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Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions
17. In a study it was discovered that 25%of the paintings of a certain gallery are not original.A collector in 15% of the cases makes a mistake in judging if a painting is authentic or a copy. If she buys a piece thinking that it is original, what is the probability that it is not?
16. Urns I, II, and III contain three pennies and four dimes, two pennies and five dimes, three pennies and one dime, respectively. One coin is selected at random from each urn.If two of the three coins are dimes, what is the probability that the coin selected from urn I is a dime?
15. At a grocery store, an absent-minded, honest professor,Dexter, hands the playful cashier, Hans, a dollar bill. Hans puts the bill in the cash register drawer and gives Dexter $2.75 in change. Dexter, who really does not remember whether he handed him a $10 bill or a$20 bill, claims that he gave
14. A certain cancer is found in one person in 5000. If a person does have the disease, in 92% of the cases the diagnostic procedure will show that he or she actually has it. If a person does not have the disease, the diagnostic procedure in one out of 500 cases gives a false positive result.
13. An actuary has calculated that, for the age group 16-25, the probability is 0.08 that a driver whose car is insured by her company gets involved in a car accident within a year. For the age groups 26-35, 36-65, and 66-100, the corresponding probabilities are 0.03, 0.02, and 0.04, respectively.
12. Based on an insurance company’s evaluation, with probability 0.85, Whitney is a safe driver, and with probability 0.15 she is not. Suppose that a safe driver avoids at-fault accidents in the next year with probability 0.87, and an unsafe driver avoids at-fault accidents in that period with
11. Suppose that currently it is a bull market, and in the stock market, share prices are rising. Furthermore, suppose that the Nasdaq Composite index closes at higher points 86% of the trading days. Iniko is a Nasdaq financial expert, and when she predicts that the Nasdaq Composite index closes at
10. A stack of cards consists of six red and five blue cards. A second stack of cards consists of nine red cards. A stack is selected at random and three of its cards are drawn. If all of them are red, what is the probability that the first stack was selected?
9. There are three dice in a small box. The first die is unbiased; the second one is loaded, and, when tossed, the probability of obtaining 6 is 3/8, and the probability of obtaining each of the other faces is 1/8. The third die is also loaded, but the probability of obtaining 6 when tossed is 2/7,
8. Suppose that 5% of the men and 2% of the women working for a corporation make over$120,000 a year. If 30% of the employees of the corporation are women, what percent of those who make over $120,000 a year are women?
7. In a trial, the judge is 65% sure that Susan has committed a crime. Julie and Robert are two witnesses who know whether Susan is innocent or guilty. However, Robert is Susan’s friend and will lie with probability 0.25 if Susan is guilty. He will tell the truth if she is innocent. Julie is
6. When traveling from Springfield, Massachusetts, to Baltimore, Maryland, 30% of the time Charles takes I-91 south and then I-95 south, and 70% of the time he takes I-91 south, then I-84 west followed by I-81 south and I-83 south. Choosing the first option takes Charles a random time between 5
5. When Professor Wagoner teaches calculus, he only grades 25% of the students’ exam papers, randomly selected. The remaining papers are graded by his teaching assistants(TA’s). Suppose that 78% of the papers graded by Professor Wagoner and 86% of the papers graded by the TA’s get a passing
4. A judge is 65% sure that a suspect has committed a crime. During the course of the trial, a witness convinces the judge that there is an 85% chance that the criminal is lefthanded.If 23% of the population is left-handed and the suspect is also left-handed, with this new information, how certain
3. Suppose that 20%of the e-mails Derek receives are spam. Suppose that 12%of the spam e-mails and 0.05%of the non-spam e-mails are concerning the e-mail storage in Derek’s e-mail account. Derek has just received an e-mail concerning his e-mail storage. What is the probability that it is spam?
2. On a multiple-choice exam with four choices for each question, a student either knows the answer to a question or marks it at random. If the probability that he or she knows the answers is 2/3, what is the probability that an answer that was marked correctly was not marked randomly?
1. In transmitting dot and dash signals, a communication system changes 1/4 of the dots to dashes and 1/3 of the dashes to dots. If 40% of the signals transmitted are dots and 60% are dashes, what is the probability that a dot received was actually a transmitted dot?
A box contains seven red and 13 blue balls. Two balls are selected at random and are discardedwithout their colors being seen. If a third ball is drawn randomly and observed to be red, what is the probability that both of the discarded balls were blue?
On the basis of reconnaissance reports, Colonel Smith decides that the probability of an enemy attack against the left is 0.20, against the center is 0.50, and against the right is 0.30. A flurry of enemy radio traffic occurs in preparation for the attack. Since deception is normal as a prelude to
During a double homicide murder trial, based on circumstantial evidence alone, the jury becomes 15% certain that a suspect is guilty. DNA samples recovered from the murder scene are then compared with DNA samples extracted from the suspect. Given the size and conditions of the recovered samples, a
In a study conducted three years ago, 82% of the people in a randomly selected sample were found to have “good” financial credit ratings, while the remaining 18%were found to have “bad” financial credit ratings. Current records of the people from that sample show that 30% of those with bad
3. Seventy percent of all students of a college participate in a study abroad program. If 60% of the students of this college are male and 55% of the male students attend a study abroad program, what percentage of female students participate in a study abroad program?
2. There are three dice in a small box. The first die is unbiased; the second one is loaded, and, when tossed, the probability of obtaining 6 is 3/8, and the probability of obtaining each of the other faces is 1/8. The third die is also loaded, but the probability of obtaining 6 when tossed is 2/7,
1. At a kindergarten, there is a box full of balls. Each child draws a ball at random to play with and is not allowed to return it to the box to draw another one. When it is Natalie’s turn to draw a ball, there are 3 red and 7 blue balls left in the box. Natalie loves to play with a red ball and
27. Suppose that three numbers are selected one by one, at random and without replacement from the set of numbers {1, 2, 3, . . . , n}. What is the probability that the third number falls between the first two if the first number is smaller than the second?
26. From families with three children, a child is selected at random and found to be a girl.What is the probability that she has an older sister? Assume that in a three-child family all sex distributions are equally probable.Hint: Let G be the event that the randomly selected child is a girl, A be
25. A box contains 18 tennis balls, of which eight are new. Suppose that three balls are selected randomly, played with, and after play are returned to the box. If another three balls are selected for play a second time, what is the probability that they are all new?
24. Suppose that 10 good and three dead batteries are mixed up. Jack tests them one by one, at random and without replacement. But before testing the fifth battery he realizes that he does not remember whether the first one tested is good or is dead. All he remembers is that the last three that
23. For n ≥ 1, let E1, E2, . . . , En be events of a sample space. Find an n-element partition of Sn i=1 Ei. That is, find a set of mutually exclusive events {F1, F2, . . . , Fn} that satisfies Sn i=1 Fi =Sn i=1 Ei.
22. Suppose that the probability that a new seed planted in a specific farm germinates is equal to the proportion of all planted seeds that germinated in that farm previously.Suppose that the first seed planted in the farm germinated, but the second seed planted did not germinate. For positive
21. Suppose that 40% of the students on a campus, who are married to students on the same campus, are female. Moreover, suppose that 30% of those who are married, but not to students at this campus, are also female. If one-third of the married students on this campus are married to other students
20. Let B be an event of a sample space S with P(B) > 0. For a subset A of S, define Q(A) = P(A | B). By Theorem 3.1 we know that Q is a probability function. For E and F, events of SP(FB) > 0, show that Q(E | F) = P(E | FB).
19. Suppose that there exist N families on the earth and that the maximum number of children a family has isc. Let αj????0 ≤ j ≤ c, Pc j=0 αj = 1be the fraction of families with j children. Find the fraction of all children in the world who are the kth born of their families (k = 1, 2, . . .
18. A number is selected at random from the set {1, 2, . . . , 20}. Then a second number is selected randomly between 1 and the first number selected. What is the probability that the second number is 5?
17. An actuary has discovered that, in her company, 65% of those who have only income protection insurance and 80%of those who have only legal expense insurance will renew their policies next year. Furthermore, she has observed that 87% of those who have both of these policies will renew at least
16. A child gets lost in the Disneyland at the Epcot Center in Florida. The father of the child believes that the probability of his being lost in the east wing of the center is 0.75 and in the west wing is 0.25. The security department sends an officer to the east and an officer to the west to
15. In a town, 7/9th of the men and 3/5th of the women are married. In that town, what fraction of the adults are married? Assume that all married adults are the residents of the town.
14. Suppose that five coins, of which exactly three are gold, are distributed among five persons, one each, at random, and one by one. Are the chances of getting a gold coin equal for all participants?Why or why not?
