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probability and stochastic modeling

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

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 thatHint: Use Boole’s inequality, discussed in Exercise 42. P(A) 1-P(A). n=1 n=1
42. Let A1, A2, A3, . . . be a sequence of events of a sample space. Prove thatThis is called Boole’s inequality. P(UA) P(An). n=1 n=1
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(EUF)+P(EUF) + P(EUF) + P(EUF) = 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?
5. The probability that an earthquake will damage a certain structure during a year is 0.015.The probability that a hurricane will damage the same structure during a year is 0.025. If the probability that both an earthquake and a hurricane will damage the structure during a year is 0.0073, what is
4. Sixty-eight minutes through Morituri, the 1965 Marlon Brando movie, a group of underground anti-Nazis wanting to escape from a Nazi ship to a nearby island state that the chances of their succeeding are 15-to-1. Eighty-two minutes through the movie, the same group refers to this estimate stating
3. Suppose that 33%of the people have O+ blood and 7%have O−.What is the probability that the next president of the United States has type O blood?
2. Show that if A and B are mutually exclusive, then P(A) + P(B) ≤ 1.
In a community, 32% of the population are male smokers; 27% are female smokers.What percentage of the population of this community smoke
Suppose that 25% of the population of a city read newspaper A, 20% read newspaper B, 13% read C, 10% read both A and B, 8% read both A and C, 5% read B and C, and 4% read all three. If a person from this city is selected at random, what is the probability that he or she does not read any of these
A number is chosen at random from the set of integers {1, 2, . . . , 1000}.What is the probability that it is divisible by 3 or 5 (i.e., either 3 or 5 or both)?
A number is selected at random from the set {1, 2, . . . ,N}. What is the probability that the number is divisible by k, 1 ≤ k ≤ N?
A number is selected at random from the set of integers1, 2, . . . , 1000.What is the probability that the number is divisible by 3?
Example 1.13 An elevator with two passengers stops at the second, third, and fourth floors.If it is equally likely that a passenger gets off at any of the three floors, what is the probability that the passengers get off at different floors?
4. Consider the system shown by the diagram of Figure 1.3, consisting of 7 components denoted by 1, 2, . . . , 7. Suppose that each component is either functioning or not functioning, with no other capabilities. Suppose that the system itself also has two performance capabilities, functioning and
3. Find the simplest possible expression for the event (E ∪ F)(F ∪ G)(EG ∪ Fc).
2. In a large hospital, there are 100 patients scheduled to have heart bypass surgery. Let Ei, 1 ≤ i ≤ 100, be the event that the ith patient lives through the postoperative period of the surgery. In terms of Ei’s, (a) describe the event that all patients survive the critical postoperative
1. Jody, Ann, Bill, and Karl line up in a random order to get their photo taken. Describe the event that, on the line, males and females alternate.
34. Let {A1,A2,A3, . . .} be a sequence of events of a sample space S. Find a sequence{B1,B2,B3, . . .} of mutually exclusive events such that for all n ≥ 1, Sn S i=1 Ai = n i=1 Bi.
33. Let {A1,A2,A3, . . .} be a sequence of events. Find an expression for the event that infinitely many of the Ai’s occur.
32. In a mathematics department of 31 voting faculty members, there are three candidates running for the chair position. The voting procedure adopted by the department is approval voting, in which the eligible voters can vote for as many candidates as they wish. The candidate with themaximumvotes
31. Define a sample space for the experiment of putting in a random order seven different books on a shelf. If three of these seven books are a three-volume dictionary, describe the event that these volumes stand in increasing order side by side (i.e., volume I precedes volume II and volume II
30. Let {An}∞n=1 be a sequence of events. Prove that for every event B,(a) B????S∞ i=1 Ai=S∞ i=1 BAi.(b) B S????T∞ i=1 Ai=T∞ i=1(B∪Ai).
29. Let A and B be two events. Prove the following relations by the elementwise method.(a) (A − AB) ∪ B = A ∪ B.(b) (A ∪ B) − AB = ABc ∪ AcB.
28. Prove De Morgan’s second law, (AB)c = Ac ∪ Bc, (a) by elementwise proof; (b) by applying De Morgan’s first law to Ac and Bc.
27. A point is chosen at random from the interval (−1, 1). Let E1 be the event that it falls in the interval (−1/2, 1/2), E2 be the event that it falls in the interval (−1/4, 1/4) and, in general, for 1 ≤ i < ∞, Ei be the event that the point is in the interval (−1/2i, 1/2i).Find S ∞
