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probability and stochastic modeling

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

An absentminded professor wrote n letters and sealed them in envelopes before writing the addresses on the envelopes. Then he wrote the n addresses on the envelopes at random.What is the probability that at least one letter was addressed correctly?
A stockholder is considering whether to invest $150,000 in five new stocks, each in multiples of $10,000. In how many ways can he invest in these stocks if (a) he is determined to buy at least $10,000 of each stock; (b) he does not necessarily select to invest in all five stocks?
Let n be a positive integer, and let x1+x2+· · ·+xk = n be a given equation.A vector (x1, x2, . . . , xk) satisfying x1+x2+· · ·+xk = n is said to be a nonnegative integer solution of the equation if for each i, 1 ≤ i ≤ k, xi is a nonnegative integer. It is said to be a positive integer
Show that the number of different ways n indistinguishable objects can be placed into k distinguishable cells is n+k- n n+k-1 k-1
What is the probability that a poker hand is a full house? A poker hand consists of five randomly selected cards from an ordinary deck of 52 cards. It is a full house if three cards are of one denomination and two cards are of another denomination: for example, three queens and two 4’s.
From an ordinary deck of 52 cards, seven cards are drawn at random and without replacement.What is the probability that at least one of the cards is a king?
In Maryland’s lottery, players pick six different integers between 1 and 49, order of selection being irrelevant. The lottery commission then randomly selects six of these as the winning numbers. A player wins the grand prize if all six numbers that he or she has selected match the winning
In a small town, 11 of the 25 schoolteachers are pro-life, eight are pro-choice, and the rest are indifferent. A random sample of five schoolteachers is selected for an interview.What is the probability that (a) all of themare pro-choice; (b) all of themhave the same opinion?
A random sample of 45 instructors from different state universities were selected randomly and asked whether they are happy with their teaching loads. The responses of 32 were negative. If Drs. Smith, Brown, and Jones were among those questioned, what is the probability that all three of them gave
In how many ways can two mathematics and three biology books be selected from eight mathematics and six biology books?
4. A psychologist specializes in dissocial personality disorder and, for each patient, she lists in order of strength, the 4 strongest traits of impulsivity, recklessness, deceitfulness, exploitative behavior, low conscientiousness, irresponsible behavior, and high negative emotionality. Suppose
3. One of Isabella’s iPhone playlists has 70 songs, 15 of which are by Lionel Richie. She sets the iPhone to play the songs on that playlist. However, before doing that, Isabella checks the shuffle button of her iPhone, and the iPhones begins to play the playlist’s songs in a random order
2. In an Extrasensory Perception (ESP) experiment, five people are each asked to think of a card within the suit of hearts in an ordinary deck of 52 cards. What is the probability that (a) no two think of the same card? (b) At least two of them think of the same card?
1. At a university, all phone numbers begin with 782. If the remaining four digits are equally likely to be 0, 1, . . . , 9, what is the probability that a randomly selected faculty member has a phone number consisting of 7 distinct digits?
37. A box contains five blue and eight red balls. Jim and Jack start drawing balls from the box, respectively, one at a time, at random, and without replacement until a blue ball is drawn.What is the probability that Jack draws the blue ball?
34. A fair die is tossed eight times. What is the probability of exactly two 3’s, exactly three 1’s, and exactly two 6’s?
33. Let S and T be finite sets with n and m elements, respectively.(a) How many functions f : S → T can be defined?(b) Ifm ≥ n, how many injective (one-to-one) functions f : S → T can be defined?(c) If m = n, how many surjective (onto) functions f : S → T can be defined?
31. A list of all permutations of abcdef is put in alphabetical order.What is the 601st entry in the list?Hint: The first 5! entries all begin with a.
30. A town has six parks. On a Saturday, six classmates, who are unaware of each other’s decision, choose a park at random and go there at the same time. What is the probability that at least two of them go to the same park? Convince yourself that this exercise is the same as Exercise 15, only
29. If n balls are randomly placed into n cells, what is the probability that each cell will be occupied?
26. If we put five math, six biology, eight history, and three literature books on a bookshelf at random, what is the probability that all the math books are together?
25. There are 12 students in a class. What is the probability that their birthdays fall in 12 different months? Assume that all months have the same probability of including the birthday of a randomly selected person.
