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probability and stochastic modeling

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

We draw eight cards at random from an ordinary deck of 52 cards. Given that three of them are spades, what is the probability that the remaining five are also spades?
An English class consists of 10 Koreans, 5 Italians, and 15 Hispanics. A paper is found belonging to one of the students of this class. If the name on the paper is not Korean, what is the probability that it is Italian? Assume that names completely identify ethnic groups
From the set of all families with two children, a child is selected at random and is found to be a girl. What is the probability that the second child of this girl’s family is also a girl? Assume that in a two-child family all sex distributions are equally probable.
From the set of all families with two children, a family is selected at random and is found to have a girl. What is the probability that the other child of the family is a girl?Assume that in a two-child family all sex distributions are equally probable.
In a certain region of Russia, the probability that a person lives at least 80 years is 0.75, and the probability that he or she lives at least 90 years is 0.63. What is the probability that a randomly selected 80-year-old person from this region will survive to become 90?
10. From a faculty of six professors, six associate professors, ten assistant professors, and twelve instructors, a committee of size six is formed randomly. What is the probability that there is at least one person from each rank on the committee?Hint: Be careful, the answer is notTo find the
9. Let ℓ < k < m < n be positive integers. There will be n new movies released next month. Suppose that a prominent movie critic, Mr. Wilbert, whose reviews are syndicated to more than 200 newspapers, is required to review exactly m movies next month for publication.(a) How many choices does he
8. There are 48 students to be randomly distributed among 3 different jewelry classes, 16 students per class. If 3 of the students are visually impaired, find the probability that(a) each class gets one of the them; (b) all 3 end up in the same class.
7. Give a combinatorial proof for the relationUsing this relation, show that n n n k 24 -1
6. There are 15 trees in a row of which 5 adjacent ones are infected by a fungal disease. Is this logical evidence for an arborist to conclude that the fungus spread from one tree to next through root-to-root contact?Hint: Suppose that the disease strikes randomly, say, through airborne spores, and
5. Each state of the 50 in the United States has two senators. For each state, the senator with greater longevity in the chamber is the senior senator, and the other one is the junior senator of that state. Since no state has both senators up for election in the same year, every state always has a
4. Possible passwords for a computer account are strings of length 6 of numbers, capital letters, and small letters, with no repetition allowed (all capital letters are distinguishable from all small letters. So a string such as AaBz8b has no repetition). What is the probability that a password
3. The card game, bridge, featuring two teams of two players each, is played with an ordinary deck of 52 cards. The cards are dealt among the players, 13 each, randomly. What is the probability that exactly 3 of the aces are in the hands of one team?
2. In a ternary code, each code has 7 bits, each of which is 0, 1, or 2.What is the probability that a random code ends in 2 and has exactly three 0’s?
1. What is the probability that exactly two students of a class of size 23 have the same birthday? Assume that the birth rates are constant throughout the year and each year has 365 days.
40. Show that for n = 1, 2, 3, . . . , 2n
39. From the set of integers1, 2, 3, . . . , 100000a number is selected at random.What is the probability that the sum of its digits is 8?Hint: Establish a one-to-one correspondence between the set of integers from{1, 2, . . . , 100000} the sum of whose digits is 8, and the set of possible ways 8
37. To test if a computer program works properly, we run it with 12 different datasets, using four computers, each running three datasets. If the datasets are distributed randomly among different computers, how many possibilities are there?
36. An ordinary deck of 52 cards is dealt, 13 each, at random among A, B, C, and D. What is the probability that (a) A and B together get two aces; (b) A gets all the face cards; (c)A gets five hearts and B gets the remaining eight hearts?
35. The chair of an academic department needs to form 3 search committees each consisting of 5 tenured full professors. If the department has 18 such faculty, how many choices does the chair have if he decides that no one can serve on more than one search committee?Note that a set of 5 professors
33. An urn contains 15 white and 15 black balls. Suppose that 15 persons each draw two balls blindfolded from the urn without replacement.What is the probability that each of them draws one white ball and one black ball?
32. In a bridge game, each of the four players gets 13 random cards.What is the probability that every player has an ace?
30. Suppose that four women and two men enter a restaurant and sit at random around a table that has four chairs on one side and another four on the other side. What is the probability that the men are not all sitting on one side?
29. In a lottery, players pick 5 different integers between 1 and 47, and the order of selection is irrelevant. The lottery commission then randomly selects 5 of these as the winning numbers. A player wins the grand prize if all 5 numbers that he or she has selected match the winning numbers.
27. A broadcasting company recruits college students to help with their Olympic coverage.Suppose that three of their finalists are female athletes, and the remaining four are male athletes. If they choose four of them randomly, what is the probability that (a) two of them are female and two are
26. How many eight-digit numbers without two identical successive digits are there?
24. For a card game tournament, 16 individuals must be divided into 4 groups of 4 each. In how many ways can this be done?
23. For his calculus final exam, Dr. Channing, whose classes have a reputation for being very easy, gives his students 20 problems with their complete solutions, and tells them that he will choose 10 of them randomly for the final exam. Students who solve all 10 problems correctly will get an A,
21. In how many arrangements of the letters BERKELEY are all three E’s adjacent?
20. From a group of 8 male and 16 female potential jurors, a jury of 12 is selected at random.What is the probability that all males are included?
18. One of Zoey’s iPhone playlists has 12 songs, 5 by Beyonc´e, 3 by Adele, and 4 by C´eline Dion. She sets the iPhone to play the songs of that playlist. However, before doing that Zoey checks the shuffle button of her iPhone, and the iPhone plays the playlist’s songs in a random order,
16. Korina is returning from a trip to the United Kingdom and is packing 12 wrapped chocolate bars of which 3 are Aero Caramels, 4 are Chomps, 3 are Duncans, and 2 are Freddos.The bars of the same brand are indistinguishable. In how many distinguishable ways can Korina pack these chocolate bars?
15. Judy has three sets of classics in literature, each set having four volumes. In how many ways can she put them on a bookshelf so that books of each set are not separated?
14. After each lecture, a professor assigns 10 problems. However, he grades only 4 of them randomly. If Salina’s solution to only one of the recent assignment problems is wrong, what is the probability that she still gets a perfect score?
13. In a soccer tryout, 25 players compete. The coach has decided to cut 8 of the players, assign 1 as a goalie, 4 as defenders, 3 as midfielders, 3 as forwards, and the remaining 6 as substitutes. How many choices does he have?
12. At a departmental party, Dr. James P. Coughlin distributes 4 baby photos of 4 of his colleagues among the guests with a list of the names of those colleagues. Each guest is invited to identify the babies in the photos.What is the probability that a participant can identify exactly 2 of the 4
11. A list of all permutations of 13579 is put in increasing order. What is the 100th number in the list?Hint: The first 4! = 24 numbers all begin with 1.
10. In a Napa Valley winery, guests are invited to a tasting room, and, for each of them, an affable staff member pours 6 types of wine, which are sold at different prices, in 6 small 3-ounce stylish glasses, at random and one at a time.What is the probability that the first glass of wine they
9. A student club has 50 members, of which 9 have volunteered themselves to serve on the executive committee. One of the volunteers, Ruth, is known to be a bully. If the faculty members in charge of the club choose the executive committee members at random from the volunteers, what is the
5. If four fair dice are tossed, what is the probability that they will show four different faces?
4. To enhance children’s STEM skills, a teacher gives each student 12 Lego bricks but asks them to use only 8 of them to construct a toy. If no two of the 12 bricks are identical, how many choices does each child have for his or her 8 Lego bricks?
2. In how many ways can we divide 9 toys among 3 children evenly?
1. Use Stirling’s formula to approximate (27) and [(2n)!] [(4n)! (n!)2] for large n.
Approximate the value of2n (n!)2(2n)! for large n.
4. For a chess tournament, 16 individuals are to be divided into 8 groups of 2 each. In how many ways can this be done?
3. There are 8 chairs in a row, on the upper deck of a boat, attached to the floor by screws and nails. A sociologist observed that from the first 4 passengers who sat in that row no two sat next to each other. Based on this observation, can she conclude that, when possible, passengers avoid taking
2. A broadcasting company recruits college students to help with their Olympic coverage.If they choose 4 athletes randomly from their 7 finalists, what is the probability that(a) the 4 tallest of the finalists are selected? (b) Exactly 3 of the 4 athletes selected are the tallest of the finalists?
