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elementary probability for applications

Understanding Probability 2nd Edition Henk Tijms - Solutions

1. Problem 7.17 A small transport company has two vehicles, a truck and a van.The truck is used 75% of the time. Both vehicles are used 30% of the time and neither of the vehicles is used for 10% of the time. What is the probability that the van is used on any given day?
1. Problem 7.16 The event A has probability 2 3and there is a probability of 3 4that at least one of the events A and B occurs. What are the smallest and largest possible values for the probability of event B?
1.Problem 7.15 The probability that the events A and B both occur is 0.3. The individual probabilities of the events A and B are 0.7 and 0.5. What is the probability that neither event A nor event B occurs?
1. Problem7.14Inrepeatedlyrollingtwodice,whatistheprobabilityofgetting atotalof6beforeatotalof7?Whataboutan8anda7?Whatistheprobability ofgettingatotalof6andatotalof8inanyorderbeforetwo7’s?
1. Problem7.13Twopeopletaketurnsselectingaballat randomfromabowl containingthreewhiteballsandsevenredones.Thewinneristhepersonwho isthefirst toselectawhiteball. It isassumedthat theballsareselectedwith replacement.Defineanappropriatesamplespaceandcalculatetheprobability
1.Problem7.12Ina tennis tournament between threeplayers A, B, andC, eachplayerplaystheothersonce.Thestrengthsoftheplayerareasfollows:P(AbeatsB)=0.5,P(AbeatsC)=0.7,andP(BbeatsC)=0.4.Assuming independenceofthematchresults,calculatetheprobabilitythatplayerAwins atleastasmanygamesasanyotherplayer.
1. Problem 7.11 Let A1, A2,... be an infinite sequence of subsets with the property that k=1 P(Ak) < ∞. Define the set C as C ={ω: ω ∈ Ak for infinitely many k}. Use the continuity property of probabilities to prove that P(C) = 0 (this result is known as the Borel-Cantelli lemma).
1.Problem 7.10 Use the axioms to prove the following results:(a) P(A) ≤ P(B) if the set A is contained in the set B.(b) P ∪∞k=1 Ak ≤ ∞k=1 P(Ak)for any sequence of subsets A1, A2,...(this result is known as Boole’s inequality).∞
1.Problem 7.9 A point is chosen at random inside a triangle with height h and base of lengthb. What is the probability that the perpendicular distance from the point to the base is larger than d? What is the probability that the randomly chosen point and the base of the triangle will form a
1. Problem 7.8 A dart is thrown at random on a rectangular board. The board measures 20 cm by 50 cm. A hit occurs if the dart lands within 5 cm of any of the four corner points of the board. What is the probability of a hit?
1. Problem 7.7 The numbers B and C are chosen at random between–1 and 1, independently of each other. What is the probability that the quadratic equation x2 + Bx+C =0 has real roots? Also, derive a general expression for this probability when B and C are chosen at random from the interval
1. Problem7.6Twopeoplehaveagreedtomeetatthetrain station between 12.00 and 1.00 p.m. Independently of one another, each person is to appear at a completely random moment between the hours of 12.00 and 1.00. What is the probability that the two persons will meet within 10 minutes of one another?
1. Problem 7.5 The game of franc-carreau was a popular game in eighteenth century France. In this game, a coin is tossed on a chessboard. The player wins if the coin does not fall on one of the lines of the board. Suppose now that a round coin with a diameter of d is blindly tossed on a large
1. Problem 7.4 Three friends go to the cinema together on a weekly basis. Before buying their tickets, all three friends toss a fair coin into the air once. If one of the three gets a different outcome than the other two, that one pays for all three tickets; otherwise, everyone pays his own way.
1.Problem 7.3 Sixteen bridge teams including the teams Johnson and Smith participate in a tournament. The tournament is organized as a knock-out tour nament and has four rounds. The 16 teams are evenly matched. In each round the remaining teams are paired by drawing lots.(a) What is the probability
1. Problem7.2Independentlyofeachother,twopeoplethinkofanumberbetween 1 and 10. What is the probability that five or more numbers will separate the two numbers chosen at random by the two people?
