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elementary probability for applications
Understanding Probability 2nd Edition Henk Tijms - Solutions
1. 4.9 In a game called “26” a player chooses one number from the numbers 1,...,6 as point number. After this, the player rolls a collection of ten dice 13 times in succession. If the player’s point number comes up 26 times or more, the player receives five times the amount staked on the
1. 4.8 A military early-warning installation is constructed in a desert. The installation consists of five main detectors and a number of reserve detectors. If fewer than fivedetectors are working, the installation ceases to function. Every two months an
1. 4.7 Thekeeper of a certain king’s treasure receives the task of filling each of 100 urns with 100 gold coins. While fulfilling this task, he substitutes one lead coin for one gold coin in each urn. The king suspects deceit on the part of the sentry and has two methods at his disposal of
1. 4.6 In 1989, American investment publication Money Magazine assessed the perfor manceof277importantmutualfundsovertheprevioustenyears.Foreachofthose ten years they looked at which mutual funds performed better than the S&P index.Research showed that five of the 277 funds performed better than
1. 4.5 One hundred and twenty-five mutual funds have agreed to take part in an elim ination competition being sponsored by Four Leaf Clover investment magazine.The competition will last for two years and will consist of seven rounds. At the beginning ofeachquarter,
1. 4.4 AgameofchanceplayedhistoricallybyCanadianIndiansinvolvedthrowingeight flat beans into the air and seeing how they fell. The beans were symmetrical and were painted white on one side and black on the other. The bean thrower would winonepointif
1. 4.3 In an attempt to increase his market share, the maker of Aha Cola has formulated an advertising campaign to be released during the upcoming European soccer championship. The imageofanorangeball hasbeenimprinted ontheunderside of approximately one out of every one thousand cola can pop-tops.
1. 4.2 What are the chances of getting at least one six in one throw of six dice, at least two sixes in one throw of 12 dice, and at least three sixes in one throw of 18 dice?†Do you think these chances are the same?
1.4.1 DuringWorldWarII,LondonwasheavilybombedbyV-2guidedballisticrockets.Theserockets, luckily, were not particularly accurate at hitting targets. The number of direct hits in the southern section of London has been analyzed by splitting the area up into 576 sectors measuring one quarter of a
1. 3.32 Jeu de Treize was a popular card game in seventeenth century France. This game was played as follows. One person is chosen as dealer and the others are players.Each player puts up a stake. The dealer takes a full deck of 52 cards and shuffles them thoroughly. Then the dealer turns over the
1. 3.31 In the last 250 drawings of Lotto 6/45, the numbers 1,...,45 were drawn 46,31,27,32,35,44,34,33,37,42,35,26,41,38,40, 38,23,27,31,37,28,25,37,33,36,32,32,36,33,36, 22,31,29,28,32,40,31,30,28,31,37,40,38,34,24 times, respectively. Using simulation, determine whether these results are suspi
1. 3.30 Twenty-five persons attended a “reverse raffle,” in which everyone bought a num ber. Numbered balls were then drawn out of a bin one at a time at random. The last ball in the bin would be the winner. But when the organizers got down to the last ball, they discovered that three numbered
1. 3.29 You have $800 but you desperately needs $1,000 before midnight. The casino must bring help. You decide for bold play at European roulette. You bet on red each time. The stake is $200 if your bankroll is $200 or $800 and is $400 if your bankroll is $400 or $600. You quit as soon as you have
1. 3.28 A drunkard is wandering back and forth on a road. At each step he moves two units distance to the north with a probability of 1 2, or one unit to the south with a probability 1 2. Let ak denote the probability of the drunkard ever returning to his point of origin if the drunkard is k units
1. 3.27 Suppose you go to the local casino with $50 in your pocket, and it is your goal to multiply your capital to $250. You are playing (European) roulette, and you stake a fixed amount on red for each spin of the wheel. What is the probability of your reaching your goal when you stake fixed
1. 3.26 You are playing rounds of a certain game against an opponent until one of you has won all of the other one’s betting money. At the start of each round, each of you stakes one dollar. The probability of winning any given round is equal to p, and the winner of a round gets the other