26. In an experiment, cards are drawn, one by one, at random and successively from an ordinary deck of 52 cards. Let An be the event that no face card or ace appears on the first n − 1 drawings, and the nth draw is an ace. In terms of An’s, find an expression for the event that an ace appears
25. For the experiment of flipping a coin until a heads occurs, (a) describe the sample space;(b) describe the event that it takes an odd number of flips until a heads occurs.
24. Travis picks up darts to shoot toward an 18′′-diameter dartboard aiming at the bullseye on the board. Describe a sample space for the point at which a dart hits the board.
23. Let E, F, and G be three events. Determine which of the following statements are correct and which are incorrect. Justify your answers.(a) (E − EF) ∪ F = E ∪ F.(b) FcG ∪ EcG = G(F ∪ E)c.(c) (E ∪ F)cG = EcFcG.(d) EF ∪ EG ∪ FG ⊂ E ∪ F ∪ G.
22. Prove that the event B is impossible if and only if for every event A, A = (B ∩ Ac) ∪ (Bc ∩ A).
21. A device that has n, n > 1, components fails to operate if at least one of its components breaks down. The device is observed at a random time. Let Ai, 1 < i ≤ n, denote the event that the ith component is operative at the random time. In terms of Ai’s, describe the event that the device is
20. At a certain university, every year eight to 12 professors are granted University Merit Awards. This year among the nominated faculty are Drs. Jones, Smith, and Brown. Let A, B, and C denote the events, respectively, that these professors will be given awards.In terms of A, B, and C, find an
19. Find the simplest possible expression for the following events.(a) (E ∪ F)(F ∪ G).(b) (E ∪ F)(Ec ∪ F)(E ∪ Fc).
18. An insurance company sells a joint life insurance policy to Alexia and her husband, Roy.Define a sample space for the death or survival of this couple in five years. What is the event that at that time only one of them lives?
17. 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;and for hole saws, it is between 300 and 600 thousands. Define a sample space for the demands for these three types of
16. A psychologist, who is interested in human complexion, in a study of hues and shades, shows her subjects three pieces of wood colored almond, lemon, and flax, respectively.She then asks them to identify their favorite colors. Define a sample space for the answer given by a random subject.
15. A limousine that carries passengers from an airport to three different hotels just left the airport with two passengers. Describe the sample space of the stops and the event that both of the passengers get off at the same hotel.
14. Let E, F, and G be three events; explain the meaning of the relations E ∪ F ∪ G = G and EFG = G.
13. When flipping a coin more than once, what experiment has a sample space defined by the following?S = {TT, HTT, THTT, HHTT, HHHTT, HTHTT, THHTT, . . . }.
12. A device that has two components fails if at least one of its components breaks down.The device is observed at a random time. Let Ai, 1 ≤ i ≤ 2, denote the outcome that the ith component is operative at the random time. In terms of Ai’s, (a) define a sample space for the status of the
11. A telephone call from a certain person is received some time between 7:00 A.M. and 9:10 A.M. every day. Define a sample space for this phenomenon, and describe the event that the call arrives within 15 minutes of the hour.
10. Define a sample space for the experiment of drawing two coins from a purse that contains two quarters, three nickels, one dime, and four pennies. For the same experiment describe the following events:(a) drawing 26 cents;(b) drawing more than 9 but less than 25 cents;(c) drawing 29 cents.
9. Two dice are rolled. Let E be the event that the sum of the outcomes is odd and F be the event of at least one 1. Interpret the events EF, EcF, and EcFc.
8. Define a sample space for the experiment of putting three different books on a shelf in random order. If two of these three books are a two-volume dictionary, describe the event that these volumes stand in increasing order side-by-side (i.e., volume I precedes volume II).
7. Define a sample space for the experiment of choosing a number from the interval (0, 20).Describe the event that such a number is an integer.
6. A box contains three red and five blue balls. Define a sample space for the experiment of recording the colors of three balls that are drawn from the box, one by one, with replacement.
5. A deck of six cards consists of three black cards numbered 1, 2, 3, and three red cards numbered 1, 2, 3. First, Vann draws a card at random and without replacement. Then Paul draws a card at random and without replacement from the remaining cards. Let A be the event that Paul’s card has a
4. In the experiment of tossing two dice, what do the following events represent?E =(1, 3), (2, 6), (3, 1), (6, 2)and F =(1, 5), (2, 4), (3, 3), (4, 2), (5, 1).

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