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probability and stochastic modeling
Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions
24. Experience shows that 75% of certain kinds of seeds germinate when planted under normal conditions. Determine the minimum number of seeds to be planted so that the chance of at least five of them germinating is more than 90%.
23. A bowl contains w white and b blue chips. Chips are drawn at random and with replacement until a blue chip is drawn. What is the probability that (a) exactly n draws are required; (b) at least n draws are required?
22. The probability that a child of a certain family inherits a certain disease is 0.23 independently of other children inheriting the disease. If the family has five children and the disease is detected in one child, what is the probability that exactly two more children have the disease as well?
21. From the set {x : 0 ≤ x ≤ 1} numbers are selected at random and independently and rounded to three decimal places. What is the probability that 0.345 is obtained (a) for the first time on the 1000th selections; (b) for the third time on the 3000th selections?
20. An insurance company claims that only 70% of the drivers regularly use seat belts. In a statistical survey, it was found that out of 20 randomly selected drivers, 12 regularly used seat belts. Is this sufficient evidence to conclude that the insurance company’s claim is false?
19. Passengers are making reservations for a particular flight on a small commuter plane 24 hours a day at a Poisson rate of 3 reservations per 8 hours. If 24 seats are available for the flight, what is the probability that by the end of the second day all the plane seats are reserved?
18. Suppose that 6% of the claims received by an insurance company are for damages from vandalism. What is the probability that at least three of the 20 claims received on a certain day are for vandalism damages?
17. A certain type of seed when planted fails to germinate with probability 0.06. If 40 of such seeds are planted, what is the probability that at least 36 of them germinate?
16. A bowl contains 10 red and six blue chips. What is the expected number of blue chips among five randomly selected chips?
15. Of the 28 professors in a certain department, 18 drive foreign and 10 drive domestic cars. If five of these professors are selected at random, what is the probability that at least three of them drive foreign cars?
14. Past experience shows that 30% of the customers entering Harry’s Clothing Store will make a purchase. Of the customers who make a purchase, 85% use credit cards. Let X be the number of the next six customers who enter the store, make a purchase, and use a credit card. Find the probability
13. In data communication, one method for error control makes the receiver send a positive acknowledgment for every message received with no detected error and a negative acknowledgment for all the others. Suppose that (i) acknowledgments are transmitted error free, (ii) a negative acknowledgment
12. What is the probability that the sixth toss of a die is the first 6?
11. A fair coin is tossed successively until a head occurs. If N is the number of tosses required, what are the expected value and the variance of N?
10. The policy of the quality control division of a certain corporation is to reject a shipment if more than 5% of its items are defective. A shipment of 500 items is received, 30 of them are randomly tested, and two have been found defective. Should that shipment be rejected?
9. Suppose that a certain bank returns bad checks at a Poisson rate of three per day. What is the probability that this bank returns at most four bad checks during the next two days?
8. Suppose that 10 trains arrive independently at a station every day, each at a random time between 10:00 A.M. and 11:00 A.M.. What is the expected number and the variance of those that arrive between 10:15 A.M. and 10:28 A.M.?
7. From a panel of prospective jurors, 12 are selected at random. If there are 200 men and 160 women on the panel, what is the probability that more than half of the jury selected are women?
6. In a community, the chance of a set of triplets is 1 in 1000 births. Determine the probability that the second set of triplets in this community occurs before the 2000th birth.
5. A doctor has five patients with migraine headaches. He prescribes for all five a drug that relieves the headaches of 82% of such patients. What is the probability that the medicine will not relieve the headaches of two of these patients?
4. A university has n students, 70% of whom will finish an aptitude test in less than 25 minutes. If 12 students are selected at random, what is the probability that at most two of them will not finish in 25 minutes?
3. A restaurant serves 8 fish entrées, 12 beef, and 10 poultry. If customers select from these entrées randomly, what is the expected number of fish entrées ordered by the next four customers?
