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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

Professors Davidson and Johnson from the University of Victoria in Canada gave the following problem to their students in a finite math course:An urn contains N balls of which B are black and N −B
Every month, a large lot of 2000 fuses manufactured by a certain company is shipped to a major retail store. The policy of the store is to return a lot if more than 2% of its fuses are defective.
In a community of a + b potential voters, a are pro-choice and b (b < a)are pro-life. Suppose that a vote is taken to determine the will of the majority with regard to legalizing abortion. If n (n
In 500 independent calculations a scientist has made 25 errors. If a second scientist checks seven of these calculations randomly, what is the probability that he detects two errors? Assume that the
A smoking mathematician carries two matchboxes, one in his right pocket and one in his left pocket. Whenever he wants to smoke, he selects a pocket at random and takes a match from the box in that
Two gamblers play a game in which in each play gambler A beats B with probability p, 0 < p < 1, and loses to B with probability q = 1−p. Suppose that each play results in a forfeiture of $1 for the
Sharon and Ann play a series of backgammon games until one of them wins five games. Suppose that the games are independent and the probability that Sharon wins a game is 0.58.(a) Find the probability
A father asks his sons to cut their backyard lawn. Since he does not specify which of the three sons is to do the job, each boy tosses a coin to determine the odd person, who must then cut the lawn.
From an ordinary deck of 52 cards we draw cards at random, with replacement, and successively until an ace is drawn. What is the probability that at least 10 draws are needed?
34. 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
33. 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
31. LetN(t), t ≥ 0be 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
30. LetN(t), t ≥ 0be a Poisson process.What is the probability of (a) an even number of events in (t, t + α); (b) an odd number of events in (t, t + α)?
29. 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
26. 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
25. 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
24. Let X be a Poisson random variable with parameter λ. LetFind the probability mass function of Y . 0 if X is zero or even Y If X is odd.
23. 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
22. A company is located in a region that is prone to dust storms. The company has purchased an insurance policy that pays $200,000 for each dust storm in a year except the first one. If the number
20. 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?
18. 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.
17. A pharmaceutical company has developed an Ebola vaccine, which is 98.5% of the time effective. If 5000 people have received the vaccine, and sooner or later all of them will be exposed to the
15. 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. 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 IndependenceDay (a) using the binomial
4. By Example 2.24, 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?
Let N(t) be the number of earthquakes that occur at or prior to time t worldwide.Suppose thatN(t) : t ≥ 0is a Poisson process with rate λ and the probability that the magnitude of an earthquake
A fisherman catches fish at a Poisson rate of two per hour from a large lake with lots of fish. Yesterday, he went fishing at 10:00 A.M. and caught just one fish by 10:30 and a total of three by
Suppose that earthquakes occur in a certain region of California, in accordance with a Poisson process, at a rate of seven per year.(a) What is the probability of no earthquakes in one year?(b) What
Suppose that children are born at a Poisson rate of five per day in a certain hospital. What is the probability that (a) at least two babies are born during the next six hours;(b) no babies are born
Consider a class of size k of unrelated students. Assuming that the birth rates are constant throughout the year and each year has 365 days, with probability p = 1/365, each pair of the students has
Suppose that n raisins are thoroughly mixed in dough. If we bake k raisin cookies of equal sizes from this mixture, what is the probability that a given cookie contains at least one raisin?
Every week the average number of wrong-number phone calls received by a certain mail-order house is seven. What is the probability that they will receive (a) two wrong calls tomorrow; (b) at least
2. Suppose that a prescription drug causes 7 side effects each with probability of 0.05 independently of the others.What is the probability that a patient taking this drug experiences at most 2 side
1. 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
38. 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
37. An urn contains n balls whose colors, red or blue, are equally probable.For example, the probability that all of the balls are red is (1/2)n.If in drawing k balls from the urn, successively
36. (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
35. The post office 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
34. 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 35 of Section 2.4 for a definition of a straight
