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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions

2. There are 14 marketing 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?
1. In a trial, the judge is 65% sure that Susan has committed a crime. Robert is a witness who knows 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. What is the probability
23. 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
22. 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 are a total of eight French and six Korean students in classes A and B, what is the probability that class C has
21. 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
20. 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?
19. 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?
18. 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?
17. Suppose that 28 crayons, of which four are red, are divided randomly among Jack, Marty, Sharon, and Martha (seven each). If Sharon has exactly one red crayon, what is the probability that Marty has the remaining three?
16. 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.
15. Prove that if P (E | F ) ≥ P (G | F ) and P (E | Fc) ≥ P (G | Fc), then P (E) ≥ P (G).
14. Prove Theorem 3.1.
13. Prove that if P (A) = a and P (B) =b, then P (A | B) ≥ (a + b − 1)/b.
12. Show that if P (A) = 1, then P (B | A) = P (B).
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
10. From 100 cards numbered 00, 01, . . . , 99, one card is drawn. Suppose that α and β are 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?
8. In throwing two fair dice, what is the probability of a sum of 5 if they land on different numbers?
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
6. Prove that P (A | B) > P (A) if and only if P (B | A) > P (B). In probability, if for two events A and B, P (A | B) > P (A), we say that A and B are positively correlated. If P (A | B) < P (A), A and B are said to be negatively correlated.
5. What is the probability that both of them have landed 5? 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
4. Suppose that two fair dice have been tossed and the total of their top faces is found to be divisible by
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
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.
1. Suppose that 15% of the population of a country are unemployed women, and a total of 25% are unemployed. What percent of the unemployed are women?
28. From the set of integers $ 1, 2, 3, . . . , 100000% a 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
27. A four-digit number is selected at random. What is the probability that its ones place is greater than its tens place, its tens place is greater than its hundreds place, and its hundreds place is greater than its thousands place? Note that the first digit of an n-digit number is nonzero.
26. A fair die is tossed eight times. What is the probability that the eighth outcome is not a repetition?
25. To test if a computer program works properly, we run it with 12 different data sets, using four computers, each running three data sets. If the data sets are distributed randomly among different computers, how many possibilities are there?
24. 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?
23. In a lottery the tickets are numbered 1 through N. A person purchases n (1 ≤ n ≤ N) tickets at random. What is the probability that the ticket numbers are consecutive? (This is a special case of a problem posed by Euler in 1763.)
22. 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?
21. 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 not ; 6 1
20. In a bridge game, each of the four players gets 13 random cards. What is the probability that every player has an ace?
19. The chairperson of the industry-academic partnership of a town invites all 12 members of the board and their spouses to his house for a Christmas party. If a board member may attend without his spouse, but not vice versa, how many different groups can the chairperson get?
18. If five Americans, five Italians, and five Mexicans sit randomly at a round table, what is the probability that the persons of the same nationality sit together?
17. Cyrus and 27 other students are taking a course in probability this semester. If their professor chooses eight students at random and with replacement to ask them eight different questions, what is the probability that one of them is Cyrus?
16. 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?
15. By mistake, a student who is running a computer program enters with negative signs two of the six positive numbers and with positive signs two of the four negative numbers. If at some stage the program chooses three distinct numbers from these 10 at random and multiplies them, what is the
14. A palindrome is a sequence of characters that reads the same forward or backward. For example, rotator, madam, Hannah, the German name Otto, and an Indian language, Malayalam, are palindromes. So are the following expressions: “Put up,” “Madam I’m Adam,” “Was it a cat I saw?” and
13. How many eight-digit numbers without two identical successive digits are there?
12. In a small town, both of the accidents that occurred during the week of June 8, 1988, were on Friday the 13th. Is this a good excuse for a superstitious person to argue that Friday the 13th’s are inauspicious?
11. Bill and John play in a backgammon tournament. A player is the winner if he wins three games in a row or four games altogether. In what percent of all possible cases does the tournament end because John wins four games without winning three in a row?
10. In how many arrangements of the letters BERKELEY are all three E’s adjacent?
9. In how many ways can 23 identical refrigerators be allocated among four stores so that one store gets eight refrigerators, another four, a third store five, and the last one six refrigerators?
8. Suppose that 30 lawn mowers, of which seven have defects, are sold to a hardware store. If the store manager inspects six of the lawn mowers randomly, what is the probability that he finds at least one defective lawn mower?
7. 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?
6. A window dresser has decided to display five different dresses in a circular arrangement. How many choices does she have?
5. A father buys nine different toys for his four children. In how many ways can he give one child three toys and the remaining three children two toys each?
4. From the 10 points that are placed on a circumference, two are selected randomly. What is the probability that they are adjacent?
3. If four fair dice are tossed, what is the probability that they will show four different faces?
2. Virginia has 1 one-dollar bill, 1 two-dollar bill, 1 five-dollar bill, 1 ten-dollar bill, and 1 twenty-dollar bill. She decides to give some money to her son Brian without asking for change. How many choices does she have?
1. Albert goes to the grocery store to buy fruit. There are seven different varieties of fruit, and Albert is determined to buy no more than one of any variety. How many different orders can he place?
52. What is the probability that the birthdays of at least two students of a class of size n are at most k days apart? Assume that the birthrates are constants throughout the year and that each year has 365 days.
51. A fair coin is tossed n times. Calculate the probability of getting no successive heads. Hint: Let xi be the number of sequences of H’s and T’s of length i with no successive H’s. Show that xi satisfies xi = xi−1 + xi−2, i ≥ 2, where x0 = 1 and x1 = 2. The answer is xn/2n. Note that
50. An absentminded professor wrote n letters and sealed them in envelopes without 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.
