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probability and stochastic modeling

Fundamentals Of Probability 2nd Edition Saeed Ghahramani - Solutions

9 A fair die is thrown twice. If the second outcome is 6, what is the proba- bility that the first one is 6 as well?
8 From an ordinary deck of 52 cards, 10 cards are drawn at random. If exactly four of them are hearts, what is the probability of at least one spade being among them?
7 Roads A, B, and C are the only escape routes from a state prison. Prison records show that, of the prisoners who tried to escape, 30% used road A, 50% used road B, and 20% used road C. These records also show that 80% of those who tried to escape via A, 75% of those who try to escape via B, and
6 A bus traveling from Baltimore to New York breaks down at a random location. If the bus was seen running at Wilmington, what is the probability that the breakdown occurred after passing through Philadelphia? The distances from New York, Philadelphia, and Wilmington to Baltimore are, respectively,
5 Professor Stern has three cars. The probability that on a given day car 1 is operative is 0.95, that car 2 is operative is 0.97, and that car 3 is operative is 0.85. If Professor Stern's cars operate independently, find the probability that next Thanksgiving day (a) all three of his cars are
4 Diseases D1, D2, and D; cause symptom A with probabilities 0.5, 0.7, and 0.8, respectively. If 5% of a population have disease D, 2% have disease D, and 3.5% have disease D3, what percent of the population have symptom A? Assume that the only possible causes of symptom A are D, D, and D, and that
3 In statistical surveys where individuals are selected randomly and are asked questions, experience has shown that only 48% of those under 25 years of age, 67% between 25 and 50, and 89% above 50 will respond. A social scientist is about to send a questionnaire to a group of randomly selected
2 A polygraph operator detects innocent suspects as being guilty 3% of the time. If during a crime investigation six innocent suspects are examined by the operator, what is the probability that at least one of them is detected as guilty?
1 Two persons arrive at a train station, independently of each other, at random times between 1:00 PM. and 1:30 PM. What is the probability that one will arrive between 1:00 PM. and 1:12 PM., and the other between 1:17 PM. and 1:30 PM.?
42 (Laplace's Law of Succession) Suppose that +1 urns are numbered 0 through n, and the ith urn contains i red and n-i white balls, 0 in. An urn is selected at random, and then the balls it contains are removed one by one, at random, and with replacement. If the first m balls are all red, what is
41 Hemophilia is a hereditary disease. If a mother has it, then with probability 1/2, any of her sons independently will inherit it. Otherwise, none of the sons becomes hemophilic. Julie is the mother of two sons, and from her family's medical history it is known that, with the probability 1/4, she
40 In a contest, contestants A, B, and C are each asked, in turn, a general scientific question. If a contestant gives a wrong answer to a question, he drops out of the game. The remaining two will continue to compete until one of them drops out. The last person remaining is the winner. Suppose
39 From a population of people with unrelated birthdays, 30 people are se- lected at random. What is the probability that exactly four people of this group have the same birthday and that all the others have different birth- days (exactly 27 birthdays altogether)? Assume that the birthrates are
38 Figure 3.10 shows an electric circuit in which each of the switches located at 1, 2, 3, 4, and 5 is independently closed or open with probabilities p and 1-p, respectively. If a signal is fed to the input, what is the probability that it is transmitted to the output?
37 Suppose that n 2 missiles are fired at a target and hit it independently. If the probability that the ith missile hits it is p., i = 1, 2, ..., n, find the probability that at least two missiles will hit the target.
36 An urn contains nine red and one blue balls. A second urn contains one red and five blue balls. One ball is removed from each urn at random and without replacement, and all of the remaining balls are put into a third urn. If we draw two balls randomly from the third urn, what is the probability
35 In the experiment of rolling two fair dice successively, what is the proba- bility that a sum of 5 appears before a sum of 7? Hint: See Exercise 34.
34 Let S be the sample space of a repeatable experiment. Let A and B be mutually exclusive events of S. Prove that, in independent trials of this experiment, the event A occurs before the event B with probability P(A)/[P(A) + P(B)]. Hint: See Example 3.30; this example can be done the same way.
33 An experiment consists of first tossing a fair coin and then drawing a card randomly from an ordinary deck of 52 cards with replacement. If we perform this experiment successively, what is the probability of obtaining heads on the coin before an ace from the cards? Hint: See Example 3.30.
32 Suppose that an airplane passenger whose itinerary requires a change of airplanes in Ankara, Turkey, has a 4% chance of losing each piece of his or her luggage independently. Suppose that the probability of losing each piece of luggage in this way is 5% at Da Vinci airport in Rome, 5% at Kennedy
31 If two fair dice are tossed six times, what is the probability that the sixth sum obtained is not a repetition?
30 A fair coin is flipped indefinitely. What is the probability of (a) at least one head in the first n flips; (b) exactly k heads in the first n flips; (c) getting heads in all of the flips indefinitely?
29 A fair coin is tossed n times. Show that the events "at least two heads" and "one or two tails" are independent if n = 3, but dependent if n = 4.
28 An urn contains two red and four white balls. Balls are drawn from the urn successively, at random and with replacement. What is the probability that exactly three whites occur in the first five trials?
27 Figure 3.9 shows an electric circuit in which each of the switches located at 1, 2, 3, 4, 5, and 6 is independently closed or open with probabilities p and 1-p, respectively. If a signal is fed to the input, what is the probability that it is transmitted to the output? 3 5
26 Let (A1, A2, P. 1in. A,) be an independent set of events and P(A,) (a) What is the probability that at least one of the events A1, A2...... A occurs? (b) What is the probability that none of the events A1, A2, A occurs?
25 An event occurs at least once in four independent trials with probability 0.59. What is the probability of its occurrence in one trial?
24 From the set of all families with three children a family is selected at random. Let A be the event that "the family has children of both sexes" and B be the event that "there is at most one girl in the family." Are A and B independent? Answer the same question for families with two children and
23 If the events A and B are independent and the events B and C are inde- pendent, is it true that the events A and C are also independent? Why or why not?
22 In a certain county, 15% of patients suffering heart attacks are younger than 40, 20% are between 40 and 50, 30% are between 50 and 60, and 35% are above 60. On a certain day, 10 unrelated patients suffering heart attacks are transferred to a county hospital. If among them there is at least one
21 There are n cards in a box numbered 1 through n. We draw cards suc- cessively and at random with replacement. If the ith draw is the card numbered i, we say that a match has occurred. What is the probability of at least one match in n trials? What happens if n increases without bound?
20 A fair die is rolled six times. If on the ith roll, 1 i 6, the outcome is i, we say that a match has occurred. What is the probability that at least one match occurs?
19 Prove that if A, B, and C are independent, then A and B UC are inde- pendent. Also show that A B and C are independent.
18 In a community of M men and W women, m men and w women smoke (mM, w W). If a person is selected at random and A and B are the events that the person is a man and smokes, respectively, under what conditions are A and B independent?
17 In a tire factory, the quality control inspector examines a randomly chosen sample of 15 tires. When more than one defective tire is found, production is halted, the existing tires are recycled, and production is then resumed. The purpose of this process is to ensure that the defect rate is no
16 In his book, Probability 1, published by Harcourt Brace, 1998, Amir Aczel estimates that the probability of life for any one given star in the known universe is 0.00000000000005 independently of life for any other star. Assuming that there are 100 billion galaxies in the universe and each galaxy
15 Three missiles are fired at a target and hit it independently, with probabil- ities 0.7, 0.8, and 0.9, respectively. What is the probability that the target is hit?
14 Suppose that 55% of the customers of a shoestore buy black shoes. Find the probability that at least one of the next six customers who purchase a pair of shoes from this store will buy black shoes. Assume that these customers decide independently.
13 Show that if A and B are independent and AC B, then either P(A) = 0 or P(B) = 1.
12 Show that if an event A is independent of itself, then P(A) = 0 or 1.
11 (a) Show that if P(A) = 1, then P(AB) = P(B). (b) Prove that any event A with P(A) = 0 or P(A) = 1 is inde- pendent of every event B.
