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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

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. In a metropolis, consecutive traffic lights are coordinated so that a driver who finds the first light green will find the second light also green with probability 0.75. Similarly, a driver who
5. Suppose that before death, an individual in a population of living organisms gives birth to 0, 1, or 2 new individuals, independently of other organisms with probabilities 1/4, 1/2, and 1/4,
4. Tasha has two dice, one is unbiased, but the other one is loaded so that the probability of 6 is twice the probability of any of the other five faces, which are all equally likely.Tasha picks one
3. In Helsinki, Finland, Hunter is the fourth in line at a station to board a city tour minibus having 16 passenger seats. A minibus with all empty passenger seats arrives at the station.Suppose that
2. Suppose that a system of five components is functional if at least one of the components A1, A2, and A3 and both components B1 and B2 are operable. Suppose that a component is operable,
1. Currently, negotiations on complicated border disputes are going on between Chernarus and Carpathia. Suppose that a summit between the foreign ministers of the two countries will resolve all the
26. A student at a certain university will pass the oral Ph.D. qualifying examination if at least two of the three examiners pass her or him. Past experience shows that (a) 15% of the students who
25. Solve the following problem, asked ofMarilyn Vos Savant in the “Ask Marilyn” column of Parade Magazine, August 9, 1992.Three of us couples are going to Lava Hot Springs next weekend. We’re
24. A child is lost at 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 wing 0.25.The security
23. A fair coin is tossed. If the outcome is heads, a red hat is placed on Lorna’s head. If it is tails, a blue hat is placed on her head. Lorna cannot see the hat. She is asked to guess the color
22. 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
21. Urn I contains 25 white and 20 black balls. Urn II contains 15 white and 10 black balls.An urn is selected at random and one of its balls is drawn randomly and observed to be black and then
20. Suppose that the Dow-Jones Industrial Average (DJIA) rises 52% of trading days and falls 48% of those days. Elmer is a stock market expert, and his forecasts will turn out to be true 68% of the
19. An experiment consists of first tossing an unbiased coin and then rolling a fair die. If we perform this experiment successively, what is the probability of obtaining a heads on the coin before a
18. Six fair dice are tossed independently. Find the probability that the number of 1’s minus the number of 2’s will be 3.
17. There are 49 unbiased dice and one loaded die in a basket. For the loaded die, when tossed the probability of obtaining 6 is 3/8, and the probability of obtaining each of the other five faces is
16. Urns I and II contain three pennies and four dimes, and two pennies and five dimes, respectively. One coin is selected at random from each urn. If exactly one of them is a dime, what is the
15. Suppose that 10 dice are thrown and we are told that among them at least one has landed 6. What is the probability that there are two or more sixes?
14. A fair die is thrown twice. If the second outcome is 6, what is the probability that the first one is 6 as well?
13. 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?
12. 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
11. Stacy and George are playing the heads or tails game with a fair coin. The coin is flipped repeatedly until either the fifth heads or the fifth tails appears. If the fifth heads occurs first,
10. A bus traveling from Baltimore to New York breaks down at a random location. If the bus was seen running atWilmington, what is the probability that the breakdown occurred after passing through
9. 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
8. Diseases D1, D2, and D3 cause symptom A with probabilities 0.5, 0.7, and 0.8, respectively.If 5% of a population have disease D1, 2% have disease D2, and 3.5% have diseaseD3, what percent of the
7. Every morning, Galya flips a fair coin to decide whether she wants to take the bus to school or she wants to drive there. No matter what her means of transport, if she leaves her house at or
6. Let n > 1 be an integer. Suppose that a fair coin is tossed independently and repeatedly.Find the probability of tails on the first toss or heads on the nth toss.
5. 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
4. Suppose that 5% of men and 0.25% of women are color blind. In Belavia, only 42%of the persons 65 and older are male. What is the probability that a randomly selected person from this age group is
3. 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
2. From the students of a college that does not offer graduate programs, a student is selected at random. Let E1, E2, E3, and E4 denote the events that the student is a freshman, sophomore, junior,
1. Two persons arrive at a train station, independently of each other, at random times between 1:00 P.M. and 1:30 P.M.What is the probability that one will arrive between 1:00 P.M.and 1:12 P.M., and
4. Suppose that 48% of Dr. Darabi’s patients visit his office for a dental cleaning, 18% visit his office for orthodontic work, 12% for an extraction, 10% for a root canal, and the rest for other
3. In a ball bearing manufacturing process, after an item is manufactured, two inspectors will inspect it for defects. Suppose that each inspector finds a defect, independently of the other
2. A specific type of missile fired at a target hits it with probability 0.6. Find the minimum number of such missiles to be fired to have a probability of at least 0.95 of hitting the target.
1. Suppose that the Dow-Jones Industrial Average (DJIA) rises 52% of trading days and falls 48% of those days. What is the probability that, in the next 5 trading days, the DJIA does not rise on any
50. (Laplace’s Law of Succession) Suppose that n + 1 urns are numbered 0 through n, and the ith urn contains i red and n − i white balls, 0 ≤ i ≤ n. An urn is selected at random, and then the
49. 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
48. Figure 3.11 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 asignal is fed
47. 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
46. From a population of people with unrelated birthdays, 30 people are selected at random.What is the probability that exactly four people of this group have the same birthday and that all the
45. 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
44. (The Game of Craps) In the game of craps, the player rolls two unbiased dice. If the sum of the outcomes is 2, 3, or 12, she loses. If it is 7 or 11, she wins. However, if the sum is one of the
43. 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
42. If two fair dice are tossed six times, what is the probability that the sixth sum obtained is not a repetition?
41. 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
40. 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.
39. 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
38. Cards are drawn at random and with replacement from an ordinary deck of 52 cards, successively and indefinitely. Using Theorem 1.8, show that the probability that the ace of hearts never occurs
37. Figure 3.10 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
36. From the set of all families with two children, a family is selected at random and is found to have a girl who was born in January. What is the probability that the other child of the family is a
35. Let {A1,A2, . . . ,An} be an independent set of events and P(Ai) = pi, 1 ≤ i ≤ n.(a) What is the probability that at least one of the events A1, A2, . . . , An occurs?(b) What is the
34. An event occurs at least once in four independent trials with probability 0.59. What is the probability of its occurrence in one trial?
33. 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
32. If the events A and B are independent and the events B and C are independent, is it true that the events A and C are also independent?Why or why not?
31. When a loaded die is rolled, the outcome is 6 with probability p. Find the probability that in successive rolls of the die, all outcomes are 6’s.
30. 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
29. There are n cards in a box numbered 1 through n. We draw cards successively and at random with replacement. If the ith draw is the card numbered i, we say that a match has occurred. (a) What is
28. 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?
27. Prove that if A, B, and C are independent, then A and B ∪ C are independent. Also show that A − B and C are independent.
26. In a community ofM men andW women,mmen and w women smoke, wherem ≤ M and w ≤ W. If a person is selected at randomand A and B are the events that the person is a man and smokes, respectively,
25. 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
24. In the experiment of rolling two fair dice successively, what is the probability that a sum of 5 appears before a sum of 7?Hint: See Example 3.35.
23. 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,
22. In his book, Probability 1, published by Harcourt Brace and Company, 1998, Amir Aczel estimates that the probability of life for any one given star in the known universe is 0.000,000,000,000,05
21. In the ball bearingmanufacturing process, for each itembeingmade, two types of defects will occur, independently of each other, with probabilities 0.03 and 0.05, respectively.What is the
20. Three missiles are fired at a target and hit it independently, with probabilities 0.7, 0.8, and 0.9, respectively.What is the probability that the target is hit?
19. An actuary studying the insurance preferences of homeowners in California discovers that the event that a homeowner purchases earthquake coverage is independent of the event that he or she
18. 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
17. Show that if A and B are independent and A ⊆ B, then either P(A) = 0 or P(B) = 1.
16. (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 independent of every event B.
15. Show that if an event A is independent of itself, then P(A) = 0 or 1.
14. Find an example in which P(AB) < P(A)P(B).
13. 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
(Chevalier de M´er ´e’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
11. Consider the four “unfolded” dice in Figure 3.9 designed by Stanford professor Bradley Effron. Clearly, none of these dice is an ordinary die with sides numbered 1 through 6.A game consists
10. 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
9. According to a recent mortality table, the probability that a 35-year-old U.S. citizen will live to age 65 is 0.725. (a) What is the probability that John and Jim, two 35-year-old Americans who
8. 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?
7. 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?
6. Cal Ripken is a distinguished American baseball player who played 21 seasons in Major League Baseball for the Baltimore Orioles (1981-2001).Among his many achievements, the most important one is
5. The only information revealed concerning the three children of the new mayor of a large town is that their names are Daly, Emmett, and Karina. Is the event that Daly is younger than Emmett
4. 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?
3. 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
2. Clark and Anthony are two old friends. Let A be the event that Clark will attend Anthony’s funeral. Let B be the event that Anthony will attend Clark’s funeral. Are A and B independent?Why or
1. Jean le Rond d’Alembert, a French mathematician, believed that in successive flips of a fair coin, after a long run of heads, a tail is more likely. Do you agree with d’Alembert on this?
We draw cards, one at a time, at random and successively from an ordinary deck of 52 cards with replacement. What is the probability that an ace appears before a face card?
Figure 3.8 shows an electric circuit in which each of the switches located at 1, 2, 3, and 4 is independently closed or open with probabilities p and 1 − p, respectively. If a signal is fed to the
3. 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
2. A box has 10 coins of which 3 are gold. Eileen selects a coin at random first. Then Bernice draws a coin randomly and finds that it is gold. What is the probability that Eileen’s coin is also
1. In a small town in Massachusetts, of the 60 students who took the SAT the last time that it was offered, 17 had attended an SAT test prep course. If taking this course increases the chances of a
25. The advantage of a certain blood test is that 90% of the time it is positive for patients having a certain disease. Its disadvantage is that 25% of the time it is also positive in healthy people.
24. 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
23. An urn contains five red and three blue chips. Suppose that four of these chips are selected at random and transferred to a second urn, which was originally empty. If a random chip from this
22. There are two stables on a farm, one that houses 20 horses and 13 mules, the other with 25 horses and eight mules. Without any pattern, animals occasionally leave their stables and then return to
21. Some studies have shown that 1 in 177 Americans have celiac disease. Nevertheless, in general, for ordinary physicians, it is hard to diagnose it and, as a result, only 20% of the patients that
20. Solve the following problem, asked of Marilyn Vos Savant in the “Ask Marilyn” column of Parade Magazine, February 18, 1996.Say I have a wallet that contains either a $2 bill or a $20 bill
19. With probability of 1/6 there are i defective fuses among 1000 fuses (i = 0, 1, 2, 3, 4, 5). If among 100 fuses selected at random, none was defective, what is the probability of no defective
18. There are three identical cards that differ only in color. Both sides of one are black, both sides of the second one are red, and one side of the third card is black and its other side is red.

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