1.3.25 Red Dog is a casino game played with a deck of 52 cards. Suit plays no role in determining the value of each card. An ace is worth 14, king 13, queen 12, jack 11, and numbered cards are worth the number indicated on the card. After staking a bet a player is dealt two cards. If these two
1. 3.24 A gang of thieves has gathered at their secret hideaway. Just outside, a beat-cop lurking about realizes that he has happened upon the notorious hideaway and takes it upon himself to arrest the gang leader. He knows that the villains, for reasons of security, will exit the premises one by
1. 3.23 The game “Casino War” is played with a deck of cards compiled of six ordinary decks of 52 playing cards. Each of the cards is worth the face value shown (color is irrelevant). The player and the dealer each receive one card. If the player’s card has a higher value than the dealer’s,
1.3.22 Go back and take another look at Problem 2.29 from Chapter 2. For ease of notation, let us rename the numbers 5,10,...,100 on the wheel as 1,2,...,20.For any a = 1,2,...,20, let S(a) denote the probability of candidate A winning if candidate A stops after the first spin giving a score of a
1. 3.21 In the popular English game of Hazard, a player must first determine which of the five numbers from 5,...,9 will be the “main” point. The player does this by rolling two dice until such time as the point sum equals one of these five numbers.The player then rolls again. He/she wins if
1.3.20 In a particular game, you begin by tossing a die. If the toss results in i points, then you go on to toss i dice together. If the sum of the points resulting from the toss of the i dice is greater than (less than) 12, you win (lose) one dollar, and if the sum of those points is equal to 12,
1. 3.19 Inatelevisiongameshow,thecontestantcanwinasmallprize,amediumprize,and a large prize. The large prize is a sports car. Each of the three prizes is “locked up”in a separate box. There are five keys randomly arranged in front of the contestant.One opens the lock to the small prize, another
1.3.18 A commercial radio station is advertising a particular call-in game that will be played in conjunction with the introduction of a new product. The game is to be played every day for a period of 30 days. The game is only open to listeners between the ages of 15 and 30. Each caller will be the
1. 3.17 A company has 110 employees in service. Use computer simulation to find the probability of there being 12 or more separate occasions when two or more employees have the same birthday. Also, determine the probability that, in each of the 12 months, two or more employees have the same
1. 3.16 You received a tip that the management of a theater will give a free ticket to the first person in line having the same birthday as someone before him/her in line.Assuming that people enter the line one at a time and you do not know those people, what is the best position to take in the
1. 3.15 Of the unclaimed prize monies from the previous year, a lottery has purchased 500 automobiles to raffle off as bonus prizes among its 2.4 million subscribing members. Bonus winners are chosen by a computer programmed to choose 500
1. 3.14 In the Massachusetts Numbers Game, a four-digit number is drawn from the num bers 0000,0001,...,9999 every evening (except Sundays). Let’s assume that the same lottery takes place in ten other states each evening.(a) What is the probability that the same number will be drawn in two or
1. 3.13 Suppose that someone has played bridge 30 times a week on average over a period of 50years. Apply the result from Problem 3.12(b) to calculate the probability that this person has played exactly the same hand at least twice during the span of the 50 years.
1. 3.12 The birthday problem and those cited in Problems 3.9–3.11 can be described as a special case ofthefollowingmodel.Randomly,youdropn ballsinc compartments such that each ball is dropped independently of the others. It is assumed that c >n. What is the probability pn that at least two balls
1. 3.11 Agroupofsevenpeopleinahotellobbyarewaitingfortheelevatortotakethemup to their rooms. The hotel has 25 floors, each floor containing the same number of rooms.Supposethattheroomsofthesevenwaitingpeoplearerandomlydistributed around the hotel.(a) What is the probability of at least two people
1. 3.10 What is the probability that the same number will come up at least twice in the next ten spins of a roulette wheel?
1. 3.9 You bet your friend that, of the next 15 automobiles to appear, at least two will have license plates beginning and ending with the same number. What is your probability of winning?
