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Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions
12. An experiment consists of first tossing an unbiased coin and then rolling a fair die. If we perform this experiment successively, what is the probability of obtaining a heads on the coin before a 1 or 2 on the die?
11. Urns I and II contain three pennies and four dimes, and two pennies and five dimes, respectively. One coin is selected at random from each urn. If exactly one of them is a dime, what is the probability that the coin selected from urn I is the dime?
10. Suppose that 10 dice are thrown and we are told that among them at least one has landed 6. What is the probability that there are two or more sixes?
9. A fair die is thrown twice. If the second outcome is 6, what is the probability that the first one is 6 as well?
8. From an ordinary deck of 52 cards, 10 cards are drawn at random. If exactly four of them are hearts, what is the probability of at least one spade being among them?
7. Roads A, B, and C are the only escape routes from a state prison. Prison records show that, of the prisoners who tried to escape, 30% used road A, 50% used road B, and 20% used road C. These records also show that 80% of those who tried to escape via A, 75% of those who tried to escape via B,
6. A bus traveling from Baltimore to New York breaks down at a random location. If the bus was seen running at Wilmington, what is the probability that the breakdown occurred after passing through Philadelphia? The distances from NewYork, Philadelphia, and Wilmington to Baltimore are,
5. Professor Stern has three cars. The probability that on a given day car 1 is operative is 0.95, that car 2 is operative is 0.97, and that car 3 is operative is 0.85. If Professor Stern’s cars operate independently, find the probability that on next Thanksgiving day (a) all three of his cars
4. Diseases D1, D2, and D3 cause symptom A with probabilities 0.5, 0.7, and 0.8, respectively. If 5% of a population have disease D1, 2% have disease D2, and 3.5% have disease D3, what percent of the population have symptom A? Assume that the only possible causes of symptom A are D1, D2, and D3 and
3. In statistical surveys where individuals are selected randomly and are asked questions, experience has shown that only 48% of those under 25 years of age, 67% between 25 and 50, and 89% above 50 will respond. A social scientist is about to send a questionnaire to a group of randomly selected
2. A polygraph operator detects innocent suspects as being guilty 3% of the time. If during a crime investigation six innocent suspects are examined by the operator, what is the probability that at least one of them is detected as guilty?
1. Two persons arrive at a train station, independently of each other, at random times between 1:00 P.M. and 1:30 P.M. What is the probability that one will arrive between 1:00 P.M. and 1:12 P.M., and the other between 1:17 P.M. and 1:30 P.M.?
12. Let p and q be positive numbers with p + q = 1. For a gene with dominant allele A and recessive allelea, let p2, 2pq, and q2 be the probabilities that a randomly selected person has genotype AA, Aa, and aa, respectively. If a man is of genotype AA, what is the probability that his brother is
11. In a certain country, all children with cystic fibrosis die before reaching adulthood. For this lethal disease, the normal allele, denoted by C, is dominant to the mutant allele, denoted byc. In that country, the probability is p that a person is only a carrier; that is, he or she is Cc. If Mr.
10. Hemophilia is a sex-linked hereditary disease with normal allele H dominant to the mutant allele h. Kim and John are married and their daughter,Ann, has hemophilia. If John has hemophilia, and the frequencies of H and h, in the population, are 0.98 and 0.02, respectively, what is the
9. Color blindness is a sex-linked hereditary disease with normal allele C dominant to the mutant allelec. Kim and John are married and their son, Dan, is color-blind. If, in the population, the frequencies of C and c are 0.83 and 0.17, respectively, what is the probability that Kim is color-blind?
8. Hemophilia is a sex-linked disease with normal allele H dominant to the mutant allele h. Kim and John are both phenotypically normal. If the frequencies of H and h are 0.98 and 0.02, respectively, what is the probability that Dan, their son, has hemophilia?