1.Problem 7.1 Two players A and B each roll one die. The absolute difference of the outcomes is computed. Player A wins if the difference is 0, 1, or 2;otherwise, player B wins. What is the probability that player A wins?7.1 Probabilistic foundations 229
1. 6.17 Inatelevisiongameshow,youcanwin10,000dollarsbyguessingthecomposition of red and white marbles contained in a nontransparent vase. The vase contains a very large number of marbles. You must guess whether the vase has twice as many red marbles as white ones, or whether it has twice as many
1. 6.16 A sum of money is placed in each of two envelopes. The amounts differ from one another, but you do not know what the values of the two amounts are. You do know that the values lie between two boundaries m and M with 0 < m < M.You choose an envelope randomly. After inspecting its contents,
1. 6.15 Ataparticular airport, each passenger must pass through a special fire arms detec tor. An average of 1 out of every 100,000 passengers is carrying a fire arm. The detector is 100% accurate in the detection of fire arms, but in an average of 1 in 10,000 cases, it results in a false alarm
1. 6.14 Suppose that there is a DNA test that determines with 100% accuracy whether or not a particular gene for a certain disease is present. A woman would like to do the DNAtest, but wants to have the option of holding out hope that the gene is not present in her DNA even if it is determined that
1. 6.13 Consider the sushi delight problem from Section 6.1. Suppose now that both a piranha and a goldfish are added to the fishbowl alongside the original fish. What is the probability that the original fish is a piranha if a piranha is taken out of the bowl?
1. 6.12 A doctor finds evidence of a serious illness in a particular patient and must make a determination about whether or not to advise the patient to undergo a dangerous operation. If the patient does suffer from the illness in question, there is a 95%probability that he will die if he does not
1. 6.11 There are two taxicab companies in a particular city, “Yellow Cabs” and “White Cabs.”Ofallthecabsinthecity,85%are“YellowCabs”and15%are“WhiteCabs.”The issue of cab color has become relevant in a hit-and-run case before the courts in this city, in which witness testimony will
1. 6.10 You know that bowl A has three red and two white balls inside and that bowl B has four red and three white balls. Without your being aware of which one it is, one of the bowls is randomly chosen and presented to you. Blindfolded, you must pick two balls out of the bowl. You may proceed
1. 6.9 Alcohol checks are regularly conducted among drivers in a particular region.Drivers are first subjected to a breathtest. Only after a positive breathtest result is a driver taken for a blood test. This test will determine whether the driver has been driving under the influence of alcohol.
1. 6.8 Passers-by are invited to take part in the following sidewalk betting game. Three cards are placed into a hat. One card is red on both sides, one is black on both sides, and one is red on one side and black on the other side. A participant is asked to pick a card out of the hat at random,
1. 6.7 The final match of world championship soccer is to be played between England and the Netherlands. The star player for the Dutch team, Dennis Nightmare, has† In his book Calculated Risks (Simon & Schuster, 2002) Gerd Gigerenzer advocates that doctors and lawyers be educated in more
1. 6.6 Consider the following variant of the Monty Hall dilemma. There are now four doors, behind one of which there is an automobile. You first indicate a door. Then the host opens another door behind which a gag prize is to be found. You are now given the opportunity to switch doors. Regardless
1. 6.5 Consider the Monty Hall dilemma with the following twists: there are five doors, and the host promises to open two of the gag prize doors after the contestant has chosen a door. Set up a chance tree to calculate the probability of the contestant winning the automobile by switching doors.
1. 6.4 NowconsidertheMontyHalldilemmafromSection6.1withthefollowing differ ence: youlearnedbeforehandthatthere is a0.2probability of the automobilebeing behind door 1, a 0.3 probability of its being behind door 2, and a 0.5 probability of its being behind door 3. Your strategy is to choose the door
1. 6.3 You are one of 50,000 spectators at a baseball game. Upon entering the ballpark, each spectator has received a ticket bearing an individual number. A winning number will be drawn from all of these 50,000 numbers. At a certain point, five numbers are called out over the loudspeaker. These
1. 6.2 On a table before you are two bowls containing red and white marbles; the first bowl contains seven red and three white marbles, and the second bowl contains 70 red and 30 white marbles. You are asked to select one of the two bowls, from which you will blindly draw two marbles (with no
1.6.1 The roads are safer at nonrush hour times than during rush hour because fewer accidents occur outside of rush hour than during the rush hour crunch. Do you agree or do you disagree?