10 Find an example in which P(AB) < P(A)P(B).
9 In data communications, a message transmitted from one end is subject to various sources of distortion and may be received erroneously at the other end. Suppose that a message of 64 bits (a bit is the smallest unit of information and is either 1 or 0) is transmitted through a medium. If each bit
8 Chevalier de Mr's Paradox: In the seventeenth century in France there were two popular games, one to obtain at least one 6 in four throws of a fair die, and the other to bet on at least one double 6 in 24 throws of two fair dice. French nobleman and mathematician Chevalier de Mr argued that the
7 The Italian mathematician Giorlamo Cardano once wrote that if the odds in favor of an event are 3 to 1, then the odds in favor of the occurrence of that event in two consecutive independent experiments are 9 to 1. (He squared 3 and 1 to obtain 9 to 1.) Was Cardano correct?
6 According to a recent mortality table, the probability that a 35-year-old U.S. citizen will live to age 65 is 0.725. What is the probability that John and Jim, two 35-year-old Americans who are not relatives, both live to age 65? What is the probability that neither man lives to that age?
5 Suppose that two points are selected at random and independently from the interval (0, 1). What is the probability that the first one is less than 3/4, and the second one is greater than 1/4?
4 An urn has three red and five blue balls. Suppose that eight balls are selected at random and with replacement. What is the probability that the first three are red and the rest are blue balls?
3 A fair die is rolled twice. Let A denote the event that the sum of the outcomes is odd, and B denote the event that it lands 2 on the first toss. Are A and B independent? Why or why not?
2 In a certain country, the probability that a fighter plane returns from a mission without mishap is 49/50, independent of other missions. In a conversation, Mia concluded that any pilot who flew 49 consecutive mis- sions without mishap should be returned home before the fiftieth mission. But, on
1 Jean le Rond d'Alembert, a French mathematician, believed that in suc- cessive flips of a fair coin, after a long run of heads, a tail is more likely. Do you agree with d'Alembert on this? Explain.
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 differ- ent 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 t 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, 10 assistant professors, and 12 instructors, a committee of six is formed randomly. What is the probability that there is at least one person from each rank on the committee?10 OOHHE 34 6 30 = 1.397. To find the correct answer, use the
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 these two
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, I 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?
1 Use Stirling's formula to approximate (2) 2n 1 and 22 [(2n)!] [(4n)! (n!)] for large n.
48 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.
47 A fair coin is tossed n times. Calculate the probability of getting no successive heads. Hint: Let x; be the number of sequences of H's and T's of length i with no successive H's. Show that x satisfies x = x1+x-2, 2, where x = 1 and x = 2. The answer is x/2". Note that {x} is a Fibonacci-type
46 A 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. Let M be the
45 Using the binomial theorem, calculate the coefficient of x" in the expan- sion of (1+x)=(1+x)" (1+x)" to prove that (27)=2(7) 1-0 For a combinatorial proof of this relation, see Example 2.26.
44 We are given n (n>5) points in space, no three of which lie on the same straight line. Let 2 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 2 are parallel. From the set of the
43 In how many ways can 10 different photographs be placed in six different envelopes, no envelope empty? Hint: An easy way to do this problem is to use the following version of the inclusion-exclusion principle: Let A1, A2, A, be n subsets of a finite set with N elements. Let N(A;) be the number
42 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 Py, the probability of what we actually observed happen. (b) To estimate the number of trout in the lake, statisticians find the
41 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).
40 Evaluate the following sum: 3 2
39 By a combinatorial argument, prove that for rn and rm, n+m m m n m ("+")-()()+(7)(-1) ++ (77) (3).
38 Show that Hint: (") + ("+) +---+ ("+")=("+r+1). n+ (C)-(+)-(2)
37 Prove that (6)-(1)+(2)---+-1(2)+---+ (-1)*(") = 0.
36 Suppose that n indistinguishable balls are placed at random into n dis- tinguishable cells. What is the probability that exactly one cell remains empty?
35 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
34 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?
33 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?
32 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.
31 Prove the binomial expansion formula by induction. Hint: Use the identity () + (^)=(" +).
30 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, suppose
29 Each state of the 50 in the U.S. has two senators. What is the probability that in a random committee of 50 senators (a) Maryland is represented; (b) all states are represented?
28 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? Figure
27 Using induction, binomial expansion, and the identity (n-n)! prove the formula of multinomial expansion. n!

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