33. In Exercise 32, 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
32. 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
31. 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
30. Let X be a binomial random variable with the parameters (n, p). Prove that n E(X)=2 x=1 n p (1 - p)-np - np + np.
29. 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
28. In a community, a persons are pro-choice, 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
27. 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
26. 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,”
25. A woman and her husband want to have a 95% chance for at least one boy and at least one girl.What is the minimumnumber of children that they should plan to have?Assume that the events that a
24. Vincent is a patient with the life threatening blood cancer leukemia, and he is in need of a bone marrow transplant. He asks n people whether or not they are willing to donate bone marrow to him
23. 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
22. 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
21. Let X be the number of sixes obtained when a balanced die is tossed five times. Find the probability mass function of Y = (X − 3)2.
20. What are the expected value and variance of the number of full house hands in n poker hands? A poker hand consists of five randomly selected cards from an ordinary deck of 52 cards. It is a full
19. 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.
18. Dr. Willis is teaching two sections of Calculus I, each with 25 students. Suppose that each student will earn a passing grade, independently of other students, with probability of 0.82.What is
17. 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?
16. Even though there are some differences between the taste of Max Cola and the taste of Golden Cola, most people cannot tell the difference. Pedro claims that in the absence of brand information,
15. Suppose that each day the price of a stock moves up 1/8th of a point with probability 1/3 and moves down 1/8th of a point with probability 2/3. If the price fluctuations from one day to another
14. On average, how many times should Ernie play poker in order to be dealt a straight flush(royal flush included)? (See Exercise 35 of Section 2.4 for definitions of a royal and a straight flush.)
13. Let X be a binomial random variable with parameters (n, p) and probability mass function p(x). Prove that if (n + 1)p is an integer, then p(x) is maximum at two different points. Find both of
12. From the interval (0, 1), five points are selected at random and independently. What is the probability that (a) at least two of them are less than 1/3; (b) the first decimal point of exactly two
11. If two fair dice are rolled 10 times, what is the probability of at least one 6 (on either die) in exactly five of these 10 rolls?
10. Suppose that the Internal Revenue Service will audit 20%of income tax returns reporting an annual gross income of over $80,000.What is the probability that of 15 such returns, at most four will
9. Only 60% of certain kinds of seeds germinate when planted under normal conditions.Suppose that four such seeds are planted, and X denotes the number of those that will germinate. Find the
8. Let X be a discrete random variable with probability mass function p given byFind the probability mass functions of the random variables Y = |X| and Z = X2. X -1 0 1 p(x) 2/9 4/9 3/9 other values 0
7. A manufacturer of nails claims that only 3% of its nails are defective. A random sample of 24 nails is selected, and it is found that two of them are defective. Is it fair to reject the
6. A box contains 30 balls numbered 1 through 30. Suppose that five balls are drawn at random, one at a time, with replacement. What is the probability that the numbers of two of them are prime?
5. In a state where license plates contain six digits, what is the probability that the license number of a randomly selected car has two 9’s? Assume that each digit of the license number is
4. LetX be a Bernoulli random variable with parameter p, 0 < p < 1. Find the probability mass function of Y = 1 − X, E(Y ), and Var(Y ).
Two proofreaders, Ruby and Myra, read a book independently and found r and m misprints, respectively. Suppose that the probability that a misprint is noticed by Ruby is p and the probability that it
A town of 100,000 inhabitants is exposed to a contagious disease. If the probability that a person becomes infected is 0.04, what is the expected number of people who become infected?
We will now examine an elementary example of a random walk. In Chapter 12, we will revisit this concept and its applications.Suppose that a particle is at 0 on the integer number line and suppose
Let p be the probability that a randomly chosen person is pro-life, and let X be the number of persons pro-life in a random sample of size n. Suppose that, in a particular random sample of n persons,
A Canadian pharmaceutical company has developed three types of Ebola virus vaccines. Suppose that vaccines I, II, and III are effective 92%, 88%, and 96% of the time, respectively. Assume that a
Machuca’s favorite bridge hands are those with at least two aces. Determine the number of times he should play in order to have a chance of 90% or more to get at least two favorite hands? Recall
Suppose that jury members decide independently and that each with probability p (0 < p < 1) makes the correct decision. If the decision of the majority is final, which is preferable: a three-person
In a small town, out of 12 accidents that occurred in June 1986, four happened on Friday the 13th. Is this a good reason for a superstitious person to argue that Friday the 13th is inauspicious?