1. For an experiment, Sheri, a neuroscientist, needs to select 5 rats from each of the 5 groups of rats being studied. If each group has 12 rats, how many choices does Sheri have?
66. An absentminded professor wrote n letters and sealed them in envelopeswithout writing the addresses on them. Then he wrote the n addresses on the envelopes at random.What is the probability that exactly k of the envelopes were addressed correctly?Hint: Consider a particular set of k letters.
65. Using the binomial theorem, calculate the coefficient of xn in the expansion of(1 + x)2n = (1 + x)n(1 + x)n to prove thatFor a combinatorial proof of this relation, see Example 2.32. (2n) n IM- i=0 n i 2
64. For k ≥ 1, n ≥ k, how many distinct positive integer vectors (x1, x2, . . . , xk) satisfy the inequality x1 + x2 + · · · + xk ≥ n?
62. Let n be a positive integer. A random sample of four elements is taken from the set 0, 1, 2, . . . , n, one at a time and with replacement. What is the probability that the sum of the first two elements is equal to the sum of the last two elements?
61. A lake has N trout, and t of them are caught at random, tagged, and returned.We catch n trout at a later time randomly and observe that m of them are tagged.(a) Find PN, the probability of what we observed actually happening.(b) To estimate the number of trout in the lake, statisticians find
60. Suppose that five points are selected at random from the interval (0, 1). What is the probability that exactly two of them are between 0 and 1/4?Hint: For any point there are four equally likely possibilities: to fall into (0, 1/4),[1/4, 1/2), [1/2, 3/4), and [3/4, 1).
59. Evaluate the following sum: (0)+()+(2) (*) + +-+ (9) +- (9) + (9)
58. By a combinatorial argument, prove that for r ≤ n and r ≤ m, n+m ("+")-(6)()-(0) (-)--(7)(0) m r + r
57. Prove that n n n 2 n +.. (0)-(0)-(0)----+--(0)----* (") - k 0.
56. Show that Hint: = (*)+(+) +-+("+")-(*****). n+ (c)-(+)-(+). ==
55. At a departmental party, Dr. James P. Coughlin distributes 7 baby photos of 7 of his colleagues among the guests with a list of the names of those colleagues. Each guest is invited to identify the babies in the photos.What is the probability that a participant can identify exactly 3 of the 7
54. Suppose that n indistinguishable balls are placed at random into n distinguishable cells.What is the probability that exactly one cell remains empty?
52. In a box, there are 12 balls, identical in every way, except that they are numbered 1 through 12.We draw 6 balls at random and without replacement one by one.What is the probability that the numbers on the balls appear in increasing order but not necessarily consecutive?Hint: There is a
50. In a closet there are 10 pairs of shoes. If six shoes are selected at random, what is the probability of (a) no complete pairs; (b) exactly one complete pair; (c) exactly two complete pairs; (d) exactly three complete pairs?
48. Prove the binomial expansion formula by induction.Hint: Use the identity n k (k-1) + (7) = (1 + 1). k
47. (Newton–Pepys problem) In 1693, Samuel Pepys, an English Naval administrator and member of the Parliament, corresponded with Isaac Newton concerning the following wager he intended to make: Which of the following trials is more likely than the other two?(a) At least one 6 when 6 dice are
46. Each state of the 50 in the United States has two senators. What is the probability that in a random committee of 50 senators (a) Maryland is represented; (b) all states are represented?
44. Using induction, binomial expansion, and the identityprove the formula of multinomial expansion (Theorem 2.6). n nin! ng! (n-n)! nk! n! n! n! nk!
43. An ordinary deck of 52 cards is dealt, 13 each, to four players at random. What is the probability that each player receives 13 cards of the same suit?
42. What is the coefficient of x3y7 in the expansion of (2x − y + 3)13?
41. What is the coefficient of x2y3z2 in the expansion of (2x − y + 3z)7?
40. Using Theorem 2.6, expand (x + y + z)2.
37. A young businessman is obsessed with buying life insurance policies to protect his family.He decides to allocate $7200 a year toward all or some of the Term, Universal, Indexed Universal, Survivorship Universal, and Variable Universal life insurance premiums.If he decides to pay premiums in
36. There are 12 nuts and 12 bolts in a box. If the contents of the box are divided between two handymen, what is the probability that each handyman will get six nuts and six bolts?