13. At a gas station, 85% of the customers use 87 octane gasoline, 5% use 91 octane, and 10% use 93 octane. Suppose that, 70%, 85%, and 95% of 87 octane users, 91 octane users, and 93 octane users fill their tanks, respectively. What is the probability that the next customer entering this station
12. Solve the following problem, from the “Ask Marilyn” column of Parade Magazine, October 29, 2000.I recently returned from a trip to China, where the government is so concerned about population growth that it has instituted strict laws about family size. In the cities, a couple is permitted
11. When traveling from Springfield, Massachusetts, to Baltimore, Maryland, 30% of the time Charles takes I-91 south and then I-95 south, and 70% of the time he takes I-91 south, then I-84 west followed by I-81 south and I-83 south. Choosing the first option takes Charles a random time between 5
10. A factory produces its entire output with three machines.Machines I, II, and III produce 50%, 30%, and 20% of the output, but 4%, 2%, and 4% of their outputs are defective, respectively.What fraction of the total output is defective?
9. A person has six guns. The probability of hitting a target when these guns are properly aimed and fired is 0.6, 0.5, 0.7, 0.9, 0.7, and 0.8, respectively.What is the probability of hitting a target if a gun is selected at random, properly aimed, and fired?
8. Suppose that 37% of a community are at least 45 years old. If 80% of the time a person who is 45 or older tells the truth, and 65% of the time a person below 45 tells the truth, what is the probability that a randomly selected person answers a question truthfully?
7. Of the patients in a hospital, 20% of those with, and 35% of those without myocardial infarction have had strokes. If 40% of the patients have had myocardial infarction, what percent of the patients have had strokes?
6. Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade?
5. One of the cards of an ordinary deck of 52 cards is lost. What is the probability that a random card drawn from this deck is a spade?
4. Jim has three cars of different models: A, B, and C. The probabilities that models A, B, and C use over 3 gallons of gasoline from Jim’s house to his work are 0.25, 0.32, and 0.53, respectively. On a certain day, all three of Jim’s cars have 3 gallons of gasoline each. Jim chooses one of his
3. A random number is selected from the interval (0, 1]. Which one of the following is a partition of the sample space of this experiment?Which one is not? (a) {(0, 1/2), (1/2,1]}, (c) {(0,2/5], [2/5, 1]}, (b) {(0, 1/3], (1/3,1]}, (d) {(-1,1]: ]: n 1 is given and 1in}. n n
An urn contains 10 white and 12 red chips. Two chips are drawn at random and, without looking at their colors, are discarded. What is the probability that a third chip drawn is red?
Suppose that the only parasite living in an aquatic habitat is a single-celled organism, which after a second, with equal probabilities, either splits into two organisms, remains as is, or dies. Suppose that in subsequent seconds, all the living parasites in the habitat follow the same behavior as
Suppose that 80%of the seniors, 70%of the juniors, 50%of the sophomores, and 30% of the freshmen of a college use the library of their campus frequently. If 30% of all students are freshmen, 25% are sophomores, 25% are juniors, and 20% are seniors, what percent of all students use the library
Two gamblers play the game of “heads or tails,” in which each time a fair coin lands heads up player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that player A initially has a dollars and player B has b dollars. If they continue to play this game
In a trial, the judge is 65% sure that Susan has committed a crime. Julie and Robert are two witnesses who know whether Susan is innocent or guilty. However, Robert is Susan’s friend and will lie with probability 0.25 if Susan is guilty. He will tell the truth if she is innocent. Julie is
An insurance company rents 35% of the cars for its customers from agency I and 65% from agency II. If 8% of the cars of agency I and 5% of the cars of agency II break down during the rental periods, what is the probability that a car rented by this insurance company breaks down?
2. On a given day, the first item produced by a manufacturer is defective with probability p and non-defective with probability 1 − p. However, whether or not an item produced afterward is defective depends only on the item that was produced right before it. It is defective with probability p1 if
1. An urn contains 6 blue and 4 red balls. Three balls are drawn at random and without replacement. What is the probability that the balls drawn are alternatively of different colors?
14. In a series of games, the winning number of the nth game, n = 1, 2, 3, . . . , is a number selected at random from the set of integers {1, 2, . . . , n + 2}. Don bets on 1 in each game and says that he will quit as soon as he wins.What is the probability that he has to play indefinitely?Hint:
13. From an ordinary deck of 52 cards, cards are drawn one by one, at random and without replacement.What is the probability that the fourth heart is drawn on the tenth draw?Hint: Let F denote the event that in the first nine draws there are exactly three hearts, and E be the event that the tenth
12. Cards are drawn at random from an ordinary deck of 52, one by one and without replacement.What is the probability that no heart is drawn before the ace of spades is drawn?