7. In the United States, among the Caucasian population, cystic fibrosis is a serious genetic disease. For cystic fibrosis, the normal allele is dominant to the mutant allele. Suppose that, in a certain region of the country, 5.29% of the people have cystic fibrosis. Under the Hardy-Weinberg
6. In a population, 1% of the people suffer from hereditary genetic deafness. The hearing allele is dominant and is denoted by D. The recessive allele for deafness is denoted byd. Kim and John, from the population, are married and have a son named Dan. If Kim and Dan are both deaf, what is the
5. For Drosophila (a kind of fruit fly), B, the gray body, is dominant overb, the black body, and V , the wild-type wing is dominant over v, vestigial (a very small wing). A geneticist, T. H. Morgan, when mating Drosophila of genotype BbV v with Drosophila of genotype bbvv, observed that 42% of the
4. In humans, the presence of freckles and having free earlobes are independently inherited dominant traits. Kim and John both have free earlobes and both have freckles, but their son, Dan, has attached earlobes and no freckles. What is the probability that Kim and John’s next child has free
3. For the shape of pea seed, the allele for round shape, denoted by R, is dominant to the allele for wrinkled shape, denoted by r. A pea plant with round seed shape is crossed with a pea plant with wrinkled seed shape. If half of the offspring have round seed shape and half have wrinkled seed
2. Suppose that a gene has k (k > 2) alleles. For that gene, how many genotypes can an individual have?
1. Kim has blood type O and John has blood type A. The blood type of their son, Dan, is O. What is the probability that John’s genotype is AO?
44. (Laplace’s Law of Succession) Suppose that n + 1 urns are numbered 0 through n, and the ith urn contains i red and n − i white balls, 0 ≤ i ≤ n. An urn is selected at random, and then the balls it contains are removed one by one, at random, and with replacement. If the first m balls are
43. Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons becomes hemophilic. Julie is the mother of two sons, and from her family’s medical history it is known that, with the probability 1/4,
42. In a contest, contestants A, B, and C are each asked, in turn, a general scientific question. If a contestant gives a wrong answer to a question, he drops out of the game. The remaining two will continue to compete until one of them drops out. The last person remaining is the winner. Suppose
41. Figure 3.11 shows an electric circuit in which each of the switches located at 1, 2, 3, 4, and 5 is independently closed or open with probabilitiesp and 1−p, respectively. If a signal is fed to the input, what is the probability that it is transmitted to the output?
40. From a population of people with unrelated birthdays, 30 people are selected at random. What is the probability that exactly four people of this group have the same birthday and that all the others have different birthdays (exactly 27 birthdays altogether)? Assume that the birthrates are
39. Suppose that n ≥ 2 missiles are fired at a target and hit it independently. If the probability that the ith missile hits it is pi, i = 1, 2, . . . , n, find the probability that at least two missiles will hit the target.
38. An urn contains nine red and one blue balls. A second urn contains one red and five blue balls. One ball is removed from each urn at random and without replacement, and all of the remaining balls are put into a third urn. If we draw two balls randomly from the third urn, what is the probability
37. In the experiment of rolling two fair dice successively, what is the probability that a sum of 5 appears before a sum of 7? Hint: See Exercise 36.
36. Let S be the sample space of a repeatable experiment. Let A and B be mutually exclusive events of S. Prove that, in independent trials of this experiment, the event A occurs before the event B with probability P (A)/[P (A) + P (B)]. Hint: See Example 3.31; this exercise can be done the same way.
35. An experiment consists of first tossing a fair coin and then drawing a card randomly from an ordinary deck of 52 cards with replacement. If we perform this experiment successively, what is the probability of obtaining heads on the coin before an ace from the cards? Hint: See Example 3.31.
34. Suppose that an airplane passenger whose itinerary requires a change of airplanes in Ankara, Turkey, has a 4% chance of losing each piece of his or her luggage independently. Suppose that the probability of losing each piece of luggage in this way is 5% at Da Vinci airport in Rome, 5% at
33. If two fair dice are tossed six times, what is the probability that the sixth sum obtained is not a repetition?
32. A fair coin is flipped indefinitely. What is the probability of (a) at least one head in the first n flips; (b) exactly k heads in the first n flips; (c) getting heads in all of the flips indefinitely?