34. How many all capital, different words of any length, meaningful or meaningless, can we create from the letters of “JUSTICE” without repeating a letter?
33. Suppose that 12 married couples take part in a contest. If 12 persons each win a prize, what is the probability that from every couple one of them is a winner? Assume that all of the24 12!possible sets of winners are equally probable.
32. A fair die is tossed six times. What is the probability of getting exactly two 6’s?
31. Find the values of TL i=0 n i n and i=0 n 2
30. A round-robin tournament, as defined in the previous exercise, is a competition in which each participant plays against every other participant exactly once. Using a round-robin tournament with n + 1 contestants, give a combinatorial argument for the following identity: 1+2++ n = +n= (n+1). 2
29. A round-robin tournament is a competition in which each participant plays against every other participant exactly once. In a round-robin tournament with n contestants, what is the total number of possible outcomes? An outcome lists out the winner and the loser in each competition.
28. A lake contains 200 trout; 50 of them are caught randomly, tagged, and returned. If, again, we catch 50 trout at random, what is the probability of getting exactly five tagged trout?
27. In a medical study, 25 patients with a duodenal ulcer have volunteered to be treated by an experimental drug. The researcher has decided to treat 18 of the patients, randomly selected, by the new drug and to give the remaining 7 a placebo as controls. If the duodenal ulcers of 17 of these
25. A soccer coach assigns his players randomly, 1 as a goalie, 4 as defenders, 3 as midfielders, and 3 as forwards. (a) How many choices does he have? (b) If only one player is experienced enough to play as a goalie, what is the probability that in the random division, he ends up to be the goalie?
24. In North America, in the World Series, which is the most important baseball event each year, the American League champion team plays a series of games with the National League champion team. The first team to win four games will be the winner of the World Series championship and is awarded the
23. If five numbers are selected at random from the set {1, 2, 3, . . . , 20}, what is the probability that their minimum is larger than 5?
22. A fair coin is tossed 10 times. What is the probability of (a) five heads; (b) at least five heads?
21. There are 23 girls and 19 boys lined up randomly for a play in a kindergarten classroom.What is the probability that the 12th child in the line is a girl?
19. Find the 4th term of the binomial expansion of x+ I
18. Find the 9th term of the binomial expansion of 13 (1+)
17. Find the coefficient of x3y4 in the expansion of (2x − 4y)7.
16. Find the coefficient of x9 in the expansion of (2 + x)12.
15. In addition to many scientific experiments that astronauts should operate while on the space station, they must make sure that the station is in perfect condition. Suppose that there are 13 astronauts on board, and on a specific day, three astronauts are needed to replace broken equipment, five
14. In front of Ilaria’s office there is a parking lot with 13 parking spots in a row.When cars arrive at this lot, they park randomly at one of the empty spots. Ilaria parks her car in the only empty spot that is left. Then she goes to her office. On her return she finds that there are seven
13. In a game, there are 12 tiles in a bag, each bearing one of the numbers 1 through 12.Lida draws 6 tiles at random. What is the probability that the largest number she picks out is 9?
12. In a binary code, each code word has 6 bits, each of which is 0 or 1. What is the probability that a random code word (a) has three 0’s and three 1’s? (b) Begins with two 0’s?(c) Ends in three 1’s?
10. Ann puts at most one piece of fruit in her child’s lunch bag every day. If she has only three oranges and two apples for the next eight lunches of her child, in how many ways can she do this?
8. In a box of 12 fuses, there are 3 that are defective. If a quality control engineer tests 4 of these fuses at random, what is the probability that he does not find any of the defective fuses?
6. As part of an English department’s requirements, students must take 5 courses out of a list of 9, of which 3 are advanced courses in Shakespeare. If it is required that at least one of these 5 courses be a Shakespeare course, how many choices does an English student have for this particular
5. From an ordinary deck of 52 cards, five are drawn randomly. What is the probability of drawing exactly three face cards?
4. There are 24 eggs in a rectangular egg carton arranged in 4 rows of 6 eggs each. Three eggs are selected at random from the carton to make an omelet. What is the probability that the eggs are selected from the corners of the carton?
Prove that (2n n n i=0 n 2
What is the coefficient of x2y3 in the expansion of (2x + 3y)5?

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