24. Even with all the new advanced technological methods to send signals, some ships still use flags for that purpose. Each flag has its own significance as a signal, and each arrangement of two or more flags conveys a special message. If a ship has 8 flags and 4 flagpoles, how many signals can it
23. Five fair dice are tossed. What is the probability of getting a sum of 27?
22. There are 20 chairs in a room numbered 1 through 20. If eight girls and 12 boys sit on these chairs at random, what is the probability that the thirteenth chair is occupied by a boy?
20. 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 three glasses of wine
18. The letters in the word SUPERCALIFRAGILISTICEXPIALIDOCIOUS are arranged randomly. (a) How many of the distinguishable arrangements begin with G and end with X? (b) What is the probability that the outcome begins with G and ends with X?
17. A fair die is tossed eight times. What is the probability of exactly two 3’s, three 1’s, and three 6’s?
15. Six fair dice are tossed. What is the probability that at least two of them show the same face?
11. There are 25 hard metal bunk beds in an open bay barracks at a military base. In how many ways can newly recruited soldiers be assigned to the 50 beds in the barracks? If the sergeant divides the soldiers into two groups of 25 each and asks one group to occupy the lower bunks and the other the
6. A three-year old child, Sheridan, has 8 square tiles, each bearing one of the letters in her name. If she arranges the tiles randomly in a row all facing in the correct direction, what is the probability that they show her name?
5. Cynthia’s husband is awayMonday through Friday, and she does not know how to cook.If every evening she dines at one of her 8 favorite restaurants randomly, and during that period she dines at a restaurant at most once, what is the probability that, one evening she will eat at Wilbraham
4. How many different messages can be sent by five dashes and three dots?
3. How many permutations of the set {a,b, c,d, e} begin with a and end with c?
2. In New York City, a subway train arrives at the platform, and 12 people enter a subway car that has only 6 empty seats. In how many ways can these passengers be seated?
A fair coin is flipped 10 times. What is the probability of obtaining exactly three heads?
In howmany ways can we paint 11 offices so that four of themwill be painted green, three yellow, two white, and the remaining two pink?
How many different 10-letter codes can be made using three a’s, four b’s, and three c’s?
If five boys and five girls sit in a rowin a randomorder,what is the probability that no two children of the same sex sit together?
Suppose that two anthropology, four computer science, three statistics, three biology, and five music books are put on a bookshelf with a random arrangement. What is the probability that the books of the same subject are together?
Three people, Brown, Smith, and Jones, must be scheduled for job interviews.In how many different orders can this be done?
3. In a pharmacy, there are 8 unrelated people standing in line to pick up their prescriptions.Before the pharmacist gives the patients their prescription medications, following the pharmacy’s policy, she asks them the month and day of their birthdays. What is the probability that at least two of
2. The chair 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 membermay attend without his spouse, but not vice versa, how many different groups can the chair get?
1. In 2010, three unrelated faculty members, Professors Rodriguez, Bucs, and Beineke ended up sitting on three adjacent seats in a row at the spring commencement ceremony of a university, where faculty occupied the chairs randomly. These professors somehow got into a discussion about birthday
36. What is the probability that a random r-digit number (r ≥ 3) contains at least one 0, at least one 1, and at least one 2?
35. A number is selected randomly from the set {0000, 0001, 0002, . . . , 9999}. What is the probability that the sum of the first two digits of the number selected is equal to the sum of its last two digits?
34. The elevator of a four-floor building leaves the first floor with six passengers and stops at all of the remaining three floors. If it is equally likely that a passenger gets off at any of these three floors, what is the probability that, at each stop of the elevator, at least one passenger
33. One of the five elevators in a building leaves the basement with eight passengers and stops at all of the remaining 11 floors. If it is equally likely that a passenger gets off at any of these 11 floors, what is the probability that no two of these eight passengers will get off at the same
32. A delicatessen advertises that it offers over 3000 varieties of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, and ham with or without Swiss and/or American cheese on French, white, or whole wheat bread, and possible additions of
31. In tossing four fair dice, what is the probability of tossing, at most, one 3?
30. How many divisors does a natural number N have?Hint: A natural number N can be written as pn1 1 pn2 2 · · · pnk k , where p1, p2, . . . , pk are distinct primes.