1. 3.8 Suppose that a large group of people are undergoing a blood test for a particular illness. The probability that a random person has the illness in question is equal to 0.001. In order to save on the work, it is decided to split the group into smaller groups each consisting of r people. The
1. 3.7 Five friends go out to a pub together. They agree to let a roll of the dice determine who pays for each round. Each friend rolls one die, and the one getting the lowest number of points picks up the tab for that round. If the low number is rolled by more than one friend in any given round,
1. 3.6 The Yankees and the Mets are playing a best-four-of-seven series. The winner takes all of the prize money of one million dollars. Unexpectedly, the competition must be suspended when the Yankees lead two games to one. How should the prize money be divided between the two teams if the
1. 3.5 What is the probability of a randomly chosen five-digit number lining up in the same order from right to left as it does from left to right?
1. 3.4 The national lottery is promoting a special, introductory offer for the upcoming summer season. Advertisements claim that, during the four scheduled summer drawings, it will hardly be possible not to win a prize, because four of every ten tickets will win at each drawing. What do you think
1. 3.3 InboththeMassachusettsNumbersGameandtheNewHampshireLottery,afour digit number is drawn each evening from the sequence 0000, 0001,...,9999. On Tuesday evening, September 9, 1981, the number 8092 was drawn in both lottery games. Lottery officials declared that the probability of both lotteries
1. 3.2 Is the probability of a randomly chosen person having his/her birthday fall on a Monday equal to the probability of two randomly chosen people having their birthdays fall on the same day of the week?
1.3.1 Is it credible if a local newspaper somewhere in the world reports on a given day that a member of the local bridge club was dealt a hand containing a full suit of 13 clubs?
1. 2.45 Center court at Wimbledon is buzzing with excitement. The dream finale between Alassi and Bicker is about to begin. The weather is fine, and both players are in top condition. In the past, these two players have competed multiple times under similar conditions. On the basis of past
1. 2.44 In a certain betting contest you may choose between two games A and B at the start of every turn. In game A you always toss the same coin, while in game B you toss either coin 1 or coin 2 depending on your bankroll. In game B you must toss coin 1 if your bankroll is a multiple of three;
1.2.43 Independently of each other, ten numbers are randomly drawn from the interval(0, 1). You mayviewthenumbersonebyoneintheorderinwhichtheyaredrawn.After viewing each individual number, you are given the opportunity to take it or let it pass. You are not allowed to go back to numbers you have
1. 2.42 Aqueue of 50 people is waiting at a box office in order to buy a ticket. The tickets cost five euros each. For any person, there is a probability of 1 2that she/he will pay with a five-euro note and a probability of 1 2that she/he will pay with a ten-euro note. When the box opens there is
1. 2.41 Each of seven dwarfs has his own bed in a common dormitory. Every night, they retire to bed one at a time, always in the same sequential order. On a particular evening, the youngest dwarf, who always retires first, has had too much to drink.He randomly chooses one of the seven beds to fall
1. 2.40 Onehundredpassengerslineuptoboardanairplanewith100seats.Eachpassenger is to board the plane individually, and must take his or her assigned seat before the next passenger may board. However, the passenger first in line has lost his boarding pass and takes a random seat instead. This
1. 2.39 Sixteen teams remain in a soccer tournament. A drawing of lots will determine which eight matches will be played. Before the drawing takes place, it is possible to place bets with bookmakers over the outcome of the drawing. Use computer simulation to find the probability of correctly
1. 2.38 You have received a reliable tip that in the local casino the roulette wheel is not exactly fair. The probability of the ball landing on the number 13 is twice what it should be. The roulette table in question will be in use that evening. In that casino, European roulette is played. You go
1. 2.37 Aparticle movesoverthe flat surface of a grid such that an equal unit of distance is measured withevery step. The particle begins at the origin (0,0). The first step may be to the left, right, up or down, with equal probability 1 4. The particle cannot move back in the direction that the
1. 2.36 A drunkard is standing in the middle of a very large town square. He begins to walk. Each step he takes is a unit distance in a randomly chosen direction.The direction for each step taken is chosen independently of the direction of the others. Suppose that the drunkard takes a total of n
1. 2.35 You are playing the following game: a fair coin is tossed until it lands heads three times in a row. You get 12 dollars when this occurs, but you must pay one dollar for each toss. Use computer simulation to find out whether this is a fair contest.
