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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

17. In a study it was discovered that 25%of the paintings of a certain gallery are not original.A collector in 15% of the cases makes a mistake in judging if a painting is authentic or a copy. If she
16. Urns I, II, and III contain three pennies and four dimes, two pennies and five dimes, three pennies and one dime, respectively. One coin is selected at random from each urn.If two of the three
15. At a grocery store, an absent-minded, honest professor,Dexter, hands the playful cashier, Hans, a dollar bill. Hans puts the bill in the cash register drawer and gives Dexter $2.75 in change.
14. A certain cancer is found in one person in 5000. If a person does have the disease, in 92% of the cases the diagnostic procedure will show that he or she actually has it. If a person does not
13. An actuary has calculated that, for the age group 16-25, the probability is 0.08 that a driver whose car is insured by her company gets involved in a car accident within a year. For the age
12. Based on an insurance company’s evaluation, with probability 0.85, Whitney is a safe driver, and with probability 0.15 she is not. Suppose that a safe driver avoids at-fault accidents in the
11. Suppose that currently it is a bull market, and in the stock market, share prices are rising. Furthermore, suppose that the Nasdaq Composite index closes at higher points 86% of the trading days.
10. A stack of cards consists of six red and five blue cards. A second stack of cards consists of nine red cards. A stack is selected at random and three of its cards are drawn. If all of them are
9. There are three dice in a small box. The first die is unbiased; the second one is loaded, and, when tossed, the probability of obtaining 6 is 3/8, and the probability of obtaining each of the
8. Suppose that 5% of the men and 2% of the women working for a corporation make over$120,000 a year. If 30% of the employees of the corporation are women, what percent of those who make over
7. In a trial, the judge is 65% sure that Susan has committed a crime. Julie and Robert are two witnesses who know whether Susan is innocent or guilty. However, Robert is Susan’s friend and will
6. When traveling from Springfield, Massachusetts, to Baltimore, Maryland, 30% of the time Charles takes I-91 south and then I-95 south, and 70% of the time he takes I-91 south, then I-84 west
5. When Professor Wagoner teaches calculus, he only grades 25% of the students’ exam papers, randomly selected. The remaining papers are graded by his teaching assistants(TA’s). Suppose that 78%
4. A judge is 65% sure that a suspect has committed a crime. During the course of the trial, a witness convinces the judge that there is an 85% chance that the criminal is lefthanded.If 23% of the
3. Suppose that 20%of the e-mails Derek receives are spam. Suppose that 12%of the spam e-mails and 0.05%of the non-spam e-mails are concerning the e-mail storage in Derek’s e-mail account. Derek
2. On a multiple-choice exam with four choices for each question, a student either knows the answer to a question or marks it at random. If the probability that he or she knows the answers is 2/3,
1. In transmitting dot and dash signals, a communication system changes 1/4 of the dots to dashes and 1/3 of the dashes to dots. If 40% of the signals transmitted are dots and 60% are dashes, what is
A box contains seven red and 13 blue balls. Two balls are selected at random and are discardedwithout their colors being seen. If a third ball is drawn randomly and observed to be red, what is the
On the basis of reconnaissance reports, Colonel Smith decides that the probability of an enemy attack against the left is 0.20, against the center is 0.50, and against the right is 0.30. A flurry of
During a double homicide murder trial, based on circumstantial evidence alone, the jury becomes 15% certain that a suspect is guilty. DNA samples recovered from the murder scene are then compared
In a study conducted three years ago, 82% of the people in a randomly selected sample were found to have “good” financial credit ratings, while the remaining 18%were found to have “bad”
3. Seventy percent of all students of a college participate in a study abroad program. If 60% of the students of this college are male and 55% of the male students attend a study abroad program, what
2. There are three dice in a small box. The first die is unbiased; the second one is loaded, and, when tossed, the probability of obtaining 6 is 3/8, and the probability of obtaining each of the