The time between the arrival of two consecutive customers at a post office is 3 minutes, on average. Assuming that customers arrive in accordance with a Poisson process, find the probability that tomorrow during the lunch hour (between noon and 12:30 P.M.) fewer than seven customers arrive.
2. The time between the arrival of two consecutive customers at a postoffice is 3 minutes, on average. Assuming that customers arrive in accordance with a Poisson process, find the probability that tomorrow during the lunch hour (between noon and 12:30 P.M.) fewer than seven customers arrive.
1. Of police academy applicants, only 25% will pass all the examinations. Suppose that 12 successful candidates are needed. What is the probability that, by examining 20 candidates, the academy finds all of the 12 persons needed?
29. Suppose that, from a box containing D defective and N − D nondefective items, n (≤ D) are drawn one by one, at random and without replacement. (a) Find the probability that the kth item drawn is defective. (b) If the (k − 1)st item drawn is defective, what is the probability that the kth
28. To estimate the number of trout in a lake, we caught 50 trout, tagged and returned them. Later we caught 50 trout and found that four of them were tagged. From this experiment estimate n, the total number of trout in the lake. Hint: Let pn be the probability of four tagged trout among the 50
27. Adam rolls a well-balanced die until he gets a 6. Andrew rolls the same die until he rolls an odd number. What is the probability that Andrew rolls the die more than Adam does?
26. In the Banach matchbox problem, Example 5.22, find the probability that the box which is emptied first is not the one that is first found empty.
25. In the Banach matchbox problem, Example 5.22, find the probability that when the first matchbox is emptied (not found empty) there are exactly m matches in the other box.
24. Twelve hundred eggs, of which 200 are rotten, are distributed randomly in 100 cartons, each containing a dozen eggs. These cartons are then sold to a restaurant. How many cartons should we expect the chef of the restaurant to open before finding one without rotten eggs?
23. In data communication, messages are usually combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either 1 or 0. Suppose that the length of a character (in bits) is a geometric random variable with parameter p. Furthermore,
22. Let X be a geometric random variable with parameter p, and n and m be nonnegative integers (a) For what values of n is P (X = n) maximum? (b) What is the probability that X is even? (c) Show that the geometric is the only distribution on the positive integers with the memoryless property: P (X
21. On average, how many independent games of poker are required until a preassigned player is dealt a straight? (See Exercise 20 of Section 2.4 for a definition of a straight. The cards have distinct consecutive values that are not of the same suit: for example, 3 of hearts, 4 of hearts, 5 of
20. A fair coin is flipped repeatedly. What is the probability that the fifth tail occurs before the tenth head?
19. A computer network consists of several stations connected by various media (usually cables). There are certain instances when no message is being transmitted. At such “suitable instances,” each station will send a message with probability p, independently of the other stations. However,
18. A vending machine contains cans of grapefruit juice that cost 75 cents each, but it is not working properly. The probability that it accepts a coin is 10%. Angela has a quarter and five dimes. Determine the probability that she should try the coins at least 50 times before she gets a can of
17. The probability is p that a message sent over a communication channel is garbled. If the message is sent repeatedly until it is transmitted reliably, and if each time it takes 2 minutes to process it, what is the probability that the transmission of a message takes more than t minutes?
16. In an annual charity drive, 35% of a population of 560 make contributions. If, in a statistical survey, 15 people are selected at random and without replacement, what is the probability that at least two persons have contributed?
15. Florence is moving and wishes to sell her package of 100 computer diskettes. Unknown to her, 10 of those diskettes are defective. Sharon will purchase them if a random sample of 10 contains no more than one defective disk. What is the probability that she buys them?
14. Suppose that 15% of the population of a town are senior citizens. Let X be the number of nonsenior citizens who enter a mall before the tenth senior citizen arrives. Find the probability mass function of X. Assume that each customer who enters the mall is a random person from the entire
13. Solve the Banach matchbox problem (Example 5.22) for the case where the matchbox in the right pocket contains M matches, the matchbox in his left pocket contains N matches, and m ≤ min(M, N ).