29. An integer is selected at random from the set {1, 2, . . . , 1, 000, 000}. What is the probability that it contains the digit 5?
28. In a large town, Kennedy Avenue is a long north-south avenue with many intersections.A drunken man is wandering along the avenue and does not really know which way he is going. He is currently at an intersection O somewhere in the middle of the avenue.Suppose that, at the end of each block, he
27. A fair die is tossed eight times. What is the probability that the eighth outcome is not a repetition?
26. From an ordinary deck of 52 cards, 15 cards are drawn randomly and without replacement.What is the probability that the first four cards are all clubs?
25. To log into their computer accounts, Alec and Mildred must create passwords that begin with a letter followed by 5 to 7 letters or numbers. If Mildred creates a password herself, but Alec uses a random password creation tool, what is the probability that they end up with the same password?
24. In North America, in the World Series, which is the most important baseball event each year, the American League champion team plays with the National League champion team a series of games. The first team to win four games will be the winner of the World Series championship and is awarded the
23. A salesperson covers islands A, B, . . . , I. These islands are connected by the bridges shown in the Figure 2.3. While on an island, the salesperson takes one of the possible bridges at random and goes to another one. She does her business on this new island and then takes a bridge at random
22. There are N types of drugs sold to reduce acid indigestion. A random sample of n drugs is taken with replacement.What is the probability that brand A is included? A B. C H I D G F E Figure 2.3 Islands and connecting bridges of Exercise 23.
21. A campus telephone extension has four digits. How many different extensions with no repeated digits exist? Of these, (a) how many do not start with a 0; (b) how many do not have 01 as the first two digits?
20. Suppose that four cards are drawn successively from an ordinary deck of 52 cards, with replacement and at random.What is the probability of drawing at least one king?
19. In a mental health clinic there are 12 patients. A therapist invites all these patients to join her for group therapy. How many possible groups could she get?
18. How many four-digit numbers can be formed by using only the digits 2, 4, 6, 8, and 9?How many of these have some digit repeated?
17. For an international traveler, the only Automated Teller Machine (ATM) personal identification numbers (PINs) acceptable are those that do not begin with 0 and those that do not consist of identical digits. If PINs allowed can have 4, 5, or 6 digits, what is the probability that a 4-, 5-, or
16. A delicatessen has advertised that it offers over 500 varieties of sandwiches. If at this deli it is possible to have any combination of salami, turkey, bologna, corned beef, ham, and cheese on French bread with the possible additions of lettuce, tomato, and mayonnaise, is the deli’s
15. How many divisors does 55,125 have?Hint: 55,125 = 325372
14. How many n × m arrays (matrices) with entries 0 or 1 are there?
13. A library has 800,000 books, and the librarian wants to encode each by using a code word consisting of three letters followed by two numbers. Are there enough code words to encode all of these books with different code words?
12. Suppose that in a state, license plates have three letters followed by three numbers, in a way that no letter or number is repeated in a single plate. Determine the number of possible license plates for this state.
11. A multiple-choice test has 15 questions, each having four possible answers, of which only one is correct. If the questions are answered at random, what is the probability of getting all of them right?
10. Mr. Smith has 12 shirts, eight pairs of slacks, eight ties, and four jackets. Suppose that four shirts, three pairs of slacks, two ties, and two jackets are blue. (a) What is the probability that an all-blue outfit is the result of a random selection? (b) What is the probability that he wears
9. From an ordinary deck of 52 cards, 4 cards are selected at random and placed on a table face down. A boy, who would like to be a magician, guesses the denomination and suit of all four cards randomly. What is the probability that he guesses at least one card correctly?
8. Two fair dice are thrown. What is the probability that the outcome is a 6 and an odd number?
7. In a word game, there are 6 tiles in a bag, each bearing one of the letters A, C, E, G, L, N. Zanya draws 8 tiles, one at a time, randomly, writes down the letter each time, and then returns the tile to the bag.What is the probability that, in the order drawn, the letters form the word ELEGANCE?
6. In how many ways can we draw five cards from an ordinary deck of 52 cards (a) with replacement; (b) without replacement?
5. In flipping a fair coin 23 times, what is the probability of all heads or all tails?
4. In how many different ways can 15 offices be painted with four different colors?
3. The population of a town is 20,000. If each resident has three initials, is it true that at least two people have the same initials?