1. 5.34 Youhaveaneconomywithariskyassetandarisklessasset.Yourstrategyistohold always a constant proportion α of your wealth in the risky asset and the remaining proportion of your wealth in the riskless asset, where 0
1. 5.33 Consider the stock price process {St} from Section 5.8.6. Verify that the proba bility of the stock price increasing to aS0 without falling down first to bS0 equals[1 −b2(μ−1 2σ2)/σ2]/[1 − (b/a)2(μ−1 2σ2)/σ2] for 0 < b < 1 < a.
1. 5.32 Use the gambler’s ruin formula from Problem 3.26 to make plausible that, for anyc, d > 0, P(process hits c before −d) = 1−e−2dμ/σ2 1 −e−2(d+c)μ/σ2 for a Brownian motion process with drift parameter μ= 0 and variance parameter σ2 (the probability is d/(d + c)ifμ = 0).
1. 5.31 Inorder to test a new pseudo-random number generator, we let it generate 100,000 random numbers. From this result, we go on to form a binary sequence in which the ith element will be equal to 0 if the ith randomly generated number is smaller than 1 2, and will otherwise be equal to 1. The
1. 5.30 You are interested in assembling a random sample of young people that occasion ally use soft drugs. To prevent people from falsely claiming not to use soft drugs, you have thought of the following procedure. The interviewer asks each young person to toss a coin, keeping the result of the
1. 5.29 In 1986, an article appeared on the front page of the New York Times about the results of a research project on the effect of a light dose of aspirin on the incidence of heart attacks. By means of a carefully selected randomization method, a group of 22,000 healthy middle-aged males was
1. 5.28 Six million voters are expected to vote in the upcoming presidential election.There are two candidates, A and B. The voters cast their ballots independently of one another and each voter will vote for candidate A with probability p and for candidate B with probability 1 − p. Calculate for
1. 5.27 The Dutch Ministry of Education has taken a random sampling of the student population of 400. The students in the sample group were asked if they were in favor of the introduction of a weekend pass for public transportation. Suppose that 208 students were in favor of the pass. Give a 95%
1. 5.26 The Nero Palace casino has a new, exciting gambling machine: the multiplying bandit. How does it work? The bandit has a lever or “arm” that the player may depress up to ten times. After each pull, an H (heads) or a T (tails) appears, each with probability 1 2. The game is over when
1. 5.25 A damage claims insurance company has 20,000 policyholders. The amount claimed yearly by policyholders has an expected value of $150 and a standard deviation of $750. Give an approximation for the probability that the total amount claimed in the coming year will be larger than 3.3 million
1. 5.24 In a particular small hospital, approximately 25 babies per week are born, while in a large hospital approximately 75 babies per week are born. Which hospital, do you think, has a higher percentage of weeks during which more than 60% of the newborn babies are boys? Argue your answer without
1. 5.23 The Dutch lotto formerly consisted of drawing six numbers from the numbers 1,...,45 but the rules were changed. In addition to six numbers from 1,...,45, a colored ball is drawn from six distinct colored balls. A statistical analysis of the lotto drawings in the first two years of the new
1. 5.22 In the 52 drawings of Lotto 6/45 in Orange Country last year an even number was drawn 162 times and an odd number 150 times. Does this outcome cast doubts on the unbiased nature of the drawings? Hint: the number of even numbers obtained in a single drawing of Lotto 6/45 has a hypergeometric
1. 5.21 A gambler claims to have rolled an average of 3.25 points per roll in 1,000 rolls of a fair die. Do you believe this?
1. 5.20 A large table is marked with parallel and equidistant lines a distance D apart. A needle of length L(≤ D) is tossed in the air and falls at random onto the table.The eighteenth century French scientist Georges-Louis Buffon proved that the probability of the needle falling across one of
1. 5.19 A national information line gets approximately 100 telephone calls per day. On a particular day, only 70 calls come in. Is this extraordinary?