12. On average, how many games of bridge are necessary before a player is dealt three aces? A bridge hand is 13 randomly selected cards from an ordinary deck of 52 cards
11. The digits after the decimal point of a random number between 0 and 1 are numbers selected at random, with replacement, independently, and successively from the set {0, 1, . . . , 9}. In a random number from (0, 1), on the average, how many digits are there before the fifth 3?
10. In rural Ireland, a century ago, the students had to form a line. The student at the front of the line would be asked to spell a word. If he spelled it correctly, he was allowed to sit down. If not, he received a whack on the hand with a switch and was sent to the end of the line. Suppose that
9. Suppose that independent Bernoulli trials with parameter p are performed successively. Let N be the number of trials needed to get x successes, and X be the number of successes in the first n trials. Show that P (N = n) = x n P (X = x). Remark: By this relation, in coin tossing, for example, we
8. The probability is p that a randomly chosen light bulb is defective. We screw a bulb into a lamp and switch on the current. If the bulb works, we stop; otherwise, we try another and continue until a good bulb is found. What is the probability that at least n bulbs are required?
7. A store has 50 light bulbs available for sale. Of these, five are defective. A customer buys eight light bulbs randomly from this store. What is the probability that he finds exactly one defective light bulb among them?
6. A certain basketball player makes a foul shot with probability 0.45. What is the probability that (a) his first basket occurs on the sixth shot; (b) his first and second baskets occur on his fourth and eighth shots, respectively?
4. The probability is p that Marty hits target M when he fires at it. The probability is q that Alvie hits target A when he fires at it. Marty and Alvie fire one shot each at their targets. If both of them hit their targets, they stop; otherwise, they will continue.(a) What is the probability that
3. An absentminded professor does not remember which of his 12 keys will open his office door. If he tries them at random and with replacement: (a) On average, how many keys should he try before his door opens? (b) What is the probability that he opens his office door after only three tries?
2. Define a sample space for the experiment of performing independent Bernoulli trials until the second success occurs.
1. Define a sample space for the experiment that, from a box containing two defective and five nondefective items, three items are drawn at random and without replacement.
27. Let X be a Poisson random variable with parameter λ. Show that the maximum of P (X = i) occurs at [λ], where [λ] is the greatest integer less than or equal to λ. Hint: Let p be the probability mass function of X. Prove that p(i) = λ i p(i − 1). Use this to find the values of i at which p
26. In a forest, the number of trees that grow in a region of area R has a Poisson distribution with mean λR, where λ is a given positive number. (a) Find the probability that the distance from a certain tree to the nearest tree is more thand. (b) Find the probability that the distance from a
25. Customers arrive at a grocery store at a Poisson rate of one per minute. If 2/3 of the customers are female and 1/3 are male, what is the probability that 15 females enter the store between 10:30 and 10:45? Hint: Use the result of Exercise 24.
24. Let $ N (t), t ≥ 0 % be a Poisson process with rate λ. Suppose that N (t) is the total number of two types of events that have occurred in [0, t]. Let N1(t) and N2(t) be the total number of events of type 1 and events of type 2 that have occurred in [0, t], respectively. If events of type 1
22. Balls numbered 1,2, ... , and n are randomly placed into cells numbered 1, 2, ... , and n. Therefore, for 1 ≤ i ≤ n and 1 ≤ j ≤ n, the probability that ball i is in cell j is 1/n. For each i, 1 ≤ i ≤ n, if ball i is in cell i, we say that a match has occurred at cell i. (a) What is
21. Suppose that in Maryland, on a certain day, N lottery tickets are sold and M win. To have a probability of at least α of winning on that day, approximately how many tickets should be purchased?
