Fundamentals Of Probability 2nd Edition Saeed Ghahramani - Solutions

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28. 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?
27. At a party, 15 married couples are seated at random at a round table. What is the probability that all men are sitting next to their wives? Suppose that of these married couples, five husbands and their wives are older than 50 and the remaining husbands and wives are all younger than 50. What
26. Suppose that 20 sticks are broken, each into one long and one short part. By pairing them randomly, the 40 parts are then used to make 20 new sticks. (a) What is the probability that long parts are all paired with short ones? (b) What is the probability that the new sticks are exactly the same
25. 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?
24. Let S and T be finite sets with n and m elements, respectively. (a) How many functions f : S → T can be defined? (b) If m ≥ 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?
23. A club of 136 members is in the process of choosing a president, a vice president, a secretary, and a treasurer. If two of the members are not on speaking terms and do not serve together, in how many ways can these four people be chosen?
22. 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 11, only
21. If n balls are randomly placed into n cells, what is the probability that each cell will be occupied?
20. Five boys and five girls sit at random in a row. What is the probability that the boys are together and the girls are together?
19. One of the five elevators in a building starts with seven passengers and stops at nine floors. Assuming that it is equally likely that a passenger gets off at any of these nine floors, find the probability that at least two of these passengers will get off at the same floor.
18. 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?
17. 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.
16. 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?
15. At a party, n men and m women put their drinks on a table and go out on the floor to dance. When they return, none of them recognizes his or her drink, so everyone takes a drink at random. What is the probability that each man selects his own drink?
14. In drawing nine cards with replacement from an ordinary deck of 52 cards, what is the probability of three aces of spades, three queens of hearts, and three kings of clubs?
13. A fair die is tossed eight times. What is the probability of exactly two 3’s, three 1’s, and three 6’s?
12. (a) Find the number of distinguishable permutations of the lettersMI SSI SSIP P I . (b) In how many of these permutations P’s are together? (c) In how many I ’s are together? (d) In how many P’s are together, and I ’s are together? (e) In a random order of the letters MI SSI SSIP P I ,
11. Six fair dice are tossed. What is the probability that at least two of them show the same face?
10. A dancing contest has 11 competitors, of whom three are Americans, two are Mexicans, three are Russians, and three are Italians. If the contest result lists only the nationality of the dancers, how many outcomes are possible?
9. In a conference, Dr. Richman’s lecture is related to Dr. Chollet’s and should not precede it. If there are six more speakers, how many schedules could be arranged? Warning: Dr. Richman’s lecture is not necessarily scheduled right after Dr. Chollet’s lecture.
8. At various yard sales, a woman has acquired five forks, of which no two are alike. The same applies to her four knives and seven spoons. In how many different ways can three place settings be chosen if each place setting consists of exactly one fork, one knife, and one spoon? Assume that the
7. In an exhibition, 20 cars of the same style that are distinguishable only by their colors, are to be parked in a row, all facing a certain window. If four of the cars are blue, three are black, five are yellow, and eight are white, how many choices are there?
6. Professor Haste is somewhat familiar with six languages. To translate texts from one language into another directly, how many one-way dictionaries does he need?
5. Let A be the set of all sequences of 0’s, 1’s, and 2’s of length 12. (a) How many elements are there in A? (b) How many elements of A have exactly six 0’s and six 1’s? (c) How many elements of A have exactly three 0’s, four 1’s, and five 2’s?
4. Robert has eight guests, two of whom are Jim and John. If the guests will arrive in a random order, what is the probability that John will not arrive right after Jim?
3. How many different messages can be sent by five dashes and three dots?
2. How many permutations of the set {a,b, c,d, e} begin with a and end with c?
1. In the popular TV show Who Wants to Be a Millionaire, contestants are asked to sort four items in accordance with some norm: for example, landmarks in geographical order, movies in the order of date of release, singers in the order of date of birth. What is the probability that a contestant can
19. The coefficient 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.
18. A bus traveling from Baltimore to New York has 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.
17. 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
16. 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 21, Section 1.4, for the definition of relatively prime numbers.
15. 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?
14. 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
13. 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
12. Let A and B be two events. The event (A−B)∪(B −A) is called the symmetric difference of A and B and is denoted by A )B . Clearly, A )B is the event that exactly one of the two events A and B occurs. Show that P (A ) B) = P (A) + P (B) − 2P (AB).
11. 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?
10. 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
9. 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?
8. 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.
7. Let A, B, and C be three events. Prove that P (A ∪ B ∪ C) ≤ P (A) + P (B) + P (C).
6. 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?
4. 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).
3. 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
2. 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
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.
14. 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 of
13. Show that the result of Exercise 8 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.