1. At a kindergarten, there is a box full of balls. Each child draws a ball at random to play with and is not allowed to return it to the box to draw another one. When it is Natalie’s turn to draw
27. Suppose that three numbers are selected one by one, at random and without replacement from the set of numbers {1, 2, 3, . . . , n}. What is the probability that the third number falls between the
26. From families with three children, a child is selected at random and found to be a girl.What is the probability that she has an older sister? Assume that in a three-child family all sex
25. A box contains 18 tennis balls, of which eight are new. Suppose that three balls are selected randomly, played with, and after play are returned to the box. If another three balls are selected
24. Suppose that 10 good and three dead batteries are mixed up. Jack tests them one by one, at random and without replacement. But before testing the fifth battery he realizes that he does not
23. For n ≥ 1, let E1, E2, . . . , En be events of a sample space. Find an n-element partition of Sn i=1 Ei. That is, find a set of mutually exclusive events {F1, F2, . . . , Fn} that satisfies Sn
22. Suppose that the probability that a new seed planted in a specific farm germinates is equal to the proportion of all planted seeds that germinated in that farm previously.Suppose that the first
21. Suppose that 40% of the students on a campus, who are married to students on the same campus, are female. Moreover, suppose that 30% of those who are married, but not to students at this campus,
20. Let B be an event of a sample space S with P(B) > 0. For a subset A of S, define Q(A) = P(A | B). By Theorem 3.1 we know that Q is a probability function. For E and F, events of SP(FB) > 0,
19. Suppose that there exist N families on the earth and that the maximum number of children a family has isc. Let αj????0 ≤ j ≤ c, Pc j=0 αj = 1be the fraction of families with j children.
18. A number is selected at random from the set {1, 2, . . . , 20}. Then a second number is selected randomly between 1 and the first number selected. What is the probability that the second number
17. An actuary has discovered that, in her company, 65% of those who have only income protection insurance and 80%of those who have only legal expense insurance will renew their policies next year.
16. A child gets lost in the Disneyland at the Epcot Center in Florida. The father of the child believes that the probability of his being lost in the east wing of the center is 0.75 and in the west
15. In a town, 7/9th of the men and 3/5th of the women are married. In that town, what fraction of the adults are married? Assume that all married adults are the residents of the town.
14. Suppose that five coins, of which exactly three are gold, are distributed among five persons, one each, at random, and one by one. Are the chances of getting a gold coin equal for all
13. At a gas station, 85% of the customers use 87 octane gasoline, 5% use 91 octane, and 10% use 93 octane. Suppose that, 70%, 85%, and 95% of 87 octane users, 91 octane users, and 93 octane users
12. Solve the following problem, from the “Ask Marilyn” column of Parade Magazine, October 29, 2000.I recently returned from a trip to China, where the government is so concerned about population
11. When traveling from Springfield, Massachusetts, to Baltimore, Maryland, 30% of the time Charles takes I-91 south and then I-95 south, and 70% of the time he takes I-91 south, then I-84 west
10. A factory produces its entire output with three machines.Machines I, II, and III produce 50%, 30%, and 20% of the output, but 4%, 2%, and 4% of their outputs are defective, respectively.What
9. A person has six guns. The probability of hitting a target when these guns are properly aimed and fired is 0.6, 0.5, 0.7, 0.9, 0.7, and 0.8, respectively.What is the probability of hitting a
8. Suppose that 37% of a community are at least 45 years old. If 80% of the time a person who is 45 or older tells the truth, and 65% of the time a person below 45 tells the truth, what is the
7. Of the patients in a hospital, 20% of those with, and 35% of those without myocardial infarction have had strokes. If 40% of the patients have had myocardial infarction, what percent of the
6. Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade?
5. One of the cards of an ordinary deck of 52 cards is lost. What is the probability that a random card drawn from this deck is a spade?
4. Jim has three cars of different models: A, B, and C. The probabilities that models A, B, and C use over 3 gallons of gasoline from Jim’s house to his work are 0.25, 0.32, and 0.53, respectively.
3. A random number is selected from the interval (0, 1]. Which one of the following is a partition of the sample space of this experiment?Which one is not? (a) {(0, 1/2), (1/2,1]}, (c) {(0,2/5],
An urn contains 10 white and 12 red chips. Two chips are drawn at random and, without looking at their colors, are discarded. What is the probability that a third chip drawn is red?
Suppose that the only parasite living in an aquatic habitat is a single-celled organism, which after a second, with equal probabilities, either splits into two organisms, remains as is, or dies.
Suppose that 80%of the seniors, 70%of the juniors, 50%of the sophomores, and 30% of the freshmen of a college use the library of their campus frequently. If 30% of all students are freshmen, 25% are
Two gamblers play the game of “heads or tails,” in which each time a fair coin lands heads up player A wins $1 from B, and each time it lands tails up, player B wins $1 from A. Suppose that
In a trial, the judge is 65% sure that Susan has committed a crime. Julie and Robert are two witnesses who know whether Susan is innocent or guilty. However, Robert is Susan’s friend and will lie
An insurance company rents 35% of the cars for its customers from agency I and 65% from agency II. If 8% of the cars of agency I and 5% of the cars of agency II break down during the rental periods,
2. On a given day, the first item produced by a manufacturer is defective with probability p and non-defective with probability 1 − p. However, whether or not an item produced afterward is
1. An urn contains 6 blue and 4 red balls. Three balls are drawn at random and without replacement. What is the probability that the balls drawn are alternatively of different colors?
14. In a series of games, the winning number of the nth game, n = 1, 2, 3, . . . , is a number selected at random from the set of integers {1, 2, . . . , n + 2}. Don bets on 1 in each game and says
13. From an ordinary deck of 52 cards, cards are drawn one by one, at random and without replacement.What is the probability that the fourth heart is drawn on the tenth draw?Hint: Let F denote the
12. Cards are drawn at random from an ordinary deck of 52, one by one and without replacement.What is the probability that no heart is drawn before the ace of spades is drawn?
11. Suppose that 75% of all people with credit records improve their credit ratings within three years. Suppose that 18% of the population at large have poor credit records, and of those only 30%
10. In the card game, bridge, played with an ordinary deck of 52 cards, all cards are dealt among four players, 13 each, randomly.What is the probability that each player gets one ace?Hint: Let A1 be
9. The law school of a university admits all applicants who have at least a 3.5 undergraduate GPA and a Law School Admissions Test (LSAT) score of 154 or higher. Suppose that of all students with a
8. An urn contains five white and three red chips. Each time we draw a chip, we look at its color. If it is red, we replace it along with two new red chips, and if it is white, we replace it along
7. In a lottery scratch-off game, each ticket has 10 coated circles in the middle and one coated rectangle in the lower left corner. Underneath the coats of 4 of the circles, there is a dollar sign,
6. There are five boys and six girls in a class. For an oral exam, their teacher calls them one by one and randomly. (a) What is the probability that the boys and the girls alternate?(b) What is the
5. An actuary works for an auto insurance company whose customers all have insured at least one car. She discovers that of all the customers who insure more than one car, 70% of them insure at least
3. In a game of cards, two cards of the same color and denomination form a pair. For example, 8 of hearts and 8 of diamonds is one pair, king of spades and king of clubs is another. If six cards are
2. There are 14marketing firms hiring new graduates.Kate randomly found the recruitment ads of six of these firms and sent them her resume. If three of these marketing firms are in Maryland, what is
Suppose that five good and two defective fuses have been mixed up. To find the defective ones, we test them one by one, at random and without replacement. What is the probability that we find both of
A consulting firm is awarded 43% of the contracts it bids on. Suppose that Nordulf works for a division of the firm that gets to do 15% of the projects contracted for.If Nordulf directs 35% of the
Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-by-one, at random and without replacement.What is the probability that we are
4. For events E and F, suppose that P(EF) = 0.23, P????E ∪ F= 0.67, and P(E | F) = 0.46. Find P(F | E).
3. The theaters of a town are showing seven comedies and nine dramas. Marlon has seen five of the movies. If the first three movies he has seen are dramas, what is the probability that the last two
2. For a certain loaded die, the probabilities of the possible outcomes are given by the following table.If the die is tossed and the outcome is an odd number, what is the probability that it is 1?
1. In a bridge game, played with a normal deck of 52 cards, hearts is the designated trump suit. If the first 4 cards a player is dealt are ace, 5, 9, and the king, all of hearts, what is the
25. From the set of all families with two children, a family is selected at random and is found to have a girl called Mary. We want to know the probability that both children of the family are girls.
24. In an international school, 60 students, of whom 15 are Korean, 20 are French, eight are Greek, and the rest are Chinese, are divided randomly into four classes of 15 each. If there is a total of
23. There are three types of animals in a laboratory: 15 type I, 13 type II, and 12 type III.Animals of type I react to a particular stimulus in 5 seconds, animals of types II and III react to the
22. A big urn contains 1000 red chips, numbered 1 through 1000, and 1750 blue chips, numbered 1 through 1750. A chip is removed at random, and its number is found to be divisible by 3. What is the
21. A retired person chooses randomly one of the six parks of his town everyday and goes there for hiking.We are told that he was seen in one of these parks, Oregon Ridge, once during the last 10
20. A number is selected at random from the set {1, 2, . . . , 10,000} and is observed to be odd.What is the probability that it is (a) divisible by 3; (b) divisible by neither 3 nor 5?
19. An actuary studying the insurance preferences of homeowners in a region that is prone to earthquakes, hurricanes, and floods has discovered that, for each of these perils, the probability is 0.2
18. Adam and three of his friends are playing bridge. (a) If, holding a certain hand, Adam announces that he has a king, what is the probability that he has at least one more king?(b) If, for some
16. Prove that if P(E | F) ≥ P(G | F) and P(E | Fc) ≥ P(G | Fc), then P(E) ≥ P(G).
15. Prove Theorem 3.1.
14. Prove that if P(A) = a and P(B) =b, then P(A | B) ≥ (a + b − 1)/b.
13. Show that if P(A) = 1, then P(B | A) = P(B).
12. In a study of the records of 831 women who died in 2016, an actuary observed that 185 of them died of cancer. Furthermore, she discovered that the mothers of 257 of the 831 women suffered from
11. From families with three children, a family is selected at random and found to have a boy. What is the probability that the boy has (a) an older brother and a younger sister;(b) an older brother;
10. From 100 cards numbered 00, 01, . . . , 99, one card is drawn. Suppose that α and βare the sum and the product, respectively, of the digits of the card selected. Calculate P????{α = i | β =
9. In a small lake, it is estimated that there are approximately 105 fish, of which 40 are trout and 65 are carp. A fisherman caught eight fish; what is the probability that exactly two of them are
7. A spinner is mounted on a wheel of unit circumference (radius 1/2π). Arcs A,B, and C of lengths 1/3, 1/2, and 1/6, respectively, are marked off on the wheel’s perimeter(see Figure 3.1). The
5. A bus arrives at a station every day at a random time between 1:00 P.M. and 1:30 P.M. A person arrives at this station at 1:00 and waits for the bus. If at 1:15 the bus has not yet arrived, what
4. Suppose that two fair dice have been tossed and the total of their top faces is found to be divisible by 5. What is the probability that both of them have landed 5?
3. In a technical college all students are required to take calculus and physics. Statistics show that 32% of the students of this college get A’s in calculus, and 20% of them get A’s in both
2. Suppose that 41% of Americans have blood type A, and 4% have blood type AB. If in the blood of a randomly selected American soldier the A antigen is found, what is the probability that his blood
On a TV game show, there are three curtains. Behind two of the curtains there is nothing, but behind the third there is a prize that the playermight win. The probability that the prize is behind a
A farmer decides to test four fertilizers for his soybean fields. He buys 32 bags of fertilizers, eight bags from each kind, and tries them randomly on 32 plots, eight plots from each of fields A, B,
A child mixes 10 good and three dead batteries. To find the dead batteries, his father tests them one-by-one and without replacement. If the first four batteries tested are all good, what is the

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