20. According to the United States Postal Service, http:www.usps.gov, May 15, 1998, Dogs have caused problems for letter carriers for so long that the situation has become a cliché. In 1983, more than 7,000 letter carriers were bitten by dogs. ... However, the 2,795 letter carriers who were
19. Suppose that, on the Richter scale, earthquakes of magnitude 5.5 or higher have probability 0.015 of damaging certain types of bridges. Suppose that such intense earthquakes occur at a Poisson rate of 1.5 per ten years. If a bridge of this type is constructed to last at least 60 years, what is
18. On a certain two-lane north-south highway, there is a T junction. Cars arrive at the junction according to a Poisson process, on the average four per minute. For cars to turn left onto the side street, the highway is widened by the addition of a left-turn lane that is long enough to accommodate
17. A wire manufacturing company has inspectors to examine the wire for fractures as it comes out of a machine. The number of fractures is distributed in accordance with a Poisson process, having one fracture on the average for every 60 meters of wire. One day an inspector has to take an emergency
16. Customers arrive at a bookstore at a Poisson rate of six per hour. Given that the store opens at 9:30 A.M., what is the probability that exactly one customer arrives by 10:00 A.M. and 10 customers by noon?
15. Accidents occur at an intersection at a Poisson rate of three per day. What is the probability that during January there are exactly three days (not necessarily consecutive) without any accidents?
14. In a certain town, crimes occur at a Poisson rate of five per month. What is the probability of having exactly two months (not necessarily consecutive) with no crimes during the next year?
13. Suppose that, for a telephone subscriber, the number of wrong numbers is Poisson, at a rate of λ = 1 per week. A certain subscriber has not received any wrong numbers from Sunday through Friday. What is the probability that he receives no wrong numbers on Saturday either?
12. Suppose that in Japan earthquakes occur at a Poisson rate of three per week. What is the probability that the next earthquake occurs after two weeks?
11. Suppose that on a summer evening, shooting stars are observed at a Poisson rate of one every 12 minutes. What is the probability that three shooting stars are observed in 30 minutes?
10. The department of mathematics of a state university has 26 faculty members. For i = 0, 1, 2, 3, find pi, the probability that i of them were born on Independence Day (a) using the binomial distribution; (b) using the Poisson distribution. Assume that the birth rates are constant throughout the
9. The children in a small town all own slingshots. In a recent contest, 4% of them were such poor shots that they did not hit the target even once in 100 shots. If the number of times a randomly selected child has hit the target is approximately a Poisson random variable, determine the percentage
8. Suppose that n raisins have been carefully mixed with a batch of dough. If we bake k (k > 4) raisin buns of equal size from this mixture, what is the probability that two out of four randomly selected buns contain no raisins? Hint: Note that, by Example 5.13, the number of raisins in a given bun
7. Suppose that X is a Poisson random variable with P (X = 1) = P (X = 3). Find P (X = 5).
6. On average, there are three misprints in every 10 pages of a particular book. If every chapter of the book contains 35 pages, what is the probability that Chapters 1 and 5 have 10 misprints each?
5. On a random day, the number of vacant rooms of a big hotel in New York City is 35, on average. What is the probability that next Saturday this hotel has at least 30 vacant rooms?
4. By Example 2.21, the probability that a poker hand is a full house is 0.0014. What is the probability that in 500 random poker hands there are at least two full houses?
3. Suppose that 2.5% of the population of a border town are illegal immigrants. Find the probability that, in a theater of this town with 80 random viewers, there are at least two illegal immigrants.
2. Suppose that 3% of the families in a large city have an annual income of over $60,000. What is the probability that, of 60 random families, at most three have an annual income of over $60,000?
1. Jim buys 60 lottery tickets every week. If only 5% of the lottery tickets win, what is the probability that he wins next week?
34. While Rose always tells the truth, four of her friends, Albert, Brenda, Charles, and Donna, tell the truth randomly only in one out of three instances, independent of each other. Albert makes a statement. Brenda tells Charles that Albert’s statement is the truth. Charles tells Donna that
33. An urn contains n balls whose colors, red or blue, are equally probable. 4 For example, the probability that all of the balls are red is (1/2)n. 5 If in drawing k balls from the urn, successively with replacement and randomly, no red balls appear, what is the probability that the urn contains
32. (a) What is the probability of an even number of successes in n independent Bernoulli trials? Hint: Let rn be the probability of an even number of successes in n Bernoulli trials. By conditioning on the first trial and using the law of total probability (Theorem 3.3), show that for n ≥ 1, rn
31. The postoffice of a certain small town has only one clerk to wait on customers. The probability that a customer will be served in any given minute is 0.6, regardless of the time that the customer has already taken. The probability of a new customer arriving is 0.45, regardless of the number of
30. (Genetics) In a population of n diploid organisms with alternate dominant allele A and recessive allelea, those inheriting aa will not survive. Suppose that, in the population, the number of AA individuals is α, and the number of Aa individuals is n − α. Suppose that, as a result of random
29. How many games of poker occur until a preassigned player is dealt at least one straight flush with probability of at least 3/4? (See Exercise 20 of Section 2.4 for a definition of a straight flush.)
28. In Exercise 27, suppose that a message consisting of six characters is transmitted. If each character consists of seven bits, what is the probability that the message is erroneously received, but none of the errors is detected by the parity check?
27. The simplest error detection scheme used in data communication is parity checking. Usually messages sent consist of characters, each character consisting of a number of bits (a bit is the smallest unit of information and is either 1 or 0). In parity checking, a 1 or 0 is appended to the end of
26. Suppose that an aircraft engine will fail in flight with probability 1 − p independently of the plane’s other engines. Also suppose that a plane can complete the journey successfully if at least half of its engines do not fail. (a) Is it true that a four-engine plane is always preferable
25. Let X be a binomial random variable with the parameters (n, p). Prove that E(X2 ) = .n x=1 x2 ; n x < px (1 − p)n−x = n2 p2 − np2 + np.
24. A game often played in carnivals and gambling houses is called chuck-a-luck, where a player bets on any number 1 through 6. Then three fair dice are tossed. If one, two, or all three land the same number as the player’s, then he or she receives one, two, or three times the original stake plus
23. In a community, a persons are for abortion, b (b a −b) are undecided. Suppose that there will be a vote to determine the will of the majority with regard to legalizing abortion. If by then all of the undecided persons make up their minds, what is the probability that those against abortion
22. Consider the following problem posed by Michael Khoury, U.S. Math Olympiad Team Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for the Communication of Mathematics, February 1999. Bob is teaching a class with n students. There are n desks
21. A computer network consists of several stations connected by various media (usually cables). There are certain instances when no message is being transmitted. At such “suitable instances,” each station will send a message with probability p independently of the other stations. However, if
20. A woman and her husband want to have a 95% chance for at least one boy and at least one girl. What is the minimum number of children that they should plan to have? Assume that the events that a child is a girl and a boy are equiprobable and independent of the gender of other children born in
19. Edward’s experience shows that 7% of the parcels he mails will not reach their destination. He has bought two books for $20 each and wants to mail them to his brother. If he sends them in one parcel, the postage is $5.20, but if he sends them in separate parcels, the postage is $3.30 for each
18. A certain rare blood type can be found in only 0.05% of people. If the population of a randomly selected group is 3000, what is the probability that at least two persons in the group have this rare blood type?
17. What is the probability that at least two of the six members of a family are not born in the fall? Assume that all seasons have the same probability of containing the birthday of a person selected randomly.
16. What are the expected value and variance of the number of full house hands in n poker hands? (See Exercise 20 of Section 2.4 for a definition of a full house.)
15. A certain basketball player makes a foul shot with probability 0.45. Determine for what value of k the probability of k baskets in 10 shots is maximum, and find this maximum probability.
14. From the set {x : 0 ≤ x ≤ 1}, 100 independent numbers are selected at random and rounded to three decimal places. What is the probability that at least one of them is 0.345?
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