12. Let{A1, A2, A3,...} be a sequence of events. Prove that if the series/∞ n=1 P (An) converges, then P * )∞ m=1 (∞ n=m An , = 0. This is called the Borel-Cantelli lemma. It says that if /∞ n=1 P (An) < ∞, the probability that infinitely many of the An’s occur is 0. Hint: Let Bm = (∞
11. Suppose that a point is randomly selected from the interval (0, 1). Using the definition in Section 1.7, show that all numerals are equally likely to appear as the nth digit of the decimal representation of the selected point.
9. (a) Prove that )∞ n=1(1/2 − 1/2n, 1/2 + 1/2n) = {1/2}. (b) Using part (a), show that the probability of selecting 1/2 in a random selection of a point from (0, 1) is 0.
8. Let A1, A2, . . . , An be n events. Show that if P (A1) = P (A2) = · · · = P (An) = 1, then P (A1A2 ··· An) = 1.
6. Suppose that a point is randomly selected from the interval (0, 1). Using the definition in Section 1.7, show that all numerals are equally likely to appear as the first digit of the decimal representation of the selected point. B
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?
32. (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
31. Let P be a probability defined on a sample space S. For events A of S define Q(A) = 4 P (A)52 and R(A) = P (A)/2.Is Q a probability on S? Is R a probability on S? Why or why not?
30. 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
29. Let A1, A2, A3,... be a sequence of events of an experiment. Prove that P *-∞ n=1 An , ≥ 1 −.∞ n=1 P (Ac n). Hint: Use Boole’s inequality, discussed in Exercise 28.
28. Let A1, A2, A3, ... be a sequence of events of a sample space. Prove that P *+∞ n=1 An , ≤ .∞ n=1 P (An). This is called Boole’s inequality.
27. 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?
26. 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.
25. 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 ?
24. 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.
23. 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
22. 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?
21. 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.
20. 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.
19. 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
18. Among 33 students in a class, 17 of them earned A’s on the midterm exam, 14 earned A’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?
17. 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 j th row is pij / , where n i=1 /n j=1 pij = 1. In terms of pij ’s, find the probability that the ball hits the j th horizontal strip.
16. 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?
15. 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).
14. 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).
13. 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
12. 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.
11. 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?
10. Let A and B be two events. Prove that P (AB) ≥ P (A) + P (B) − 1.
9. 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
8. 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
7. 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
6. 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?
5. 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?
4. Suppose that the probability that a driver is a male, and has at least one motor vehicles 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
3. 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
2. 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?
1. Gottfried Wilhelm Leibniz (1646–1716), the German mathematician, philosopher, statesman, and one of the supreme intellects of the seventeenth century, believed that in a throw of a pair of fair dice, the probability of obtaining the sum 11 is equal to that of obtaining the sum 12. Do you agree
2 Using Monte Carlo procedure, write a program to estimate (3)-0(1) = dx. 2 Then run your program for n = 10,000 and compare the answer with that obtained from Table 1 of the Appendix.
1 Three concentric circles of radii 2, 3, and 4 are the boundaries of regions that form a circular target. If a person fires a shot at random at the target, use simulation to approximate the probability that the bullet lands in the middle region. Hint: To simulate a random point located inside a
17 An ordinary deck of 52 cards is divided randomly into 26 pairs. Using Chebyshev's inequality, find an upper bound for the probability that, at most, 10 pairs consist of a black and a red card. Hint: For i 1, 2, ..., 26, let X = 1, if the ith red card is paired with a black card, and X, = 0,
16 Show that for a nonnegative random variable X with mean , we have that Vn, nP(X)1.
15 A fair die is rolled 20 times. What is the approximate probability that the sum of the outcomes is between 65 and 75?
14 A randomly selected book from Vernon's library is X centimeters thick. where X N(3, 1). Vernon has an empty shelf 87 centimeters long. What is the probability that he can fit 31 randomly selected books on it?
13 In a multiple-choice test with false answers receiving negative scores, the mean of the grades of the students is 0 and its standard deviation is 15. Find an upper bound for the probability that a student's grade is at least 45.
12 Each time that Ed charges an expense to his credit card, he omits the cents and records only the dollar value. If this month he has charged his credit card 20 times, using Chebyshev's inequality, find an upper bound on the probability that his record shows at least $15 less than the actual
11 A psychologist wants to estimate , the mean IQ of the students of a university. To do so, she takes a sample of size n of the students and measures their IQ's. Then she finds the average of these numbers. If she believes that the IQ's of these students are independent random variables with
10 Let X and Y be independent Poisson random variables with parameters and , respectively. (a) Show that P(X+Y=n) P(X = i)P(Y=ni). 1-0 (b) Use part (a) to prove that X + Y is a Poisson random variable with parameter + M.
9 Find the moment-generating function of a random variable X with Laplace density function defined by f(x) = -x -x
8 Suppose that in a community the distributions of heights of men and women (in centimeters) are N(173, 40) and N(160, 20), respectively. Calculate the probability that the average height of 10 randomly selected men is at least 5 centimeters larger than the average height of six randomly selected

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Fundamentals Of Probability 2nd Edition Saeed Ghahramani - Solutions

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