looking for a top-floor apartment. She hears about two vacant apartments in a building with 7 floors en 8 apartments per floor. What is the probability that there is a vacant apartment on the top floor?

7E-6 You choose at random two cards from a standard deck of 52 cards. What is the probability of getting a ten and hearts?

7E-7 A box contains 7 apples and 5 oranges. The pieces of fruit are taken out of the box, one at a time and in a random order. What is the probability that the bowl will be empty after the last apple is taken from the box?

7E-8 A group of five people simultaneously enter an elevator at the ground

floor. There are 10 upper floors. The persons choose their exit floors inde-

pendently of each other. Specify an appropriate sample space and determine

the probability that they are all going to different floors when each person

randomly chooses one of the 10 floors as the exit floor. How does the answer

change

when

each

person

chooses

with

probability

1 2

the

10th

floor

as

the

exit floor and the other floors remain equally likely as the exit floor with a

probability

of

1 18

each.

7E-9 Three friends and seven other people are randomly seated in a row.

Specify an appropriate sample space to answer the following two questions.

(a) What is the probability that the three friends will sit next to each other?

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(b) What is the probability that exactly two of the three friends will sit next to each other?

7E-10 You and two of your friends are in a group of 10 people. The group is randomly split up into two groups of 5 people each. Specify an appropriate sample space and determine the probability that you and your two friends are in the same group.

7E-11 You are dealt a hand of four cards from a well-shuffled deck of 52 cards. Specify an appropriate sample space and determine the probability that you receive the four cards J, Q, K, A in any order, with suit irrelevant.

7E-12 You draw at random five cards from a standard deck of 52 cards. What is the probability that there is an ace among the five cards and a king or queen?

7E-13 Three balls are randomly dropped into three boxes, where any ball is equally likely to fall into each box. Specify an appropriate sample space and determine the probability that exactly one box will be empty.

7E-14 An electronic system has four components labeled as 1, 2, 3, and 4. The system has to be used during a given time period. The probability that component i will fail during that time period is fi for i = 1, . . . , 4. Failures of the components are physically independent of each other. A system failure occurs if component 1 fails or if at least two of the other components fail. Specify an appropriate sample space and determine the probability of a system failure.

7E-15 The Manhattan distance of a point (x, y) in the plane to the origin (0, 0) is defined as |x| + |y|. You choose at random a point in the unit square {(x, y) : 0 x, y 1}. What is the probability that the Manhattan distance of this point to the point (0, 0) is no more than a for 0 a 2?

7E-16 You choose at random a point inside a rectangle whose sides have the lengths 2 and 3. What is the probability that the distance of the point to the closest side of the rectangle is no more than a given value a with 0 < a < 1?

7E-17 Pete tosses n + 1 fair coins and John tosses n fair coins. What is the probability that Pete gets more heads than John? Answer this question first for the cases n = 1 and n = 2 before solving the general case.

7E-18 Bill and Mark take turns picking a ball at random from a bag containing four red balls and seven white balls. The balls are drawn out of the bag without replacement and Mark is the first person to start. What is the probability that Bill is the first person to pick a red ball?

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7E-19 Three desperados A, B and C play Russian roulette in which they take turns pulling the trigger of a six-cylinder revolver loaded with one bullet. Each time the magazine is spun to randomly select a new cylinder to fire as long the deadly shot has not fallen. The desperados shoot according to the order A, B, C, A, B, C, . . .. Determine for each of the three desperados the probability that this desperado will be the one to shoot himself dead.

7E-20 A fair coin is tossed 20 times. The probability of getting the three or more heads in a row is 0.7870 and the probability of getting three or more heads in a row or three or more tails in a row is 0.9791.2 What is the probability of getting three or more heads in a row and three or more tails in a row?

7E-21 The probability that a visit to a particular car dealer results in neither buying a second-hand car nor a Japanese car is 55%. Of those coming to the dealer, 25% buy a second-hand car and 30% buy a Japanese car. What is the probability that a visit leads to buying a second-hand Japanese car?

7E-22 A fair die is repeatedly rolled and accumulating counts of 1s, 2s, . . ., 6s are recorded. What is an upper bound for the probability that the six accumulating counts will ever be equal?

7E-23 A fair die is rolled six times. What is the probability that the largest number rolled is r for r = 1, . . . , 6?

7E-24 Mr. Fermat and Mr. Pascal are playing a game of chance in a cafe

in Paris. The first to win a total of ten games is the overall winner. Each

of

the

two

players

has

the

same

probability

of

1 2

to

win

any

given

game.

Suddenly the competition is interrupted and must be ended. This happens

at a moment that Fermat has won a games and Pascal has won b games with

a < 10 and b < 10. What is the probability that Fermat would have been

the overall winner when the competition would not have been interrupted?

Hint: imagine that another 10 -a+ 10 -b-1 games would have been played.

7E-25 A random number is repeatedly drawn from 1, 2, . . . , 10. What is the probability that not all of the numbers 1, 2, . . . , 10 show up in 50 drawings?

7E-26 Three couples attend a dinner. Each of the six people chooses randomly a seat at a round table. What is the probability that no couple sits together?

7E-27 You roll a fair die six times. What is the probability that three of the six possible outcomes do not show up and each of the other three possible

2These probabilities can be computed by using an absorbing Markov chain discussed in Chapter 15 of the book.

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outcomes shows up two times? What is the probability that some outcome shows up at least three times?

7E-28 In a group of n boys and n girls, each boy chooses at random a girl and each girl chooses at random a boy. The choices of the boys and girls are independent of each other. If a boy and a girl have chosen each other, they form a couple. What is the probability that no couple will be formed?

7E-29 Twelve married couples participate in a tournament. The group of 24 people is randomly split into eight teams of three people each, where all possible splits are equally likely. What is the probability that none of the teams has a married couple?

7E-30 An airport bus deposits 25 passengers at 7 stops. Each passenger is as likely to get off at any stop as at any other, and the passengers act independently of one another. The bus makes a stop only if someone wants to get off. What is the probability that somebody gets off at each stop?

7E-31 Consider a communication network with four nodes n1, n2, n3 and n4 and five directed links l1 = (n1, n2), l2 = (n1, n3), l3 = (n2, n3), l4 = (n3, n2), l5 = (n2, n4) and l6 = (n3, n4). A message has to be sent from the source node n1 to the destination node n4. The network is unreliable. The probability that the link li is functioning is pi for i = 1, . . . , 5. The links behave physically independent of each other. A path from node n1 to node n4 is only functioning if each of its links is functioning. Use the inclusion-exclusion formula to find the probability that there is some functioning path from node n1 to node n4. How does the expression for this probability simplify when pi = p for all i?

Chapter 8

8E-1 Three fair dice are rolled. What is the probability that the sum of the three outcomes is 10 given that the three dice show different outcomes?

8E-2 A bag contains four balls. One is blue, one is white and two are red. Someone draws together two balls at random from the bag. He looks at the balls and tells you that there is a red ball among the two balls drawn out. What is the probability the the other ball drawn out is also red?

8E-3 A fair coin is tossed n times. What is the probability of heads on the first toss given that r heads were obtained in the n tosses?

8E-4 A hand of 13 cards is dealt from a standard deck of 52 cards. What is the probability that it contains more aces than tens? How does this probability change when you have the information that the hand contains at least one ace?

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