1. 5.18 In a particular area, the number of traffic accidents hovers around an average of 1,050. Last year, however, the number of accidents plunged drastically to 920.Authorities suggest that the decrease is the result of new traffic safety measures that have been in effect for one year.
1. 5.17 Each year in Houndsville an average of 81 letter carriers are bitten by dogs. In the past year, 117 such incidents were reported. Is this number exceptionally high?
1. 5.16 The owner of a casino in Las Vegas claims to have a perfectly balanced roulette wheel. Aspinofaperfectly balanced wheel stops on red an average of 18 out of 38 times. A test consisting of 2,500 trials delivers 1,105 red finishes. If the wheel is perfectly balanced, is this result plausible?
1. 5.15 Whathappensto the value of the probability of getting at least r sixes in one throw of 6r dice as r →∞?Explain your answer.
1. 5.2. Assume that there is one restroom for women only and one restroom for men only, the arrival processes of women and men are Poisson processes with equal intensities, and the coefficient of variation of the time people spend in the restroom is the same for women as for men.
1. 5.14 Women spend on average about twice as much time in the restroom as men, but why is the queue for the women’s restroom on average four or more times as long as the one for the men’s? This intriguing question was answered in the article“Ladies in waiting” by Robert Matthews in New
1. 5.13 Aninvestor decides to place $2,500 in an investment fund at the beginning of each yearforaperiodof20years.Therateofreturnonthefundwas14%fortheprevious year. If the yearly rate of return remained at 14% for each year, then, at the end of 20 years, the investor will have an amount of 20 k=1(1
1. 5.12 TheArgusInvestmentFund’sSpiderwebPlanisa60-month-longcontract accord ing to which the customer agrees to deposit a fixed amount at the beginning of eachmonth.Thecustomerchoosesbeforehandforafixeddepositof$100,$250,or$500. Argus then immediately deposits 150 times that monthly amount, to
1. 5.11 Consider the investment example from Section 5.2 in which a retiree invests$100,000 in a fund in order to reap the benefits for 20 years. The rate of return on the fund for the past year was 14%, and the retiree hopes for a yearly profit of $15,098 over the coming 20 years. If the rate of
1. 5.10 Suppose that the random variables X1, X2,...,Xn are defined on a same proba bility space. In Chapter 11, it will be seen that nσ2 i=1 nXi =i=1 n−1σ2 (Xi) +2 i=1 nj=i+1 cov(Xi, Xj).For the case that X1,...,Xn all have the same variance σ2 and cov(Xi, Xj) is equal to a constant c= 0 for
1.5.9 Youwanttoinvestintwostocks A and B.Theratesofreturnonthesestocksinthe coming year depend on the development of the economy. The economic prospects forthecomingyearconsistofthreeequallylikelycasescenarios:astrongeconomy, anormaleconomy,andaweakeconomy.Iftheeconomyisstrong,therateofreturn on
1. 5.8 You wish to invest in two funds, A and B, both having the same expected return.Thereturns of the funds are negatively correlated with correlation coefficient ρAB.The standard deviations of the returns on funds A and B are given by σA and σB.Demonstrate that you can achieve a portfolio
1. 5.7 Suppose that the rate of return on stock A takes on the values 30%, 10%, and−10%withrespective probabilities 0.25, 0.50, and 0.25 and on stock B the values 50%,10%,and−30%withthesameprobabilities0.25,0.50,and0.25.Eachstock, then, has an expected rate of return of 10%. Without calculating
1. 5.6 Inasingle-product inventory system a replenishment order will be placed as soon as the inventory on hand drops to the level s. You want to choose the reorder point s such that the probability of a stockout during the replenishment lead time is no more than 5%. Verify that s should be taken
1. 5.5 Gestation periods of humans are normally distributed with an expected value of 266 days and a standard deviation of 16 days. What is the percentage of births that are more than 20 days overdue?
1. 5.4 The cholesterol level for an adult male of a specific racial group is normally distributed with an expected value of 5.2 mmol/l and a standard deviation of 0.65 mmol/l. Which cholesterol level is exceeded by 5% of the population?