1. 2.34 A random sequence of 0’s and 1’s is generated by tossing a fair coin N times.A 0 corresponds to the outcome heads anda1totheoutcome tails. A run is an uninterrupted sequence of 0’s or 1’s only. Use computer simulation to verify experimentally that the length of the longest run
1. 2.33 In a TV program, the contestant can win one of three prizes. The prizes consist of a first prize and two lesser prizes. The dollar value of the first prize is a five-digit number and begins with 1, whereas the dollar values of the lesser prizes are three digit numbers. There are initially
1. 2.32 Solve the following problems for the coin-tossing experiment:(a) Use computer simulation to find the probability that the number of heads ever exceeds twice the number of tails if a fair coin is tossed 5 times. What is the probability if the coin is tossed 25 times. What is the probability
1. 2.31 Usingfivedice,youareplaying agameconsisting of accumulating as many points as possible in five rounds. After each round you may “freeze” one or more of the dice, i.e., a frozen die will not be rolled again in successive rounds, but the amount of points
1. 2.30 Reconsider Problem 2.29 with three candidates A, B, and C. Candidate A spins first; candidate B, second; and candidate C, last.(a) Use the optimal stopping rule found in Problem 2.29 to describe the optimal strategy of candidate B.(b)
1. 2.29 Twocandidates Aand Bremaininthefinaleofatelevisiongameshow.Atthispoint, eachcandidatemustspinawheeloffortune.The20numbers5,10,...,95,100are listed on the wheel and when the wheel has stopped spinning, a pointer randomly stops ononeofthenumbers.Eachcandidatehasachoiceofspinningthewheelone or
1. 2.28 Consider the best-choice problem from Section 2.3 with 100 slips of paper. You let the first 30 slips of paper go by and then pick the first one to come along thereafter that contains a higher number than was seen in the first 30 slips. Use computer simulation to find the probability of
1. 2.27 The card game called Ace-Jack-Two is played between one player and the bank.It goes this way: a deck of 52 cards is shuffled thoroughly, after which the bank repeatedly reveals three cards next to each other on a table. If an ace, jack or two is among the three cards revealed, the bank gets
1. 2.26 You have been asked to determine a policy for accepting reservations for an air line flight. This particular flight uses an aircraft with 15 first-class seats and 75 economy-class seats. First-class tickets on the flight cost $500 and economy-class tickets cost $250. The number of
1. 2.25 What is the probability that any two adjacent letters are different in a random permutation of the 11 letters of the word Mississippi? What is the probability that in a thoroughly shuffled deck of 52 cards no two adjacent cards are of the same rank? Use computer simulation.
1. 2.24 Seated at a round table, five friends are playing the following game. One of the five players opens the game by passing a cup to the player seated either to his left or right. That player, in turn, passes the cup to a player on his left or right and so on until the cup has progressed all
1. 2.23 You decide to bet on ten spins of the roulette wheel in European roulette and to use the double-up strategy. Under this strategy, you bet on red each time and you double your bet if red does not come up. If red comes up, you go back to your initial bet of 1 euro. Use computer simulation to
1. 2.22 Amillionaire plays European roulette every evening for pleasure. He begins every time with A = 100 chips of the same value and plays on until he has gambled away all 100 chips. When he has lost his 100 chips for that evening’s entertain ment, he quits. Use computer simulation to find the
1. 2.21 Use computer simulation to find(a) The expected value of the distance between two points that are chosen at random inside the interval (0,1).(b) The expected value of the distance between two points that are chosen at random inside the unit square.(c) The expected value of the distance
1. 2.20 Astickisbrokenatrandomintotwopieces.Youbetontheratioofthelengthofthe longer piecetothelengthofthesmallerpiece.Youreceive$k iftheratioisbetween k andk +1forsome1 ≤ k ≤ m −1,whileyoureceive$miftheratioislargerthan m. Here m is agiven positive integer. Using computer simulation, verify
1. 2.19 Usecomputersimulationtofindtheprobabilitythatthetriangle OABhasanangle larger than 90◦ when A and B are randomly chosen points within the unit circle having the point O as center. What is this probability if the unit sphere is taken instead of the unit circle? Also, simulate the
1. 2.18 Solve Problem 2.17 again for the situation in which the coefficient A is fixed at the value 1.
