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Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

2. Every day, a major ball manufacturer produces at least 800, but no more than 1300 of each of its five products: baseballs, tennis balls, softballs, basketballs, and soccer balls.Define a sample space for the production levels of these five types of balls that this manufacturer manufactures on a
1. To determine who pays for dinner, Crispin, Allison, and Terry each flip an unbiased coin.The one whose flip has a different face up will pay. If all of the flips land on the same face, they start all over again. (a) What is the probability that Crispin ends up paying for dinner; (b) what is the
35. The coefficients of the quadratic equation ax2 + bx + c = 0 are determined by tossing a fair die three times (the first outcome isa, the second oneb, and the third one c). Find the probability that the equation has no real roots.
34. A bus traveling from Baltimore to New York breaks down at a random location. What is the probability that the breakdown occurred after passing through Philadelphia? The distances from New York and Philadelphia to Baltimore are, respectively, 199 and 96 miles.
33. Suppose that each day the price of a stock moves up 1/8 of a point, moves down 1/8 of a point, or remains unchanged. For i ≥ 1, let Ui and Di be the events that the price of the stock moves up and down on the ith trading day, respectively. In terms of Ui’s and Di’s, find an expression for
32. A point is selected randomly from the interval (0, 2). For n ≥ 2, let En be the event that it is in the interval (0, n√3 ). Determine the event T∞ n=2 En.
31. A number is selected at random from the set {1, 2, 3, . . . , 150}. What is the probability that it is relatively prime to 150? See Exercise 34, Section 1.4, for the definition of relatively prime numbers.
30. A number is selected at random from the set of natural numbers {1, 2, 3, . . . , 1000}.What is the probability that it is not divisible by 4, 7, or 9?
29. Suppose that in a certain town the number of people with blood type O and blood type A are approximately the same. The number of people with blood type B is 1/10 of those with blood type A and twice the number of those with blood type AB. Find the probability that the next baby born in the town
28. Mildred, a commuter student studying at Western New England University, reported to her advisor that there are three traffic lights that she needs to pass while driving from home to school. She said that, based on her experience, 10% of the time all of the three traffic lights are green, 35% of
27. A bookstore receives six boxes of books per month on six random days of each month.Suppose that two of those boxes are from one publisher, two from another publisher, and the remaining two from a third publisher. Define a sample space for the possible orders in which the boxes are received in a
26. Let S = {ω1, ω2, ω3, . . .} be the sample space of an experiment. Suppose that P????{ω1}= 1/8 and, for a constant k, 0 < k < 1, P????{ωi+1}= k · P????{ωi}for i ≥ 1. Find P????{ωi}, for i > 1.
25. Let A and B be two events. The event (A − B) ∪ (B − A) is called the symmetric difference ofAandB and is denoted byAB. Clearly, AB is the event that exactly one of the two events A and B occurs. Show that P(AB) = P(A) + P(B) − 2P(AB).
24. Let A and B be two events. Suppose that P(A), P(B), and P(AB) are given.What is the probability that neither A nor B will occur?
23. Answer the following question, asked of Marilyn Vos Savant in the “Ask Marilyn”column of Parade Magazine, March 3, 1996.My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were
22. Five customers enter a wireless corporate store to buy smartphones. If the probability that at least two of them purchase an Android smartphone is 0.6 and the probability that all of them buy non-Android smartphones is 0.17, what is the probability that exactly one Android smartphone is
21. Anthony, Bob, and Carl, three American race car drivers, will compete in a professional Trans-Am road race. Past records show that the probability that one of these former champions wins is 10/13. If Anthony is twice as likely to win as Bob, and Bob is three times more likely than Carl to win
20. Suppose that P????E ∪ F= 0.75 and P????E ∪ Fc= 0.85. Find P(E).
19. Suppose that, in a temperate coniferous forest, 60% of randomly selected quarter-acre plots have cedar trees, 45% have cypress trees, 30% have redwoods, 40% have both cedar and cypress, 25% have cedar and redwoods, 20% have cypress and redwoods, and 80% of such plots have at least one of these
18. Suppose that 40% of the people in a community drink or serve white wine, 50% drink or serve red wine, and 70% drink or serve red or white wine. What percentage of the people in this community drink or serve both red and white wine?