31. A fair coin is tossed n times. Show that the events “at least two heads” and “one or two tails” are independent if n = 3 but dependent if n = 4.
30. An urn contains two red and four white balls. Balls are drawn from the urn successively, at random and with replacement. What is the probability that exactly three whites occur in the first five trials?
29. Figure 3.10 shows an electric circuit in which each of the switches located at 1, 2, 3, 4, 5, and 6 is independently closed or open with probabilities p and 1 − p, respectively. If a signal is fed to the input, what is the probability that it is transmitted to the output?
28. Let {A1, A2, . . . , An} be an independent set of events and P (Ai) = pi, 1 ≤ i ≤ n. (a) What is the probability that at least one of the events A1, A2, . . . , An occurs? (b) What is the probability that none of the events A1, A2, . . . , An occurs?
27. An event occurs at least once in four independent trials with probability 0.59. What is the probability of its occurrence in one trial?
26. From the set of all families with three children a family is selected at random. Let A be the event that “the family has children of both sexes” and B be the event that “there is at most one girl in the family.” Are A and B independent? Answer the same question for families with two
25. If the events A and B are independent and the events B and C are independent, is it true that the events A and C are also independent? Why or why not?
24. In a certain county, 15% of patients suffering heart attacks are younger than 40, 20% are between 40 and 50, 30% are between 50 and 60, and 35% are above 60. On a certain day, 10 unrelated patients suffering heart attacks are transferred to a county hospital. If among them there is at least one
23. There are n cards in a box numbered 1 through n. We draw cards successively and at random with replacement. If the ith draw is the card numbered i, we say that a match has occurred. (a) What is the probability of at least one match in n trials? (b) What happens if n increases without bound?
22. A fair die is rolled six times. If on the ith roll, 1 ≤ i ≤ 6, the outcome is i, we say that a match has occurred. What is the probability that at least one match occurs?
21. Prove that if A, B, and C are independent, then A and B ∪ C are independent. Also show that A − B and C are independent.
20. In a community of M men and W women, m men and w women smoke (m ≤ M, w ≤ W ). If a person is selected at random and A and B are the events that the person is a man and smokes, respectively, under what conditions are A and B independent?
19. In a tire factory, the quality control inspector examines a randomly chosen sample of 15 tires. When more than one defective tire is found, production is halted, the existing tires are recycled, and production is then resumed. The purpose of this process is to ensure that the defect rate is no
18. In his book, Probability 1, published by Harcourt Brace and Company, 1998, Amir Aczel estimates that the probability of life for any one given star in the known universe is 0.00000000000005 independently of life for any other star. Assuming that there are 100 billion galaxies in the universe
17. Three missiles are fired at a target and hit it independently, with probabilities 0.7, 0.8, and 0.9, respectively. What is the probability that the target is hit?
16. Suppose that 55% of the customers of a shoestore buy black shoes. Find the probability that at least one of the next six customers who purchase a pair of shoes from this store will buy black shoes. Assume that these customers decide independently.
15. Show that if A and B are independent and A ⊆ B, then either P (A) = 0 or P (B) = 1.
14. Show that if an event A is independent of itself, then P (A) = 0 or 1.
13. (a) Show that if P (A) = 1, then P (AB) = P (B). (b) Prove that any event A with P (A) = 0 or P (A) = 1 is independent of every event B.
12. Find an example in which P (AB) < P (A)P (B).
11. In data communications, a message transmitted from one end is subject to various sources of distortion and may be received erroneously at the other end. Suppose that a message of 64 bits (a bit is the smallest unit of information and is either 1 or 0) is transmitted through a medium. If each
10. (Chevalier de Méré’s Paradox† ) In the seventeenth century in France there were two popular games, one to obtain at least one 6 in four throws of a fair die and the other to bet on at least one double 6 in 24 throws of two fair dice. French nobleman and mathematician Chevalier de Méré
9. Consider the four “unfolded” dice in Figure 3.9 designed by Stanford professor Bradley Effron. Clearly, none of these dice is an ordinary die with sides numbered 1 through 6. A game consists of two players each choosing one of these four dice and rolling it. The player rolling a larger
8. The Italian mathematician Giorlamo Cardano once wrote that if the odds in favor of an event are 3 to 1, then the odds in favor of the occurrence of that event in two consecutive independent experiments are 9 to 1. (He squared 3 and 1 to obtain 9 to 1.) Was Cardano correct?
7. According to a recent mortality table, the probability that a 35-year-old U.S. citizen will live to age 65 is 0.725. (a) What is the probability that John and Jim, two 35-year-old Americans who are not relatives, both live to age 65? (b) What is the probability that neither John nor Jim lives to
6. Suppose that two points are selected at random and independently from the interval (0, 1). What is the probability that the first one is less than 3/4, and the second one is greater than 1/4?
5. An urn has three red and five blue balls. Suppose that eight balls are selected at random and with replacement. What is the probability that the first three are red and the rest are blue balls?
4. A fair die is rolled twice. Let A denote the event that the sum of the outcomes is odd, and B denote the event that it lands 2 on the first toss. Are A and B independent? Why or why not?
3. In a certain country, the probability that a fighter plane returns from a mission without mishap is 49/50, independent of other missions. In a conversation, Mia concluded that any pilot who flew 49 consecutive missions without mishap should be returned home before the fiftieth mission. But, on
2. Clark and Anthony are two old friends. Let A be the event that Clark will attend Anthony’s funeral. Let B be the event that Anthony will attend Clark’s funeral. Are A and B independent? Why or why not?
1. Jean le Rond d’Alembert, a French mathematician, believed that in successive flips of a fair coin, after a long run of heads, a tail is more likely. Do you agree with d’Alembert on this? Explain.
23. (Shrewd Prisoner’s Dilemma) Because of a prisoner’s constant supplication, the king grants him this favor: He is given 2N balls, which differ from each other only in that half of them are green and half are red. The king instructs the prisoner to divide the balls between two identical urns.
22. Avril has certain standards for selecting her future husband. She has n suitors and knows how to compare any two and rank them. She decides to date one suitor at a time randomly. When she knows a suitor well enough, she can marry or reject him. If she marries the suitor, she can never know the
21. 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?
20. 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 threechild family all sex distributions are equally probable. Let G be the event that the randomly selected child is a girl, A be the
19. 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?
18. 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
17. 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
16. 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
15. 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 S 4 P (FB) > 0 5 , show that Q(E | F ) = P (E | FB).
14. 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, /c j=0 αj = 1 " be 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, . . .
13. 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
12. 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.
11. 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?
10. Solve the following problem, from the “Ask Marilyn” column ofParade 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
9. 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?
8. 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?
7. 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?
6. 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?
5. 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?
4. 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?
3. 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
2. Suppose that 40% of the students of a campus are women. If 20% of the women and 16% of the men of this campus are A students, what percent of all of them are A students?
1. If 5% of men and 0.25% of women are color blind, what is the probability that a randomly selected person is color blind?
11. 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:
10. 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
9. 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?
8. 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
7. Solve the following problem, asked of Marilyn Vos Savant in the “Ask Marilyn” column of Parade Magazine, January 3, 1999. You’re at a party with 199 other guests when robbers break in and announce that they are going to rob one of you. They put 199 blank pieces of paper in a hat, plus one
6. 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
5. 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).
4. If eight defective and 12 nondefective items are inspected one by one, at random and without replacement, what is the probability that (a) the first four items inspected are defective; (b) from the first three items at least two are defective?
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?
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