1. 5.3 The annual rainfall in Amsterdam is normally distributed with an expected value of 799.5 mm and a standard deviation of 121.4 mm. Over many years, what is the proportion of years that the annual rainfall in Amsterdam is below 550 mm?
1. 5.2 Someone has written a simulation program in an attempt to estimate a particu lar probability. Five hundred simulation runs result in an estimate of 0.451 for the unknown probability with 0.451±0.021 as the corresponding 95% confidence interval. One thousand simulation runs give an estimate
1.5.1 You draw 12 random numbers from (0,1) and average these 12 random numbers.Which of the following statements is then correct?(a) the average has the same uniform distribution as each of the random numbers(b) the distribution of the average becomes more concentrated in the middle and less at
1. 4.40 In Lottoland, there is a weekly lottery in which six (standard) numbers plus one bonus numberaredrawnfromthenumbers1,...,45.Inadditiontothis, onecolor is randomly drawn out of six colors. On the lottery ticket, six numbers and one color must be chosen. The players use the computer for a
1. 4.39 In the German “Lotto am Samstag,” six regular numbers and one reserve number are drawn from the numbers 1,...,49. On the lottery ticket, players must tick six different numbers out of the numbers 1,...,49. There is also an area of the lottery ticket reserved for
1. 4.38 YouplayBingotogetherwith35otherpeople.Eachplayerpurchasesonecardwith 24 different numbers that were selected at random out of the numbers 1,...,80.The organizer of the game calls out random numbers between 1 and 80, one at a time. The first player to achieve a card with all of his/her 24
1. 4.37 Take another look at the lottery problem in Section 3.5. If we divide the lottery numbers 1,...,366 into three equal groups, then we can see from Table 3.3 that 17 or more days in December fall into the first group of low numbers 1,...,122.In a fair drawing, what would be the probability of
1.4.36 Suppose that emergency response units are distributed throughout a large area according to a two-dimensional Poisson process. That is, the number of response units in any given bounded region has a Poisson distribution whose expected value is proportional to the area of the region, and the
1. 4.35 Paying customers (i.e., those who park legally) arrive at a large parking lot accord ing to a Poisson process with an average of 45 cars per hour. Independently of this, nonpaying customers (i.e., those who park illegally) arrive at the parking lot according to a Poisson process with an
1. 4.34 During the course of a summer day, tourist buses come and go in the picturesque town of Edam according to a Poisson process with an average arrival rate of five buses per hour. Each bus stays approximately two hours in Edam, which is famous for its cheese. What is the distribution of the
1.4.33 Calls arrive at a computer-controlled exchange according to a Poisson process at a rate of two calls per second. Use computer simulation to find the probability that during the busy hour there will be some period of 30 seconds in which 90 or more calls arrive.
1. 4.32 Inthefirstfivemonthsoftheyear2000,thetramhitandkilledsevenpedestriansin Amsterdam, each case caused by the pedestrian’s own carelessness. In preceding years, such accidents occurred at a rate of 3.7 times per year. Simulate a Poisson process to estimate the probability that within a
1. 4.31 A businessman parks his car illegally for one hour, twice a day, along the banks of an Amsterdam canal. During the course of an ordinary day, parking attendants monitor the streets according to a Poisson process with an average of α rounds per hour. What is the probability that the
1. 4.30 Argue that the following two problems are manifestations of the “hat-check”problem:(a) In a particular branch of a company, the 15 employees have agreed that, for the upcoming Christmas party, each employee will bring one present without putting any name on it. The presents will be
1. 4.29 Acompanyhas75employeesinservice.Theadministratorofthecompanynotices, to hisastonishment,thattherearesevendaysonwhichtwoormoreemployeeshave birthdays. Verify, by using a Poisson approximation, whether this is so astonishing after all.
1. 4.28 Calculate a Poisson approximation for the probability that in a thoroughly shuffled deck of 52 playing cards, it will occur at least one time that two cards of the same face valuewillsucceedoneanotherinthedeck(twoaces,forexample).Inaddition, makethe same calculation for the probability of
1. 4.27 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. You are asked to predict all eight matches, paying no regard to the
1. 4.26 Calculate a Poisson approximation for the probability that in a randomly selected group of 2,287 persons all of the 365 possible birthdays will be represented.