17. Let A, B, and C be three events. Show that P(A ∪ B ∪ C) = P(A) + P(B) + P(C)if and only if P(AB) = P(AC) = P(BC) = 0.
16. Let A, B, and C be three events. Prove that P(A ∪ B ∪ C) ≤ P(A) + P(B) + P(C).
15. The number of the patients now in a hospital is 63. Of these 37 are male and 20 are for surgery. If among those who are for surgery 12 are male, how many of the 63 patients are neither male nor for surgery?
14. Let A and B be two events of an experiment with P(A) = 1/3 and P(B) = 1/4. Find the maximum value for P????A ∪ B.
13. In a midwest town, 80%of households have cable TV, 60%have an internet subscription, and 90% have at least one of these.What percentage of the households of this town have both cable TV and an internet subscription?
12. A coin is tossed until, for the first time, the same result appears twice in succession.Define a sample space for this experiment.
11. The following relations are not always true. In each case give an example to refute them.(a) P(A ∪ B) = P(A) + P(B).(b) P(AB) = P(A)P(B).
10. For a saw blademanufacturer’s products, the global demand, per month, for band saws is between 30 and 36 thousands; for reciprocating saws, it is between 28 and 33 thousands;for hole saws, it is between 300 and 600 thousands; and for hacksaws, it is between 500 and 650 thousands. Define a
9. Kayla has two cars, an Audi and a Jeep. At a given time, whether these cars are operative or totaled, due to accidents, is of concern to her insurance company. Define a sample space for the driving conditions of these cars at a random time. What is the event that at least one of the two cars is
8. A department store accepts only its own credit card or an American Express card.Customers not carrying one of these two cards must pay with cash. If 47% of the customers of this store carry American Express, 32% carry the store’s credit card, and 12%carry both, what percentage of the store
7. In a certain experiment, whenever the event A occurs, the event B also occurs. Which of the following statements is true and why?(a) If we know that A has not occurred, we can be sure that B has not occurred as well.(b) If we know that B has not occurred, we can be sure that A has not occurred
6. Aiden just bought a stock for $320. Define a sample space for the price of this stock in two years. Define the event that he makes money selling this stock at that time.
5. In a tutoring center, there are three computers that can be up or down at any given time.For 1 ≤ i ≤ 3, let Ei be the event that computer i is up at a random time. In terms of Ei’s, describe the event that at least two computers are up.
4. Let P be the set of all subsets of A = {1, 2}. We choose two distinct sets randomly from P. Define a sample space for this experiment, and describe the following events:(a) The intersection of the sets chosen at random is empty.(b) The sets are complements of each other.(c) One of the sets
3. From a phone book, a phone number is selected at random. (a)What is the event that the last digit is an odd number? (b) What is the event that the last digit is divisible by 3?
2. Two dice are rolled. What is the event that the outcomes are consecutive?
1. The number of minutes it takes for a certain animal to react to a certain stimulus is a random number between 2 and 4.3. Find the probability that the reaction time of such an animal to this stimulus is no longer than 3.25 minutes.
2. For the experiment of choosing a point at random from the interval [0, 1], let En = Applying the Continuity of Probability Function to En’s, show that P(1/3 is selected) = 0. 1 2 + n+2'3 n 1. n+2.
1. Suppose that for events A and B, P(AB) = 0. Does this imply that A and B are mutually exclusive?Why or why not?
16. Let A be the set of rational numbers in (0, 1). Since A is countable, it can be written as a sequence????i.e.,A = {rn : n = 1, 2, 3, . . .}. Prove that for any ε > 0, A can be covered by a sequence of open balls whose total length is less than ε. That is, ∀ε > 0, there exists a sequence
15. Show that the result of Exercise 11 is not true for an infinite number of events. That is, show that if {Et : 0 < t < 1} is a collection of events for which P(Et) = 1, it is not necessarily true that P \t∈(0,1)Et= 1.