1. 2.17 Use computer simulation to find the probability that the quadratic equation Ax2 +Bx+C =0 has real roots when A, B, and C are chosen at random from the interval (−q,q), independently of each other. Also, use simulation to find this probability when A, B, and C are nonzero integers that are
1. 2.16 You choose three points at random inside a square. Then choose a fourth point at random inside the square. What is the probability that the triangle formed by the first three points is obtuse? What is the probability that the fourth point will fall inside this triangle? What are the
1. 2.15 At a completely random moment between 6:30 and 7:30 a.m., the morning news paper is delivered to Mr. Johnson’s residence. Mr. Johnson leaves for work at a completely random moment between 7:00 and 8:00 a.m. regardless of whether the newspaper has been delivered. What is the probability
1. 2.14 Three players enter a room and are given a red or a blue hat to wear. The color of each hat is determined by a fair coin-toss. Players cannot see the color of their own hats, but do see the color of the other two players’ hats. The game is won when at least
1. 2.13 The following game is played in a particular carnival tent. The carnival master has two covered beakers, each containing one die. He shakes the beakers thoroughly, removes the lids and peers inside. You have agreed that whenever at least one of the two dice shows an even number of points,
1. 2.12 Three players, A, B, and C, each put ten dollars into a pot with a list on which they have, independently of one another, predicted the outcome of three successive tosses of afair coin. Thefaircoinisthentossedthreetimes.Theplayerhavingmost correctly predicted the three outcomes gets the
1.2.11 In a group of 25 people, a person tells a rumor to a second person, who in turns tells it to a third person, and so on. Each person tells the rumor to just one of the people chosen at random, excluding the person from whom he/she heard the rumor. The rumor is told 10 times. What is the
1. 2.10 A particular game pays f1 times the amount staked with a probability of p and f2 times the amount staked with a probability of 1 − p, where f1 > 1,0 ≤ f2 < 1 and pf1 +(1− p)f2 > 1. You play this game a large number of times and each time you stake the same fraction α of your
1. 2.9 ConsidertheKellybetting model fromSection 2.7. In addition to the possibility of investing in a risky project over a large number of successive periods, you can get a fixed interest rate at the bank for the portion of your capital that you do not invest.You can reinvest your money at the end
1. 2.8 Sic Bo is an ancient Chinese dice game that is played with three dice. There are many possibilities for betting on this game. Two of these are “big” and “small.”Whenyoubet “big,” you win if the total points rolled equals 11, 12, 13, 14, 15, 16 or 17, except when three 4’s or
1. 2.7 Inthedicegameknownas“seven,”twofairdicearerolledandthesumofscoresis counted. Youbeton“manque”(thatasumof2,3,4,5or6willresult)oron“passe”(that a sum of 8, 9, 10, 11 or 12 will result). The sum of 7 is a fixed winner for the house. A winner receives a payoff that is double the
1. 2.6 In the daily lottery game “Guess 3,” three different numbers are picked randomly from the numbers 0,1,...,9. The numbers are picked in order. To play this game, you must choose between “Exact order” and “Any order” on the entry form. In either case, the game costs $1 to play.
1. 2.5 Use an appropriate sample space with equiprobable elements to answer the fol lowing question. You enter a grand-prize lottery along with nine other people. Ten numbered lots, including the winning lot, go into a box. One at a time, participants draw a lot out of the box. Does it make a
1.2.4 Answer each of the following four questions by choosing an appropriate sample space and assigning probabilities to the various elements of the sample space.(a) In Leakwater township, there are two plumbers. On a particular day three Leakwater residents call village plumbers independently of
1. 2.3 Adoghasalitteroffourpuppies.Setupaprobabilitymodeltoanswerthefollowing question. Can we correctly say that the litter more likely consists of three puppies of one gender and one of the other than that it consists of two puppies of each gender?