2. How many different five-letter codes can be made usinga, b,c, d, and e? How many of them start with ab?
1. How many six-digit numbers are there? How many of them contain the digit 5? Note that the first digit of an n-digit number is nonzero.
Cheyenne has $4. She decides to bet $1 on the flip of a fair coin four times.What is the probability that (a) she breaks even; (b) she wins money?
Bill and John keep playing chess until one of them wins two games in a row or three games altogether. In what percent of all possible cases does the game end because Bill wins three games without winning two in a row? B B: B -B -J B- -J -B -B B B -J Figure 2.1 Tree diagram of Example 2.11.
A restaurant advertises that it offers over 1000 varieties of pizza. If, at the restaurant, it is possible to have on a pizza any combination of pepperoni, mushrooms, sausage, green peppers, onions, anchovies, salami, bacon, olives, and ground beef, is the restaurant’s advertisement true?
Terra was born on July 4. Assuming that the birth rates are constant throughout the year and each year has 365 days, for Terra to have a 50% chance of meeting at least one person with her birthday, how many random people does she need to meet?
What is the probability that at least two students of a class of size n have the same birthday? Compute the numerical values of such probabilities for n = 23, 30, 50, and 60. Assume that the birth rates are constant throughout the year and that each year has 365 days.
At a state university in Maryland, there is hardly enough space for students to park their cars in their own lots. Jack, a student who parks in the faculty parking lot every day, noticed that none of the last 10 tickets he got was issued on a Monday or on a Friday. Is it wise for Jack to conclude
Rose has invited n friends to her birthday party. If they all attend, and each one shakes hands with everyone else at the party exactly once, what is the number of handshakes?
A box contains 7 identical balls numbered 1 through 7. Three balls are drawn, one by one, at random, and without replacement, and their numbers are recorded. What is the probability that (a) all three outcomes are odd; (b) exactly one outcome is odd?
A box contains 7 identical balls numbered 1 through 7. Three balls are drawn, one by one, at random, and with replacement, and their numbers are recorded. What is the probability that (a) all three outcomes are odd; (b) exactly one outcome is odd?
Virginia wants to give her son, Brian, 14 different baseball cards within a 7-day period. If Virginia gives Brian cards no more than once a day, in how many ways can this be done?
In tossing four fair dice, what is the probability of at least one 3?
How many outcomes are there if we throw five dice?
10. Suppose that on a certain week, only one-third of the travelers visiting Paris took a trip to the Eiffel Tower’s top, one-half visited the LouvreMuseum, and one-third took a tour of Notre Dame Cathedral. If none of the tourists visited all three sites, and for each pair of the sites,
9. In some towns of a country, the harmful substances lead and asbestos fibers find their way into drinking water. In a study, it was found that, in 13%of those towns, the drinking water supplies have neither lead nor asbestos fibers, in 32% of them the drinking water supplies have lead, and in 43%
8. Five customers enter a wireless corporate store to purchase smartphones. If the probability that at least three of them purchase an Android smartphone is 0.54, what is the probability that at most two of them buy such a phone?Hint: For 0 ≤ i ≤ 5, let Ai be the event that exactly i of these
7. For an experiment with sample space ???? S = (0, 2), for n ≥ 1, let En =1 − 1/n, 1 + 1/nand P(En) = (2n + 1)/3n. For this experiment, find the probability that the event {1} occurs. Note that this is not the experiment of choosing a point at random from the interval (0, 2), as defined in
6. Last semester, all freshman students of a college took calculus, biology, and English. If 18% of them received an A in calculus, 10% received an A in both calculus and biology, 13% received an A in both English and calculus, and 7% received an A in all three courses, what is the probability that
5. For an experiment, E and F are two events with P(E) = 0.4 and P(EcFc) = 0.35.Calculate P????EcF.
4. Suppose that, in a particular geographical area, of people aged 55 and older, 13.4%suffer from Parkinson’s disease, 11.3%suffer from Alzheimer’s disease, and 2.26%suffer from both of these completely separate neurodegenerative illnesses. What percentage of the people from this age group in
3. A device that has three components fails if at least one of its components breaks down.The device is observed at a random time. Let Ai, 1 ≤ i ≤ 3, 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

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