1. 4.25 What is the probability of two consecutive numbers appearing in any given lotto drawing of six numbers from the numbers 1,...,45? Calculate a Poisson approximation for this probability. Also, calculate a Poisson approximation for the probability of three consecutive numbers appearing in any
1. 4.24 Three people each write down the numbers 1,...,10 in a random order. Calculate a Poisson approximation for the probability that the three people all have one number in the same position.
1. 4.23 A group of 25 students is going on a study trip of 14 days. Calculate a Poisson approximation for the probability that during this trip two or more students from the group will have birthdays on the same day.
1. 4.22 Ten married couples are invited to a bridge party. Bridge partners are chosen at random, without regard to gender. What is the probability of at least one cou ple being paired as bridge partners? Calculate a Poisson approximation for this probability.
1. 4.21 What is a Poisson approximation for the probability that in a randomly selected group of 25 persons, three or more will have birthdays on a same day. What is a Poisson approximation for the probability that three or more persons from the group will have birthdays falling within one day of
1. 4.20 In the Massachusetts Numbers Game, one number is drawn each day from the 10,000 four-digit number sequence 0000,0001,...,9999. Calculate a Poisson approximation for the probability that the same number will be chosen two or more times in the upcoming 625 drawings. Before making the
1. 4.19 AnorganizationrunningtheLotto6/45analyzes100,000tickets that were filled-in by hand. On each ticket of the Lotto 6/45 six different numbers from the numbers 1,...,45 are filled in. A particular pick of six numbers occurred eight times in the 100,000 tickets. What is the probability of the
1. 4.18 You are at an assembly where 500 other persons are also present. The organizers of the assembly are raffling off a prize to be shared by all of those present whose birthday falls on that particular day. What is the probability that you will win the prize?
1. 4.17 TheBrederodeFinanceCorporationhasbegunthefollowingadvertisingcampaign in Holland. Each new loan application submitted is accompanied by a chance to win a prize of $25,000. Every month 100 zip codes will be drawn in a lottery. In Holland each house address has a unique zip code and there are
1. 4.16 In the kingdom of Lightstone, the game of Lotto 6/42 is played. In Lotto 6/42 six numbers out of the numbers 1,...,42 are drawn. At the time of an oil sheik’s visit to Lightstone the jackpot for the next drawing is listed at 27.5 million dollars.The oil sheik decides to take a gamble and
1. 4.15 A particular game is played with five poker dice. Each die displays an ace, king, queen, jack, ten and nine. Players may bet on two of the six images displayed.Whenthe dice are thrown and the bet-on images turn up, the player receives three times the amount wagered. In all other cases, the
1. 4.14 Aparticular scratch-lottery ticket has 16 painted boxes on it, each box having one of the numbers 1, 2, 5, 10, 100 or 1,000 hidden under the paint. When the paint is scratched off and it appears that a same number shows up in seven or more boxes, the player wins an amount equal to that
1. 4.13 FortheupcomingdrawingoftheTelenetLottery,fiveextraprizes have beenadded tothepot.Eachprizeconsistsofanall-expensespaidvacationtrip.Thefivewinners of the extra prize maychoosefromamongthreepossibledestinationsi = 1,2,and 3. The winners choose independently of each other. The probability that
1. 4.12 Consider an experiment with three possible outcomes 1,2, and 3, which occur with probabilities of p1, p2, and p3 = 1 − p1 − p2. For a given value of n, n independent trials of the experiment are performed. The random variable Xi gives the number of times that the outcome i occurs for i
1. 4.11 Decco is played with an ordinary deck of 52 playing cards. It costs $1 to play this game.Havingpurchasedaticketonwhichthe52playingcardsofanordinarydeck are represented, each player ticks his choice of one card from each of the four suits(the ten of hearts, jack of clubs, two of spades and
1. 4.10 Operating from within a tax-haven, some quick-witted businessmen have started an Internet Web site called Stockgamble. Through this Web site, interested parties can playthestockmarketsinanumberofcountries.Eachoftheparticipatingstock markets lists 24 stocks available in their country. The

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