New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
probability and stochastic modeling

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

5. 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
4. Sixty-eight minutes through Morituri, the 1965 Marlon Brando movie, a group of underground anti-Nazis wanting to escape from a Nazi ship to a nearby island state that the chances of their succeeding are 15-to-1. Eighty-two minutes through the movie, the same group refers to this estimate stating
3. 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
In a community, 32% of the population are male smokers; 27% are female smokers.What percentage of the population of this community smoke?
Dr. Grossman, an internist, has 520 patients, of which (i) 230 are hypertensive,(ii) 185 are diabetic, (iii) 35 are hypochondriac and diabetic, (iv) 25 are all three, (v) 150 are none, (vi) 140 are only hypertensive, and finally, (vii) 15 are hypertensive and hypochondriac but not diabetic. Find
Suppose that 25% of the population of a city read newspaper A, 20% read newspaper B, 13% read C, 10% read both A and B, 8% read both A and C, 5% read B and C, and 4% read all three. If a person from this city is selected at random, what is the probability that he or she does not read any of these
A number is chosen at random from the set of integers {1, 2, . . . , 1000}.What is the probability that it is divisible by 3 or 5 (i.e., either 3 or 5 or both)?
Suppose that in a community of 400 adults, 300 bike or swim or do both, 160 swim, and 120 swim and bike. What is the probability that an adult, selected at random from this community, bikes?
A number is selected at random from the set {1, 2, . . . ,N}. What is the probability that the number is divisible by k, 1 ≤ k ≤ N?
A number is selected at random from the set of integers1, 2, . . . , 1000.What is the probability that the number is divisible by 3?
An elevator with two passengers stops at the second, third, and fourth floors.If it is equally likely that a passenger gets off at any of the three floors, what is the probability that the passengers get off at different floors?
4. Consider the system shown by the diagram of Figure 1.3, consisting of 7 components denoted by 1, 2, . . . , 7. Suppose that each component is either functioning or not functioning, with no other capabilities. Suppose that the system itself also has two performance capabilities, functioning and
3. Find the simplest possible expression for the event (E ∪ F)(F ∪ G)(EG ∪ Fc).
2. In a large hospital, there are 100 patients scheduled to have heart bypass surgery. Let Ei, 1 ≤ i ≤ 100, be the event that the ith patient lives through the postoperative period of the surgery. In terms of Ei’s, (a) describe the event that all patients survive the critical postoperative
1. Jody, Ann, Bill, and Karl line up in a random order to get their photo taken. Describe the event that, on the line, males and females alternate.
34. Let {A1,A2,A3, . . .} be a sequence of events of a sample space S. Find a sequence{B1,B2,B3, . . .} of mutually exclusive events such that for all n ≥ 1, Sn S i=1 Ai = n i=1 Bi.
33. Let {A1,A2,A3, . . .} be a sequence of events. Find an expression for the event that infinitely many of the Ai’s occur.
32. In a mathematics department of 31 voting faculty members, there are three candidates running for the chair position. The voting procedure adopted by the department is approval voting, in which the eligible voters can vote for as many candidates as they wish. The candidate with themaximumvotes
31. Define a sample space for the experiment of putting in a random order seven different books on a shelf. If three of these seven books are a three-volume dictionary, describe the event that these volumes stand in increasing order side by side (i.e., volume I precedes volume II and volume II
27. A point is chosen at random from the interval (−1, 1). Let E1 be the event that it falls in the interval (−1/2, 1/2), E2 be the event that it falls in the interval (−1/4, 1/4) and, in general, for 1 ≤ i < ∞, Ei be the event that the point is in the interval (−1/2i, 1/2i).Find S∞
26. In an experiment, cards are drawn, one by one, at random and successively from an ordinary deck of 52 cards. Let An be the event that no face card or ace appears on the first n − 1 drawings, and the nth draw is an ace. In terms of An’s, find an expression for the event that an ace appears
25. For the experiment of flipping a coin until a heads occurs, (a) describe the sample space;(b) describe the event that it takes an odd number of flips until a heads occurs.
24. Travis picks up darts to shoot toward an 18′′-diameter dartboard aiming at the bullseye on the board. Describe a sample space for the point at which a dart hits the board.
23. Let E, F, and G be three events. Determine which of the following statements are correct and which are incorrect. Justify your answers.(a) (E − EF) ∪ F = E ∪ F.(b) FcG ∪ EcG = G(F ∪ E)c.(c) (E ∪ F)cG = EcFcG.(d) EF ∪ EG ∪ FG ⊂ E ∪ F ∪ G.