14. Let {A1,A2,A3, . . .} be a sequence of events. Prove that if the series P∞ n=1 P(An)converges, then PT∞m=1 S∞ n=m An= 0. This is called the Borel–Cantelli lemma.It says that if P∞ n=1 P(An)
13. Suppose that a point is randomly selected from the interval (0, 1). Using Definition 1.2, show that all numerals are equally likely to appear as the nth digit of the decimal representation of the selected point.
12. A point is selected at random from the interval (0, 1). What is the probability that it is rational?What is the probability that it is irrational?
11. Let A1,A2, . . . ,An be n events. Show that if P(A1) = P(A2) = · · · = P(An) = 1, then P(A1A2 · · ·An) = 1.
10. Is it possible to define a probability on a countably infinite sample space so that the outcomes are equally probable?
9. For the experiment of choosing a point at random from the interval (0, 1), let En = (1/2 − 1/2n, 1/2 + 1/2n), n ≥ 1.(a) Prove that T∞ n=1 En ={1/2}.(b) Using part (a) and the continuity of probability function, show that the probability of selecting 1/2 in a random selection of a point
8. A point is chosen at random from the interval (−1, 1). Let E1 be the event that it falls in the interval (−1, 1/3], E2 be the event that it falls in the interval (−1, 1/9], E3 be the event that it falls in the interval (−1, 1/27] and, in general, for 1 ≤ i < ∞, Ei be the event that
7. For an experiment with sample space S = (0, 2), for n ≥ 1, let En =0, 1 +1 nand P(En) = (3n + 2)/5n. Find the probability of E = (0, 1]. Note that this is not the experiment of choosing a point at random from the interval (0, 2) as defined in Section 1.7.
6. Suppose that a point is randomly selected from the interval (0, 1). Using Definition 1.2, show that all numerals are equally likely to appear as the first digit of the decimal representation of the selected point.
5. A point is selected at random from the interval (0, 2000). What is the probability that it is an integer?
4. Let A and B be two events. Show that if P(A) = 1 and P(B) = 1, then P(AB) = 1.
3. Which of the following statements are true? If a statement is true, prove it. If it is false, give a counterexample.(a) If A is an event with probability 1, then A is the sample space.(b) If B is an event with probability 0, then B = ∅.
2. Past experience shows that every new book by a certain publisher captures randomly between 4 and 12% of the market. What is the probability that the next book by this publisher captures at most 6.35% of the market?
1. A bus arrives at a station every day at a random time between 1:00 P.M. and 1:30 P.M.What is the probability that a person arriving at this station at 1:00 P.M. will have to wait at least 10 minutes?
Suppose that some individuals in a population produce offspring of the same kind. The offspring of the initial population are called second generation, the offspring of the second generation are called third generation, and so on. Furthermore, suppose that with probability exp−(2n2+7)/(6n2)the
3. Suppose that two-thirds of Americans traveling to Europe visit at least one of the three countries France, England, and Italy. Furthermore, suppose that one-half of them visit England, one-third visit France, and one-fourth visit Italy. If for each pair of these countries, one-fifth of the
2. Zack has two weeks to read a book assigned by his English teacher and two weeks to write an essay assigned by his philosophy professor. The probability that he completes both assignments on time is 0.6, and the probability that he completes at least one of them on time is 0.95. What is the
1. Let A and B be mutually exclusive events of an experiment with P(A) = 1/3 and P(B) = 1/4. What is the probability that neither A occurs nor B? (3 points)
46. (The Hat Problem) A game begins with a team of three players entering a room one at a time. For each player, a fair coin is tossed. If the outcome is heads, a red hat is placed on the player’s head, and if it is tails, a blue hat is placed on the player’s head. The players are allowed to
45. Let P be a probability defined on a sample space S. For events A of S define Q(A) =P(A)2 and R(A) = P(A)/2. Is Q a probability on S? Is R a probability on S?Why or why not?