In a county hospital 10 babies, of whom six were boys, were born last Thursday.What is the probability that the first six births were all boys? Assume that the events that a child born is a girl or
A restaurant serves 8 entr´ees of fish, 12 of beef, and 10 of poultry. If customers select from these entr´ees randomly, what is the probability that two of the next four customers order fish
10. Under what conditions on α, β, γ, and θ is the following a distribution function? a t
9. At a department store, summer polo shirts are sold from April 1st until the end of September. Suppose that for each polo shirt the department store sells the net profit is $15.00, and for each
8. At a university, professors are allowed to check out as many books as they wish from the library. Let X be the number of books checked out by a random professor visiting the library. Suppose that
7. The season finale of a Belavian reality television series, “Belavia’s Got Talent,” was held in an oval auditorium which has 25 rows of seats. The first row has 10 seats, and each succeeding
6. Let x be a nonnegative real number. By [x], we mean the greatest integer less than or equal to x. For example, [1.2] = 1, [5.8] = 5, and [0.8] = 0. In a box, there are n identical balls numbered
5. To help defray hospitalization expenses due to accidents, the insurance company, Joseph Accident and Health, offers a hospital indemnity plan in which an injured customer is paid a lump sum daily
4. Let X be the number of bagels purchased by a random customer from the Longmeadow Bagel Shop. Let F be the distribution function of X, and suppose that F(0) = 4/33, F(1) = 16/33, F(2) = 26/33, and
3. In the front yard of a doctor’s clinic, there are exactly six parking spaces next to each other in a row. Suppose that at a time when all six parking spaces are occupied, two of the cars leave
2. A large company has w women and m men in retirement age. If a random set of n, n ≤ w+m, of these employees decides to retire next year, what is the probability mass function of the number of
1. From a leap year calendar, a month is selected at random. Let X be the number of days in that month. Find E(X) and σX. If the month is selected from a non-leap year, on average, how many days
11. If the motor of a certain commercial grade dishwasher fails within the first year of purchase, the insurance company pays $2500 for its replacement or repair. Thereafter, each year, the
10. From the set of families with three children a family is selected at random, and the number of its boys is denoted by the random variable X. Find the probability mass function and the
9. Experience shows that X, the number of customers entering a post office, during anyperiod of length t, is a random variable the probability mass function of which is of the form(a) Determine the
8. The fasting blood-glucose levels of 30 children are as follows.Let X be the fasting blood-glucose level of a child chosen randomly from this group.Find the distribution function of X. 58 62 80 58
7. Let X be the amount (in fluid ounces) of soft drink in a randomly chosen bottle from company A, and Y be the amount of soft drink in a randomly chosen bottle from company B. A study has shown that
6. The annual amount of rainfall (in centimeters) in a certain area is a random variable with the distribution functionWhat is the probability that next year it will rain (a) at least 6 centimeters;
4. An electronic system fails if both of its components fail. Let X be the time (in hours)until the system fails. Experience has shown thatWhat is the probability that the system lasts at least 200
2. The mean and standard deviation in midterm tests of a probability course are 72 and 12, respectively. These quantities for final tests are 68 and 15. What final grade is comparable to Velma’s 82
1. Mr. Norton owns two appliance stores. In store 1 the number of TV sets sold by a salesperson is, on average, 13 per week with a standard deviation of five. In store 2 the number of TV sets sold by
2. Let X be a random variable with E(X) = 3 and E(X − 3)(4 − X)= −15. Find Var(−3X + 8).
1. Two fair dice are tossed. Let X be the sum of the outcomes. Find Var(X) and σX.

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