11. Suppose that 75% of all people with credit records improve their credit ratings within three years. Suppose that 18% of the population at large have poor credit records, and of those only 30% will improve their credit ratings within three years. What percentage of the people who will improve
10. In the card game, bridge, played with an ordinary deck of 52 cards, all cards are dealt among four players, 13 each, randomly.What is the probability that each player gets one ace?Hint: Let A1 be the event that the ace of hearts is dealt to one of the four players.Let A2 be the event that the
9. The law school of a university admits all applicants who have at least a 3.5 undergraduate GPA and a Law School Admissions Test (LSAT) score of 154 or higher. Suppose that of all students with a 3.5 or higher GPA, only 45% score at least 154 the first time they take the LSAT test; of all those
8. An urn contains five white and three red chips. Each time we draw a chip, we look at its color. If it is red, we replace it along with two new red chips, and if it is white, we replace it along with three new white chips. What is the probability that, in successive drawing of chips, the colors
7. In a lottery scratch-off game, each ticket has 10 coated circles in the middle and one coated rectangle in the lower left corner. Underneath the coats of 4 of the circles, there is a dollar sign, “$,” and underneath the remaining 6 circles is blank. A winning ticket is the one with only
6. There are five boys and six girls in a class. For an oral exam, their teacher calls them one by one and randomly. (a) What is the probability that the boys and the girls alternate?(b) What is the probability that the boys are called first? Compare the answers to parts(a) and (b).
5. An actuary works for an auto insurance company whose customers all have insured at least one car. She discovers that of all the customers who insure more than one car, 70% of them insure at least one SUV. Furthermore, she observes that 47% of all the customers insure at least one SUV. If 65% of
3. In a game of cards, two cards of the same color and denomination form a pair. For example, 8 of hearts and 8 of diamonds is one pair, king of spades and king of clubs is another. If six cards are selected at random and without replacement, what is the probability that there will be no pairs?
2. There are 14marketing firms hiring new graduates.Kate randomly found the recruitment ads of six of these firms and sent them her resume. If three of these marketing firms are in Maryland, what is the probability that Kate did not apply to a marketing firm in Maryland?
Suppose that five good and two defective fuses have been mixed up. To find the defective ones, we test them one by one, at random and without replacement. What is the probability that we find both of the defective fuses in exactly three tests?
A consulting firm is awarded 43% of the contracts it bids on. Suppose that Nordulf works for a division of the firm that gets to do 15% of the projects contracted for.If Nordulf directs 35% of the projects submitted to his division, what percentage of all bids submitted by the firm will result in
Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement.What is the probability that we are lucky and find both of the defective fuses in the first two tests?
4. For events E and F, suppose that P(EF) = 0.23, P????E ∪ F= 0.67, and P(E | F) = 0.46. Find P(F | E).
3. The theaters of a town are showing seven comedies and nine dramas. Marlon has seen five of the movies. If the first three movies he has seen are dramas, what is the probability that the last two are comedies? Assume that Marlon chooses the shows at random and sees each movie at most once.Hint:
2. For a certain loaded die, the probabilities of the possible outcomes are given by the following table.If the die is tossed and the outcome is an odd number, what is the probability that it is 1? Outcome 1 2 3 4 5 6 Probability 0.27 0.15 0.17 0.2 0.05 0.16
1. In a bridge game, played with a normal deck of 52 cards, hearts is the designated trump suit. If the first 4 cards a player is dealt are ace, 5, 9, and the king, all of hearts, what is the probability that none of the remaining 9 cards the player is dealt are hearts?Hint: Reduce the sample space.
25. From the set of all families with two children, a family is selected at random and is found to have a girl called Mary. We want to know the probability that both children of the family are girls. By Example 3.2, the probability should apparently be 1/3 because presumably knowing the name of the
24. In an international school, 60 students, of whom 15 are Korean, 20 are French, eight are Greek, and the rest are Chinese, are divided randomly into four classes of 15 each. If there is a total of eight French and six Korean students in classes A and B, what is the probability that class C has
23. There are three types of animals in a laboratory: 15 type I, 13 type II, and 12 type III.Animals of type I react to a particular stimulus in 5 seconds, animals of types II and III react to the same stimulus in 4.5 and 6.2 seconds, respectively. A psychologist selects 10 of these animals at
22. A big urn contains 1000 red chips, numbered 1 through 1000, and 1750 blue chips, numbered 1 through 1750. A chip is removed at random, and its number is found to be divisible by 3. What is the probability that its number is also divisible by 5?
21. A retired person chooses randomly one of the six parks of his town everyday and goes there for hiking.We are told that he was seen in one of these parks, Oregon Ridge, once during the last 10 days. What is the probability that during this period he has hiked in this park two or more times?
20. A number is selected at random from the set {1, 2, . . . , 10,000} and is observed to be odd.What is the probability that it is (a) divisible by 3; (b) divisible by neither 3 nor 5?
19. An actuary studying the insurance preferences of homeowners in a region that is prone to earthquakes, hurricanes, and floods has discovered that, for each of these perils, the probability is 0.2 that a homeowner, selected randomly, has purchased coverage only for that peril. Moreover, she has
18. Adam and three of his friends are playing bridge. (a) If, holding a certain hand, Adam announces that he has a king, what is the probability that he has at least one more king?(b) If, for some other hand, Adam announces that he has the king of diamonds, what is the probability that he has at
16. Prove that if P(E | F) ≥ P(G | F) and P(E | Fc) ≥ P(G | Fc), then P(E) ≥ P(G).
15. Prove Theorem 3.1.
14. Prove that if P(A) = a and P(B) =b, then P(A | B) ≥ (a + b − 1)/b.
13. Show that if P(A) = 1, then P(B | A) = P(B).
12. In a study of the records of 831 women who died in 2016, an actuary observed that 185 of them died of cancer. Furthermore, she discovered that the mothers of 257 of the 831 women suffered from cancer, and from the mothers of all who died of cancer 93 had died due to cancer. Find the probability
11. From families with three children, a family is selected at random and found to have a boy. What is the probability that the boy has (a) an older brother and a younger sister;(b) an older brother; (c) a brother and a sister? Assume that in a three-child family all gender distributions have equal
10. From 100 cards numbered 00, 01, . . . , 99, one card is drawn. Suppose that α and βare the sum and the product, respectively, of the digits of the card selected. Calculate P????{α = i | β = 0}, i = 0, 1, 2, 3, . . . , 18.
9. In a small lake, it is estimated that there are approximately 105 fish, of which 40 are trout and 65 are carp. A fisherman caught eight fish; what is the probability that exactly two of them are trout if we know that at least three of them are not?
7. A spinner is mounted on a wheel of unit circumference (radius 1/2π). Arcs A,B, and C of lengths 1/3, 1/2, and 1/6, respectively, are marked off on the wheel’s perimeter(see Figure 3.1). The spinner is flicked and we know that it is not pointing toward C.What is the probability that it points
5. A bus arrives at a station every day at a random time between 1:00 P.M. and 1:30 P.M. A person arrives at this station at 1:00 and waits for the bus. If at 1:15 the bus has not yet arrived, what is the probability that the person will have to wait at least an additional 5 minutes?
4. Suppose that two fair dice have been tossed and the total of their top faces is found to be divisible by 5. What is the probability that both of them have landed 5?
3. In a technical college all students are required to take calculus and physics. Statistics show that 32% of the students of this college get A’s in calculus, and 20% of them get A’s in both calculus and physics. Gino, a randomly selected student of this college, has passed calculus with an A.
2. Suppose that 41% of Americans have blood type A, and 4% have blood type AB. If in the blood of a randomly selected American soldier the A antigen is found, what is the probability that his blood type is A? The A antigen is found only in blood types A and AB.
On a TV game show, there are three curtains. Behind two of the curtains there is nothing, but behind the third there is a prize that the playermight win. The probability that the prize is behind a given curtain is 1/3. The game begins with the contestant randomly guessing a curtain. The host of the
A farmer decides to test four fertilizers for his soybean fields. He buys 32 bags of fertilizers, eight bags from each kind, and tries them randomly on 32 plots, eight plots from each of fields A, B, C, and D, one bag per plot. If from type I fertilizer one bag is tried on field A and three on
A child mixes 10 good and three dead batteries. To find the dead batteries, his father tests them one-by-one and without replacement. If the first four batteries tested are all good, what is the probability that the fifth one is dead?
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