1.2.2 Inthe television program “Big Sisters,” 12 candidates remain. The public chooses four candidates for the final round. Each candidate has an equal probability of being chosen. The Gotham Echo reckons that the local heroine, Stella Stone, has a probability of 38.5% of getting through to the
1.2.1 On a modern die the face value 6 is opposite to the face value 1, the face value 5 to the face value 2, and the face value 4 to the face value 3. In other words, by turning a die upside down, the face value k is changed into 7 − k. This fact may be used to explain why when rolling three
1. The psychology of probability intuition is a main feature of some of these problems. Consider the birthday problem: how large must a group of randomly chosen people be such that the probability of two people having birthdays on the same day will be at least 50%? The answer to this question is
1. Question 12. A daughter-son problem (§2.9, §6.1)You are told that a family, completely unknown to you, has two children and that one of these children is a daughter. Is the chance of the other child also being a daughter equal to 1 2 or 1 3 ? Are the chances altered if, aware of the fact that
1. Question 11. The Monty Hall dilemma (§6.1)A game-show climax draws nigh. A drum-roll sounds. The game show host leads you to a wall with three closed doors. Behind one of the doors is the automobile of your dreams, and behind each of the other two is a can of dog food. The three doors all have
1. You fold up the 20 pieces of paper and place them randomly onto a tabletop.Your friend opens the papers one by one. Each time he opens one, he must decide whether to stop with that one or go on to openanother one. Your friend’s task is to single out the paper displaying the highest number.
1. Question 10. The best-choice problem (§2.3)Your friend proposes the following wager: 20 people are requested, indepen dently of one another, to write a number on a piece of paper (the papers should be evenly sized). They may write any number they like, no matter how high.
1.Question 9. A statistical test problem (§3.6)Using one die and rolling it 1,200 times, someone claims to have rolled the points 1, 2, 3, 4, 5, and 6 for a respective total of 196, 202, 199, 198, 202, and 203 times. Do you believe that these outcomes are, indeed, the result of coincidence or do
1.Question 8. A sock problem (Appendix)You have taken ten different pairs of socks to the laundromat, and during the washing, six socks are lost. In the best-case scenario, you will still have seven matching pairs left. In the worst-case scenario, you will have four matching pairs left. Do you
1. Question 7. A coincidence problem (§4.3)Two people, perfect strangers to one another, both living in the same city of one million inhabitants, meet each other. Each has approximately 500 acquain tances in the city. Assuming that for each of the two people, the acquaintances represent a random
1. Question 6. Who is the murderer? (§8.2)A murder is committed. The perpetrator is either one or the other of the two persons X andY.Bothpersonsareontherunfromauthorities,andafteraninitial investigation, both fugitives appear equally likely to be the perpetrator. Further investigation reveals
1. Question 5. Hitting the jackpot (Appendix)Is the probability of hitting the jackpot (getting all six numbers right) in a 6/45 Lottery greater or lesser than the probability of throwing heads only in 22 tosses of a fair coin?
1. Question 4. A lotto problem (§4.2.3)In eachdrawingofLotto6/45,sixdistinctnumbersaredrawnfromthenumbers 1,...,45. In an analysis of 30 such lotto drawings, it was apparent that some numbers were never drawn. This is surprising. In total, 30 × 6 = 180 numbers were drawn, and it was expected that
1.Question 3. A scratch-and-win lottery (§4.2.3)Ascratch-and-win lottery dispenses 10,000 lottery tickets per week in Andorra and ten million in Spain. In both countries, demand exceeds supply. There are two numbers, composed of multiple digits, on every lottery ticket. One of these numbers is
1. Question 2. Probability of winning streaks (§2.1.3, §5.9.1)Abasketball player has a 50% success rate in free throw shots. Assuming that the outcomes of all free throws are independent from one another, what is the probability that, within a sequence of 20 shots, the player can score five
1.You gowith a friend to a football (soccer) game. The game involves 22 players of the two teams and one referee. Your friend wagers that, among these 23 persons on the field, at least two people will have birthdays on the same day.You will receive ten dollars from your friend if this is not the
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