22. Prove that the event B is impossible if and only if for every event A, A = (B ∩ Ac) ∪ (Bc ∩ A).
21. A device that has n, n > 1, components fails to operate if at least one of its components breaks down. The device is observed at a random time. Let Ai, 1 < i ≤ n, denote the event that the ith component is operative at the random time. In terms of Ai’s, describe the event that the device is
20. At a certain university, every year eight to 12 professors are granted University Merit Awards. This year among the nominated faculty are Drs. Jones, Smith, and Brown. Let A, B, and C denote the events, respectively, that these professors will be given awards.In terms of A, B, and C, find an
19. Find the simplest possible expression for the following events.(a) (E ∪ F)(F ∪ G).(b) (E ∪ F)(Ec ∪ F)(E ∪ Fc).
18. An insurance company sells a joint life insurance policy to Alexia and her husband, Roy.Define a sample space for the death or survival of this couple in five years. What is the event that at that time only one of them lives?
17. For a saw blademanufacturer’s products, the global demand, per month, for band saws is between 30 and 36 thousands; for reciprocating saws, it is between 28 and 33 thousands;and for hole saws, it is between 300 and 600 thousands. Define a sample space for the demands for these three types of
16. A psychologist, who is interested in human complexion, in a study of hues and shades, shows her subjects three pieces of wood colored almond, lemon, and flax, respectively.She then asks them to identify their favorite colors. Define a sample space for the answer given by a random subject.
15. A limousine that carries passengers from an airport to three different hotels just left the airport with two passengers. Describe the sample space of the stops and the event that both of the passengers get off at the same hotel.
14. Let E, F, and G be three events; explain the meaning of the relations E ∪ F ∪ G = G and EFG = G.
13. When flipping a coin more than once, what experiment has a sample space defined by the following?S = {TT, HTT, THTT, HHTT, HHHTT, HTHTT, THHTT, . . . }.
12. A device that has two components fails if at least one of its components breaks down.The device is observed at a random time. Let Ai, 1 ≤ i ≤ 2, 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
11. A telephone call from a certain person is received some time between 7:00 A.M. and 9:10 A.M. every day. Define a sample space for this phenomenon, and describe the event that the call arrives within 15 minutes of the hour.
10. Define a sample space for the experiment of drawing two coins from a purse that contains two quarters, three nickels, one dime, and four pennies. For the same experiment describe the following events:(a) drawing 26 cents;(b) drawing more than 9 but less than 25 cents;(c) drawing 29 cents.
9. Two dice are rolled. Let E be the event that the sum of the outcomes is odd and F be the event of at least one 1. Interpret the events EF, EcF, and EcFc.
8. Define a sample space for the experiment of putting three different books on a shelf in random order. If two of these three books are a two-volume dictionary, describe the event that these volumes stand in increasing order side-by-side (i.e., volume I precedes volume II).
7. Define a sample space for the experiment of choosing a number from the interval (0, 20).Describe the event that such a number is an integer.
6. A box contains three red and five blue balls. Define a sample space for the experiment of recording the colors of three balls that are drawn from the box, one by one, with replacement.
5. A deck of six cards consists of three black cards numbered 1, 2, 3, and three red cards numbered 1, 2, 3. First, Vann draws a card at random and without replacement. Then Paul draws a card at random and without replacement from the remaining cards. Let A be the event that Paul’s card has a
4. In the experiment of tossing two dice, what do the following events represent?E =(1, 3), (2, 6), (3, 1), (6, 2)and F =(1, 5), (2, 4), (3, 3), (4, 2), (5, 1).
3. In the experiment of flipping a coin three times, what does the event E = {THH, HTH, HHT, HHH}represent?
2. Last month, an insurance company sold 57 life insurance policies. Define a sample space for the number of claims that the company will receive from the beneficiaries of this group within the next 15 years. What is the event that the company receives at least 3 but no more than 8 claims?
1. From the letters of the wordMISSISSIPPI, a letter is chosen at random.What is a sample space for this experiment?What is the event that the outcome is a vowel?
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
2. Every day, a major ball manufacturer produces at least 800, but no more than 1300 of each of its five products: baseballs, tennis balls, softballs, basketballs, and soccer balls.Define a sample space for the production levels of these five types of balls that this manufacturer manufactures on a
1. To determine who pays for dinner, Crispin, Allison, and Terry each flip an unbiased coin.The one whose flip has a different face up will pay. If all of the flips land on the same face, they start all over again. (a) What is the probability that Crispin ends up paying for dinner; (b) what is the
35. The coefficients 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.
34. A bus traveling from Baltimore to New York 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.
33. 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
32. A point is selected randomly from the interval (0, 2). For n ≥ 2, let En be the event that it is in the interval (0, n√3 ). Determine the event T∞ n=2 En.
31. 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 34, Section 1.4, for the definition of relatively prime numbers.
30. 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?