44. In a certain country, the probability is 49/50 that a randomly selected fighter plane returns from a mission without mishap. Mia argues that this means there is one mission with a mishap in every 50 consecutive flights. She concludes that if a fighter pilot returns safely from 49 consecutive
43. Let A1,A2,A3, . . . be a sequence of events of an experiment. Prove that P ∞\n=1 An≥ 1 −∞X n=1 P(Ac n).Hint: Use Boole’s inequality, discussed in Exercise 42.
42. Let A1, A2, A3, . . . be a sequence of events of a sample space. Prove that P ∞[n=1 An≤∞X n=1 P(An).This is called Boole’s inequality.
41. Two numbers are successively selected at random and with replacement from the set{1, 2, . . . , 100}. What is the probability that the first one is greater than the second?
40. For events E and F, show that P????E ∪ F+ P????E ∪ Fc+ P????Ec ∪ F+ P????Ec ∪ Fc= 3.
39. For a Democratic candidate to win an election, she must win districts I, II, and III. Polls have shown that the probability of winning I and III is 0.55, losing II but not I is 0.34, and losing II and III but not I is 0.15. Find the probability that this candidate will win all three districts.
38. A number is selected at random from the set of natural numbers {1, 2, . . . , 1000}. What is the probability that it is divisible by 4 but neither by 5 nor by 7?
37. From an ordinary deck of 52 cards, we draw cards at random and without replacement until only cards of one suit are left. Find the probability that the cards left are all spades.
36. The secretary of a college has calculated that from the students who took calculus, physics, and chemistry last semester, 78%passed calculus, 80%physics, 84%chemistry, 60% calculus and physics, 65% physics and chemistry, 70% calculus and chemistry, and 55% all three. Show that these numbers are
35. A number is selected randomly from the set {1, 2, . . . , 1000}. What is the probability that (a) it is divisible by 3 but not by 5; (b) it is divisible neither by 3 nor by 5?
34. Two integers m and n are called relatively prime if 1 is their only common positive divisor. Thus 8 and 5 are relatively prime, whereas 8 and 6 are not. A number is selected at random from the set {1, 2, 3, . . . , 63}. Find the probability that it is relatively prime to 63.
33. The coefficients of the quadratic equation x2 + bx + c = 0 are determined by tossing a fair die twice (the first outcome isb, the second one is c). Find the probability that the equation has real roots.
32. From a small town 120 persons were selected at random and asked the following question:Which of the three shampoos, A, B, or C, do you use? The following results were obtained: 20 use A and C, 10 use A and B but not C, 15 use all three, 30 use only C, 35 use B but not C, 25 use B and C, and 10
31. Among 33 students in a class, 17 of themearnedA’s on themidtermexam, 14 earnedA’s on the final exam, and 11 did not earn A’s on either examination.What is the probability that a randomly selected student from this class earned an A on both exams?
30. A ball is thrown at a square that is divided into n2 identical squares. The probability that the ball hits the square of the ith column and jth row is pij , where Pn i=1 Pn j=1 pij = 1.In terms of pij ’s, find the probability that the ball hits the jth horizontal strip.
29. Eleven chairs are numbered 1 through 11. Four girls and seven boys sit on these chairs at random.What is the probability that chair 5 is occupied by a boy?
28. Let A, B, and C be three events. Show that exactly two of these events will occur with probability P(AB) + P(AC) + P(BC) − 3P(ABC).
27. Let A,B, and C be three events. Prove that P(A ∪ B ∪ C)= P(A) + P(B) + P(C) − P(AB) − P(AC) − P(BC) + P(ABC).
26. Suppose that, in a temperate coniferous forest, 40% of randomly selected quarter-acre plots have both cedar and cypress trees, 25% have both cedar and redwood trees, 20%have cypress and redwoods, and 15% have all three. What is the probability that in a randomly selected quarter-acre plot in
25. Suppose that in the Baltimore metropolitan area 25% of the crimes occur during the day and 80% of the crimes occur in the city. If only 10% of the crimes occur outside the city during the day, what percent occur inside the city during the night? What percent occur outside the city during the
24. Which of the following statements is true? If a statement is true, prove it. If it is false, give a counterexample.(a) If P(A) + P(B) + P(C) = 1, then the events A,B, and C are mutually exclusive.(b) If P(A ∪ B ∪ C) = 1, then A,B, and C are mutually exclusive events.
23. A card is drawn at random from an ordinary deck of 52 cards. What is the probability that it is (a) a black ace or a red queen; (b) a face or a black card; (c) neither a heart nor a queen?
22. Let A and B be two events. Prove that P(AB) ≥ P(A) + P(B) − 1.
21. In a psychiatric hospital, the number of patients with schizophrenia is three times the number with psychoneurotic reactions, twice the number with alcohol addictions, and 10 times the number with involutional psychotic reaction. If a patient is selected randomly from the list of all patients
20. A company has only one position with three highly qualified applicants: John, Barbara, and Marty. However, because the company has only a few women employees, Barbara’s chance to be hired is 20% higher than John’s and 20% higher than Marty’s. Find the probability that Barbara will be
19. In a major state university, to study the performance relationship between calculus I and calculus II grades, the mathematics department reviewed the letter grades of all students who had taken both of the calculus courses in the last twenty years. For each pair of letter grades (X, Y ), the
18. Excerpt from the TV show The Rockford Files:Rockford: There are only two doctors in town. The chances of both autopsies being performed by the same doctor are 50–50.Reporter: No, that is only for one autopsy. For two autopsies, the chances are 25–75.Rockford: You’re right. Was Rockford
17. Let S = {ω1, ω2, ω3} be the sample space of an experiment. If P????{ω1, ω2}= 0.5 and P????{ω1, ω3}= 0.7, find P????{ω1}, P????{ω2}, and P????{ω3}.
16. Suppose that in the following table, by the probability of the interval [x, y) we mean the probability that, in a certain region, a person dies on or after his or her xth birthday but before his or her yth birthdayBased on this table of mortality rates, what is the probability that a baby just
15. A professor asks her students to present three methods of generating a random state out of US states. One of the methods a student,Walter, introduces is to draw a congressman from the list of 535 voting members of the House of Representatives and then select the state to which that member
14. In a horse race, the odds in favor of the first horse winning in an 8-horse race are 2 to 5.The odds against the second horse winning are 7 to 3. What is the probability that one of these horses will win?
13. Suppose that 75% of all investors invest in traditional annuities and 45% of them invest in the stock market. If 85% invest in the stock market and/or traditional annuities, what percentage invest in both?
12. Suppose that the probability that a driver is a male, and has at least one motor vehicle accident during a one-year period, is 0.12. Suppose that the corresponding probability for a female is 0.06.What is the probability of a randomly selected driver having at least one accident during the next
11. From an ordinary deck of 52 cards, we draw cards at random and without replacement.What is the probability that at least two cards must be drawn to obtain a face card?
10. A motorcycle insurance company has 7000 policyholders of whom 5000 are under 40.If 4100 of the policyholders are males and under 40, 1100 are married and under 40, and 550 are married males who are under 40, find the probability that the next motorcycle policyholder of this company who gets
9. Jacqueline, Bonnie, and Tina are the only contestants in an athletic race, where it is not possible to tie. The probability that Bonnie wins is 2/3 that of Jacqueline winning and 4/3 that of Tina winning. Find the probability of each of these three athletes winning.
8. The admission office of a college admits only applicants whose high school GPA is at least 3.0 or whose SAT score is 1200 or higher. If 38% of the applicants of this college have at least a 3.0 GPA, 30% have a SAT of 1200 or higher, and 15% have both, what percentage of all the applicants are
7. For events A and B, suppose that the probability that at least one of them occurs is 0.8 and the probability that both of them occur is 0.3. Find the probability that exactly one of them occurs.
6. In a probability test, for two events E and F of a sample space, Tina’s calculations resulted in P(E) = 1/4, P(F) = 1/2, and P(EF) = 3/8. Is it possible that Tina made a mistake in her calculations?Why or why not?

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