49. Using the binomial theorem, calculate the coefficient of xn in the expansion of (1 + x)2n = (1 + x)n(1 + x)n to prove that ; 2n n < = .n i=0 ; n i
48. We are given n (n > 5) points in space, no three of which lie on the same straight line. Let # be the family of planes defined by any three of these points. Suppose that the points are situated in a way that no four of them are coplanar, and no two planes of # are parallel. From the set of the
47. In how many ways can 10 different photographs be placed in six different envelopes, no envelope remaining empty? Hint: An easy way to do this problem is to use the following version of the inclusion-exclusion principle: Let A1, A2, . . . , An be n subsets of a finite set # with N elements. Let
46. For a given position with n applicants, m applicants are equally qualified and n − m applicants are not qualified at all. Assume that a recruitment process is considered to be fair if the probability that a qualified applicant is hired is 1/m, and the probability that an unqualified applicant
45. 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?
44. 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 happen. (b) To estimate the number of trout in the lake, statisticians find
43. 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).
42. Evaluate the following sum: ; n 0 < + 1 2 ; n 1 < + 1 3 ; n 2 < + · · · + 1 n + 1 ; n n < .
41. By a combinatorial argument, prove that for r ≤ n and r ≤ m, ; n + m r < = ; m 0
40. Show that ; n 0 < + ; n + 1 1 < + · · · + ; n + r r < = ; n + r + 1 r < . Hint: ; n r < = ; n + 1 r < − ; n r − 1 < .
39. Prove that ; n 0 < − ; n 1 < + ; n 2 < − · · · + (−1) k ; n k < + · · · + (−1) n ; n n < = 0.
38. Suppose that n indistinguishable balls are placed at random into n distinguishable cells. What is the probability that exactly one cell remains empty?
37. A train consists of n cars. Each of m passengers (m>n) will choose a car at random to ride in. What is the probability that (a) there will be at least one passenger in each car; (b) exactly r (r < n) cars remain unoccupied?
36. In Maryland’s lottery, players pick six different integers between 1 and 49, the order of selection being irrelevant. The lottery commission then randomly selects six of these as the winning numbers. What is the probability that at least two consecutive integers are selected among the winning
35. An ordinary deck of 52 cards is divided into two equal sets randomly. What is the probability that each set contains exactly 13 red cards?
34. 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?
33. A class contains 30 students. What is the probability that there are six months each containing the birthdays of two students, and six months each containing the birthdays of three students? Assume that all months have the same probability of including the birthday of a randomly selected person.
32. Prove the binomial expansion formula by induction. Hint: Use the identity ; n k − 1 < + ; n k < = ; n + 1 k < .
31. According to the 1998 edition of Encyclopedia Britannica, “there are at least 15,000 to as many as 35,000 species of orchids.” These species have been found naturally and are distinct from each other. Suppose that hybrids can be created by crossing any two existing species. Furthermore,
30. 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?
29. A staircase is to be constructed between M and N (see Figure 2.4). The distances from M to L, and from L to N, are 5 and 2 meters, respectively. If the height of a step is 25 centimeters and its width can be any integer multiple of 50 centimeters, how many different choices do we have? M N L
28. Using induction, binomial expansion, and the identity ; n n1 < (n − n1)! n2! n3! · · · nk! = n! n1! n2! · · · nk! , prove the formula of multinomial expansion.
27. 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?
26. What is the coefficient of x3y7 in the expansion of (2x − y + 3)13?
25. What is the coefficient of x2y3z2 in the expansion of (2x − y + 3z)7?
24. Using Theorem 2.6, expand (x + y + z)2.
23. A four-digit number is selected at random. What is the probability that its ones place is less than its tens place, its tens place is less than its hundreds place, and its hundreds place is less than its thousands place? Note that the first digit of an n-digit number is nonzero.
22. A history professor who teaches three sections of the same course every semester decides to make several tests and use them for the next 10 years (20 semesters) as final exams. The professor has two policies: (1) not to give the same test to more than one class in a semester, and (2) not to
21. 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?
20. Poker hands are classified into the following 10 nonoverlapping categories in increasing order of likelihood. Calculate the probability of the occurrence of each class separately. Recall that a poker hand consists of five cards selected randomly from an ordinary deck of 52 cards. Royal flush:
19. 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 the ; 24 12< possible sets of winners are equally probable.
17. Find the values of .n i=0 2i ; n i < and .n i=0 xi ; n i < .
16. 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?
15. 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 (a) there are exactly two professors on the committee; (b) all committee members are of the same rank?
14. 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?
12. A team consisting of three boys and four girls must be formed from a group of nine boys and eight girls. If two of the girls are feuding and refuse to play on the same team, how many possibilities do we have?
11. Find the coefficient of x3y4 in the expansion of (2x − 4y)7.
10. Find the coefficient of x9 in the expansion of (2 + x)12.
9. In front of Jeff’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. Jeff parks his car in the only empty spot that is left. Then he goes to his office. On his return he finds that there are seven empty
8. Lili has 20 friends. Among them are Kevin and Gerry, who are husband and wife. Lili wants to invite six of her friends to her birthday party. If neither Kevin nor Gerry will go to a party without the other, how many choices does Lili have?
7. 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?
6. Judy puts one piece of fruit in her child’s lunch bag every day. If she has three oranges and two apples for the next five days, in how many ways can she do this?
5. A random sample of n elements is taken from a population of size N without replacement. What is the probability that a fixed element of the population is included? Simplify your answer.
2. Each state of the 50 in the United States has two senators. In how many ways may a majority be achieved in the U.S. Senate? Ignore the possibility of absence or abstention. Assume that all senators are present and voting.

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