29. 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
28. Mildred, a commuter student studying at Western New England University, reported to her advisor that there are three traffic lights that she needs to pass while driving from home to school. She said that, based on her experience, 10% of the time all of the three traffic lights are green, 35% of
27. 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 a
26. Let S = {ω1, ω2, ω3, . . .} be the sample space of an experiment. Suppose that P????{ω1}= 1/8 and, for a constant k, 0 < k < 1, P????{ωi+1}= k · P????{ωi}for i ≥ 1. Find P????{ωi}, for i > 1.
25. Let A and B be two events. The event (A − B) ∪ (B − A) is called the symmetric difference ofAandB and is denoted byAB. Clearly, AB is the event that exactly one of the two events A and B occurs. Show that P(AB) = P(A) + P(B) − 2P(AB).
24. 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?
23. 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
22. Five customers enter a wireless corporate store to buy smartphones. If the probability that at least two of them purchase an Android smartphone is 0.6 and the probability that all of them buy non-Android smartphones is 0.17, what is the probability that exactly one Android smartphone is
21. Anthony, Bob, and Carl, three American race car drivers, will compete in a professional Trans-Am road race. Past records show that the probability that one of these former champions wins is 10/13. If Anthony is twice as likely to win as Bob, and Bob is three times more likely than Carl to win
20. Suppose that P????E ∪ F= 0.75 and P????E ∪ Fc= 0.85. Find P(E).
19. Suppose that, in a temperate coniferous forest, 60% of randomly selected quarter-acre plots have cedar trees, 45% have cypress trees, 30% have redwoods, 40% have both cedar and cypress, 25% have cedar and redwoods, 20% have cypress and redwoods, and 80% of such plots have at least one of these
18. 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?
17. 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.
16. Let A, B, and C be three events. Prove that P(A ∪ B ∪ C) ≤ P(A) + P(B) + P(C).
15. 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?
14. Let A and B be two events of an experiment with P(A) = 1/3 and P(B) = 1/4. Find the maximum value for P????A ∪ B.
13. In a midwest town, 80%of households have cable TV, 60%have an internet subscription, and 90% have at least one of these.What percentage of the households of this town have both cable TV and an internet subscription?
12. A coin is tossed until, for the first time, the same result appears twice in succession.Define a sample space for this experiment.
11. 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).
10. For a saw blademanufacturer’s products, the global demand, per month, for band saws is between 30 and 36 thousands; for reciprocating saws, it is between 28 and 33 thousands;for hole saws, it is between 300 and 600 thousands; and for hacksaws, it is between 500 and 650 thousands. Define a
9. Kayla has two cars, an Audi and a Jeep. At a given time, whether these cars are operative or totaled, due to accidents, is of concern to her insurance company. Define a sample space for the driving conditions of these cars at a random time. What is the event that at least one of the two cars is
8. A department store accepts only its own credit card or an American Express card.Customers not carrying one of these two cards must pay with cash. If 47% of the customers of this store carry American Express, 32% carry the store’s credit card, and 12%carry both, what percentage of the store
7. 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
6. Aiden just bought a stock for $320. Define a sample space for the price of this stock in two years. Define the event that he makes money selling this stock at that time.
5. In a tutoring center, there are three computers that can be up or down at any given time.For 1 ≤ i ≤ 3, let Ei be the event that computer i is up at a random time. In terms of Ei’s, describe the event that at least two computers are up.
4. 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
3. From a phone book, a phone number is selected at random. (a)What is the event that the last digit is an odd number? (b) What is the event that the last digit is divisible by 3?
2. Two dice are rolled. What is the event that the outcomes are consecutive?
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.
2. For the experiment of choosing a point at random from the interval [0, 1], let En = Applying the Continuity of Probability Function to En’s, show that P(1/3 is selected) = 0. 1 1 2 + .3 n+2 3 n+2] |, n 1. n
1. Suppose that for events A and B, P(AB) = 0. Does this imply that A and B are mutually exclusive?Why or why not?
16. 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
15. Show that the result of Exercise 11 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.
14. Let {A1,A2,A3, . . .} be a sequence of events. Prove that if the series P∞ n=1 P(An)converges, then PT∞m=1 S∞ n=m An= 0. This is called the Borel–Cantelli lemma.It says that if P∞ n=1 P(An)
13. Suppose that a point is randomly selected from the interval (0, 1). Using Definition 1.2, show that all numerals are equally likely to appear as the nth digit of the decimal representation of the selected point.
12. A point is selected at random from the interval (0, 1). What is the probability that it is rational?What is the probability that it is irrational?
11. Let A1,A2, . . . ,An be n events. Show that if P(A1) = P(A2) = · · · = P(An) = 1, then P(A1A2 · · ·An) = 1.

Showing 1700 - 1800 of 6914

First
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved