Finite Mathematics & Its Applications (12th Edition)
12th Edition
ISBN: 9780134437767
Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair
Publisher: PEARSON
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Textbook Question
Chapter 6, Problem 2P
First Paradox: Under certain circumstances, you have your best chance of winning a tennis
tournament if you play most of your games against the best possible opponent.
Alice and her two sisters, Betty and Carol, are avid tennis players. Betty is the best of the three sisters, and Carol plays at the same level as Alice. Alice defeats Carol 50% of the time but only defeats Betty 40% of the time.
Alice’s mother offers to give her $100 if she can win two consecutive games when playing three alternating games against her two sisters. Since the games will alternate, Alice has two possibilities for the sequence of opponents. One possibility is to play the first game against Betty, followed by a game with Carol, and then another game with Betty. We will refer to this sequence as BCB. The other possible sequence is CBC.
Calculate the probability of Alice getting the $100 reward if she chooses the sequence CBC.
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Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games.To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage.It turns out that with a five-point advantage Bob wins 60% of the time; he wins 70% of the time with a ten-point advantage and 90% of the time with a fifteen-point advantage.Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in the…
Chapter 6 Solutions
Finite Mathematics & Its Applications (12th Edition)
Ch. 6.1 - 1. Lightbulbs A machine produces lightbulbs. As...Ch. 6.1 - 2. Citrus Fruit Suppose that there are two crates...Ch. 6.1 - 1. Committee Selection A committee of two people...Ch. 6.1 - 2. Selecting Letters A letter is selected at...Ch. 6.1 - Heads and Tails An experiment consists of tossing...Ch. 6.1 - Four-Sided Dice A pair of four-sided dice-each...Ch. 6.1 - 5. Selecting from Urns Suppose that we have two...Ch. 6.1 - Coin Tosses An experiment consists of tossing a...Ch. 6.1 - 7. Efficiency Studies An efficiency expert records...Ch. 6.1 - Census Data A census taker records the annual...
Ch. 6.1 - Student Poll A campus survey is taken to correlate...Ch. 6.1 - 10. Automobiles An experiment consists of...Ch. 6.1 - 11. Let be a sample space,
.
a. Are E and F...Ch. 6.1 - 12. Draw the events E and E′ on two separate Venn...Ch. 6.1 - 13. Let be a sample space. Determine all possible...Ch. 6.1 - 14. Let S be a sample space with n outcomes. How...Ch. 6.1 - Let S={1,2,3,4} be a sample space, E={1}, and...Ch. 6.1 - 16. Let S be any sample space, and E, F any events...Ch. 6.1 - Coin Tosses Suppose that 10 coins are tossed and...Ch. 6.1 - Three-Digit Numbers An experiment consists of...Ch. 6.1 - Genetic Traits An experiment consists of observing...Ch. 6.1 - 20. Genetic Traits Consider the experiment and...Ch. 6.1 - 21. Shuttle Bus Suppose that you observe the...Ch. 6.1 - 22. Dice A pair of dice is rolled, and the sum of...Ch. 6.1 - Selecting Balls from an Urn An urn contains balls...Ch. 6.1 - Selecting Balls from an Urn Repeat Exercise 23 in...Ch. 6.1 - 25. NBA Draft Lottery In the NBA, the 14...Ch. 6.1 - Coin Die Suppose that a coin is tossed and a die...Ch. 6.1 - 27. The Game of Clue Clue is a board game in which...Ch. 6.2 - Solutions can be found following the section...Ch. 6.2 - Solutions can be found following the section...Ch. 6.2 - Prob. 3CYUCh. 6.2 - In Exercises 1–4, classify the type of probability...Ch. 6.2 - Prob. 2ECh. 6.2 - Prob. 3ECh. 6.2 - In Exercises 1–4, classify the type of probability...Ch. 6.2 - In Exercises 5 and 6, determine the probability...Ch. 6.2 - In Exercises 5 and 6, determine the probability...Ch. 6.2 - 7. Roulette The modern American roulette wheel has...Ch. 6.2 - U.S. States A state is selected at random from the...Ch. 6.2 - 9. Word Frequencies There are 4487 words in the...Ch. 6.2 - 10. United Nations Of the 193 member countries of...Ch. 6.2 - 11. Selecting a Letter An experiment consists of...Ch. 6.2 - 12. Selecting a Number An experiment consists of...Ch. 6.2 - Dice Suppose that a red die and a green die are...Ch. 6.2 - Children An experiment consists of observing the...Ch. 6.2 - Kind of High School The given table shows the...Ch. 6.2 - Highest Degree Planned The next table shows the...Ch. 6.2 - Grade Distributions The following table shows the...Ch. 6.2 - 18. Candy Colors The colors in a bag of...Ch. 6.2 - Prob. 19ECh. 6.2 - 20. An experiment with outcomes has the following...Ch. 6.2 - College Applications The table that follows was...Ch. 6.2 - 22. Employees’ Ages The next table summarizes the...Ch. 6.2 - 23. Which of the following probabilities are...Ch. 6.2 - 24. Which of the following probabilities are...Ch. 6.2 - Car Race Three cars, a Mazda, a Honda, and a Ford,...Ch. 6.2 - 26. Hair Color In a study, the residents of...Ch. 6.2 - 27. Political Views On a certain campus, the...Ch. 6.2 - 28. Tennis The probability that Alice beats Ben in...Ch. 6.2 - 29. Pair of Dice Suppose that a pair of dice is...Ch. 6.2 - Coin Tossing An experiment consists of tossing a...Ch. 6.2 - 31. Suppose that where E and F are mutually...Ch. 6.2 - Suppose that Pr(E)=.3andPr(EF)=.7, where E and F...Ch. 6.2 - In Exercises 33–36, consider the probabilities...Ch. 6.2 - In Exercises 3336, consider the probabilities...Ch. 6.2 - In Exercises 3336, consider the probabilities...Ch. 6.2 - In Exercises 3336, consider the probabilities...Ch. 6.2 - In Exercises 37–40, use a Venn diagram similar to...Ch. 6.2 - In Exercises 3740, use a Venn diagram similar to...Ch. 6.2 - In Exercises 3740, use a Venn diagram similar to...Ch. 6.2 - In Exercises 3740, use a Venn diagram similar to...Ch. 6.2 - 41. Convert the odds of “10 to 1” to a...Ch. 6.2 - Convert the odds of 4 to 5 to a probability.Ch. 6.2 - Convert the probability .2 to odds.Ch. 6.2 - Convert the probability 37 to odds.Ch. 6.2 - Coin Tosses The probability of getting three heads...Ch. 6.2 - Advanced Degree The probability that a graduate of...Ch. 6.2 - 47. Demographic The odds of a person in the...Ch. 6.2 - 48. Election Odds In March 2016, a betting website...Ch. 6.2 - Bookies Gamblers usually give odds against an...Ch. 6.2 - 50. Odds of an Earthquake The probability that...Ch. 6.2 - Prob. 51ECh. 6.2 - Prob. 52ECh. 6.3 - Solutions can be found following the section...Ch. 6.3 - Prob. 2CYUCh. 6.3 - 1. A number is chosen at random from the whole...Ch. 6.3 - 2. A number is chosen at random from the whole...Ch. 6.3 - 3. Balls in an Urn An urn contains five red balls...Ch. 6.3 - 4. Balls in an Urn An urn contains seven green...Ch. 6.3 - Balls in an Urn An urn contains six green balls...Ch. 6.3 - 6. Balls in an Urn An urn contains eight red balls...Ch. 6.3 - 7. Opinion Polling Two out of the seven members of...Ch. 6.3 - Opinion Polling Of the 15 members on a Senate...Ch. 6.3 - 9. Committee Selection In the 114th United States...Ch. 6.3 - 10. Committee Selection The U.S. Senate consists...Ch. 6.3 - 11. Quality Control A factory produces LCD panels,...Ch. 6.3 - Rotten Tomato A bag contains nine tomatoes, of...Ch. 6.3 - Selecting Students Exercises 13–16 refer to a...Ch. 6.3 - Selecting Students Exercises 1316 refer to a...Ch. 6.3 - Selecting Students Exercises 13–16 refer to a...Ch. 6.3 - Selecting Students Exercises 13–16 refer to a...Ch. 6.3 - 17. Birthday Three people are chosen at random....Ch. 6.3 - Birthday Four people are chosen at random. What is...Ch. 6.3 - 19. Date Conflict Without consultation with each...Ch. 6.3 - 20. Presidential Choices There were 16 presidents...Ch. 6.3 - Name Badges Eight workers need an employee number...Ch. 6.3 - Random Selection Each person in a group of 10...Ch. 6.3 - Birthday Problem What is the probability that, in...Ch. 6.3 - Birthday Problem Johnny Carson, host of The...Ch. 6.3 - Dice A die is rolled twice. What is the...Ch. 6.3 - Dice A die is rolled three times. What is the...Ch. 6.3 - Dice A die is rolled four times. What is the...Ch. 6.3 - Dice A die is rolled three times. What is the...Ch. 6.3 - 29. Coin Tosses A coin is tossed 10 times. What is...Ch. 6.3 - Coin Tosses A coin is tossed seven times. What is...Ch. 6.3 - Prob. 31ECh. 6.3 - 32. Elevator An elevator has six buttons: L, 1, 2,...Ch. 6.3 - Street Routs Figure 1 shows a partial map of the...Ch. 6.3 - Street Routes Repeat Exercise 33 for Fig. 2.Ch. 6.3 - 35. Baseball Predictions In the American League,...Ch. 6.3 - Baseball Predictions Suppose that the sportswriter...Ch. 6.3 - 37. Baseball Predictions Suppose that the...Ch. 6.3 - Baseball Predictions Suppose that the sportswriter...Ch. 6.3 - Place Settings Fred has five place settings...Ch. 6.3 - 40. Track Positions Michael and Christopher are...Ch. 6.3 - 41. Group Picture A man, a woman, and their three...Ch. 6.3 - 42. Letter Positions What is the probability that...Ch. 6.3 - Poker A poker hand consists of five cards drawn...Ch. 6.3 - Poker A poker hand consists of five cards drawn...Ch. 6.3 - Poker A poker hand consists of five cards drawn...Ch. 6.3 - Prob. 46ECh. 6.3 - Prob. 47ECh. 6.3 - Powerball Lottery The winner of the Powerball...Ch. 6.3 - Illinois Lotto Exercises 49 and 50 refer to the...Ch. 6.3 - Illinois Lotto Exercises 49 and 50 refer to the...Ch. 6.3 - 51. California Lottery In the California Fantasy 5...Ch. 6.3 - Prob. 52ECh. 6.3 - Prob. 53ECh. 6.3 - Prob. 54ECh. 6.3 - 55. Health Statistics Table 2 shows the...Ch. 6.3 - Prob. 56ECh. 6.3 - Prob. 57ECh. 6.3 - Prob. 58ECh. 6.3 - License Plate Game Johnny and Doyle are driving on...Ch. 6.3 - Prob. 60ECh. 6.3 - Prob. 61ECh. 6.3 - 62. Term Papers A political science class has 20...Ch. 6.3 - Prob. 63ECh. 6.3 - Prob. 64ECh. 6.3 - Prob. 65ECh. 6.3 - Prob. 66ECh. 6.4 - 1. Cards Suppose that there are three cards: one...Ch. 6.4 - Show that if events E and F are independent of...Ch. 6.4 - 1. The Venn diagram in Fig. 3 shows the...Ch. 6.4 - 2. The Venn diagram in Fig. 4 shows the...Ch. 6.4 - Let S be a sample space and E and F be events...Ch. 6.4 - Let S be a sample space and E and F be events...Ch. 6.4 - Let S be a sample space and E and F be events...Ch. 6.4 - 6. Let S be a sample space and E and F be events...Ch. 6.4 - Let S be a sample space and E and F be events...Ch. 6.4 - Let S be a sample space and E and F be events...Ch. 6.4 - Dice When a pair of dice is rolled, what is the...Ch. 6.4 - 10. Dice When a pair of dice is rolled, what is...Ch. 6.4 - Coins A coin is tossed three times. What is the...Ch. 6.4 - Coins A coin is tossed three times. What is the...Ch. 6.4 - Bag of Marbles A bag contains five red marbles and...Ch. 6.4 - Balls in an Urn Two balls are selected at random...Ch. 6.4 - 15. Children Suppose a family has two children and...Ch. 6.4 - Children Suppose a family has two children and at...Ch. 6.4 - 17. Value of College Twenty-five percent of...Ch. 6.4 - Advanced Degrees Sixty percent of the teachers at...Ch. 6.4 - Advanced Degrees Table 1 shows the projected...Ch. 6.4 - 20. Voting Table 2 shows the number of registered...Ch. 6.4 - Military Personnel Table 3 shows the numbers (in...Ch. 6.4 - 22. College Majors Table 4 shows the probable...Ch. 6.4 - 23. Bills in Envelopes Each of three sealed opaque...Ch. 6.4 - 24. Gold and Silver Coins Consider three boxes....Ch. 6.4 - 25. Cards A sequence of two playing cards is drawn...Ch. 6.4 - Cards A sequence of two playing cards is drawn at...Ch. 6.4 - Coin Tosses A coin is tossed five times. What is...Ch. 6.4 - Coin Tosses A coin is tossed twice. What is the...Ch. 6.4 - 29. Exit Polling According to exit polling for the...Ch. 6.4 - Population Twenty percent of the worlds population...Ch. 6.4 - 31. Basketball Suppose that your team is behind by...Ch. 6.4 - 32. Password Fred remembers all but the last...Ch. 6.4 - Let E and F be events with P(E)=.4,Pr(F)=.5, and...Ch. 6.4 - 34. Let E and F be events with , and. Are E and F...Ch. 6.4 - 35. Let E and F be independent events with . Find...Ch. 6.4 - 36. Let E and F be independent events with and ....Ch. 6.4 - In Exercises 3740, assume that E and F are...Ch. 6.4 - In Exercises 3740, assume that E and F are...Ch. 6.4 - In Exercises 37–40, assume that E and F are...Ch. 6.4 - In Exercises 3740, assume that E and F are...Ch. 6.4 - Let A, B, and C be independent events with...Ch. 6.4 - 42. Let A, B, and C be independent events with , ...Ch. 6.4 - 43. Balls in an Urn A sample of two balls is drawn...Ch. 6.4 - Balls in an Urn An urn contains two white balls...Ch. 6.4 - 45. Roll a Die Roll a die, and consider the...Ch. 6.4 - Roll a Die Roll a die, and consider the following...Ch. 6.4 - Rolling Dice Roll a pair of dice, and consider the...Ch. 6.4 - Rolling Dice Roll a pair of dice, and consider the...Ch. 6.4 - 49. Epidemiology A doctor studies the known cancer...Ch. 6.4 - 50. Blood Tests A hospital uses two tests to...Ch. 6.4 - Medical Screening A medical screening program...Ch. 6.4 - Guessing on an Exam A truefalse exam has 10...Ch. 6.4 - 53. System Reliability A TV set contains five...Ch. 6.4 - System Reliability In November 2015, Intel...Ch. 6.4 - 55. Smartphones Suppose that in Sleepy Valley, 70%...Ch. 6.4 - 56. Fishing The probability that a fisherman...Ch. 6.4 - Baseball A baseball players batting average...Ch. 6.4 - Roulette If you bet on the number 7 in roulette,...Ch. 6.4 - Free-Throws A basketball player makes each...Ch. 6.4 - 60. Free-Throws Rework Exercise 59 with a...Ch. 6.4 - Free-Throws Consider Exercise 59, but let the...Ch. 6.4 - Free-Throws Consider Exercise 59, but let the...Ch. 6.4 - Prob. 63ECh. 6.4 - 64. Coin Toss A coin is tossed five times. Is the...Ch. 6.4 - Prob. 65ECh. 6.4 - Prob. 66ECh. 6.4 - Prob. 67ECh. 6.4 - 68. Use the inclusion–exclusion principle for...Ch. 6.5 - Solutions can be found following the section...Ch. 6.5 - Solutions can be found following the section...Ch. 6.5 - Solutions can be found following the section...Ch. 6.5 - Solutions can be found following the section...Ch. 6.5 - In Exercises 1–4, draw trees representing the...Ch. 6.5 - In Exercises 1–4, draw trees representing the...Ch. 6.5 - Prob. 3ECh. 6.5 - In Exercises 1–4, draw trees representing the...Ch. 6.5 - Personnel Categories Refer to Exercise 3. What is...Ch. 6.5 - 6. Tax Returns Refer to Exercise 4. What is the...Ch. 6.5 - Personnel Categories Refer to Exercise 3. What is...Ch. 6.5 - Personnel Categories Refer to Exercise 3. What is...Ch. 6.5 - 9. Selecting from Urns Suppose that there is a...Ch. 6.5 - Cards, Coins, Dice A card is drawn from a 52-card...Ch. 6.5 - 11. Cards A card is drawn from a 52-card deck. We...Ch. 6.5 - 12. Balls in an Urn An urn contains six white...Ch. 6.5 - Quality Control Twenty percent of the library...Ch. 6.5 - Water Testing In a recent environmental study of...Ch. 6.5 - 15. Color Blindness Color blindness is a...Ch. 6.5 - Manufacturing A factory has two machines that...Ch. 6.5 - 17. T-maze A mouse is put into a T-maze (a maze...Ch. 6.5 - 18. T-maze Refer to Exercise 17. What is the...Ch. 6.5 - 19. Heads or Tails Three ordinary quarters and a...Ch. 6.5 - Prob. 20ECh. 6.5 - Tennis Kim has a strong first serve; whenever it...Ch. 6.5 - Tennis When a tennis player hits his first serve...Ch. 6.5 - 23. Accidental Nuclear War Suppose that, during...Ch. 6.5 - 24. Accidental Nuclear War Refer to Exercise 23....Ch. 6.5 - Coin Tosses A coin is to be tossed at most five...Ch. 6.5 - Cards Suppose that, instead of tossing a coin, the...Ch. 6.5 - Genetics Traits passed from generation to...Ch. 6.5 - 28. Genetics Refer to Exercise 27. Suppose that a...Ch. 6.5 - College Faculty At a local college, five sections...Ch. 6.5 - Quality Control A lightbulb manufacturer knows...Ch. 6.5 - 31. Balls in an Urn Urn I contains 5 red balls and...Ch. 6.5 - 32. Balls in an Urn An urn contains five red balls...Ch. 6.5 - Prob. 33ECh. 6.5 - 34. Selecting from Urns An urn contains four red...Ch. 6.5 - Industrial Production A factory that produces...Ch. 6.5 - Golf Bud is a very consistent golfer. On par-three...Ch. 6.5 - Nontransitive Dice Consider three dice: one red,...Ch. 6.5 - U.S. Car Production Car production in North...Ch. 6.5 - Prob. 39ECh. 6.5 - Prob. 40ECh. 6.5 - Prob. 41ECh. 6.5 - Prob. 42ECh. 6.5 - Prob. 43ECh. 6.5 - Prob. 44ECh. 6.5 - Medical Screening Suppose that a test for...Ch. 6.5 - Medical Screening The probability .0002 (or .02%)...Ch. 6.5 - 47. Medical Screening The results of a trial used...Ch. 6.5 - 48. Medical Screening The results of a trial used...Ch. 6.5 - Drug Testing Suppose that 500 athletes are tested...Ch. 6.5 - Polygraph Test Recent studies have indicated that...Ch. 6.6 - 1. Quality Control Refer to Example 2. Suppose...Ch. 6.6 - 2. Political Polling Use the method of natural...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - Exercises 11–15 refer to diagnostic tests. A false...Ch. 6.6 - Exercises 11–15 refer to diagnostic tests. A false...Ch. 6.6 - Exercises 11–15 refer to diagnostic tests. A false...Ch. 6.6 - Exercises 11–15 refer to diagnostic tests. A false...Ch. 6.6 - Exercises 11–15 refer to diagnostic tests. A false...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - Prob. 19ECh. 6.6 - In Exercises 1–22, use Bayes’ theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 122, use Bayes theorem to calculate...Ch. 6.6 - In Exercises 2330, use the method of natural...Ch. 6.6 - In Exercises 23–30, use the method of natural...Ch. 6.6 - Prob. 25ECh. 6.6 - Prob. 26ECh. 6.6 - In Exercises 23–30, use the method of natural...Ch. 6.6 - In Exercises 2330, use the method of natural...Ch. 6.6 - Prob. 29ECh. 6.6 - In Exercises 2330, use the method of natural...Ch. 6.7 - 1. Rolling a Die Simulate 36 rolls of a fair die....Ch. 6.7 - Prob. 2ECh. 6.7 - Free-Throws Simulate 10 free-throws for Kobe...Ch. 6.7 - Prob. 4ECh. 6.7 - Prob. 5ECh. 6.7 - Prob. 6ECh. 6.7 - Prob. 7ECh. 6.7 - Prob. 8ECh. 6.7 - 9. Gas Queue A gas station with four self-serve...Ch. 6.7 - Prob. 10ECh. 6 - 1. What is the sample space of an experiment?
Ch. 6 - 2. Using the language of sets and assuming that A...Ch. 6 - In a sample space, what is the probability of the...Ch. 6 - 4. What subset in a sample space corresponds to...Ch. 6 - Prob. 5FCCECh. 6 - Prob. 6FCCECh. 6 - Prob. 7FCCECh. 6 - Prob. 8FCCECh. 6 - Prob. 9FCCECh. 6 - Prob. 10FCCECh. 6 - Prob. 11FCCECh. 6 - Prob. 12FCCECh. 6 - Prob. 13FCCECh. 6 - Coins A box contains a penny, a nickel, a dime, a...Ch. 6 - Prob. 2RECh. 6 - 3. Suppose that E and F are events with . Find .
Ch. 6 - Suppose that E and F are mutually exclusive events...Ch. 6 - 5. Languages Of the 120 students in a class, 30...Ch. 6 - Prob. 6RECh. 6 - Prob. 7RECh. 6 - Prob. 8RECh. 6 - 9. Demographics Twenty-six percent of all...Ch. 6 - Prob. 10RECh. 6 - Prob. 11RECh. 6 - Prob. 12RECh. 6 - 13. Barrel of Apples Five of the apples in a...Ch. 6 - 14. Opinion Sampling Of the nine city council...Ch. 6 - Exam Questions Prior to taking an essay...Ch. 6 - 16. Craps In the casino game of craps, a player...Ch. 6 - Coin Tosses A coin is to be tossed five times....Ch. 6 - Coin Tosses Two players each toss a coin three...Ch. 6 - Olympic Swimmers In an Olympic swimming event, two...Ch. 6 - Prob. 20RECh. 6 - Prob. 21RECh. 6 - Drawing Cards A card is drawn at random from a...Ch. 6 - 23. Dice What is the probability of having each of...Ch. 6 - 24. Dice Find the odds in favor of getting four...Ch. 6 - Birthdays What is the probability that, out of a...Ch. 6 - Birthdays Four people are chosen at random. What...Ch. 6 - Let E and F be events with Pr(E)=.4,Pr(F)=.3, and...Ch. 6 - 28. Let E and F be events with . Find .
Ch. 6 - Coin Tosses When a coin is tossed three times,...Ch. 6 - 30. Dice Suppose that a pair of dice is rolled....Ch. 6 - Prob. 31RECh. 6 - Prob. 32RECh. 6 - 33. Premed Majors Suppose that a certain college...Ch. 6 - Prob. 34RECh. 6 - Prob. 35RECh. 6 - Coin Tosses Suppose that we toss a coin three...Ch. 6 - Prob. 37RECh. 6 - Prob. 38RECh. 6 - 39. Archery Two archers shoot at a moving target....Ch. 6 - 40. Final Exam Fred will do well on his final exam...Ch. 6 - Let A and B be independent events for which the...Ch. 6 - Let A and B be independent events with Pr(A)=.3...Ch. 6 - Prob. 43RECh. 6 - Prob. 44RECh. 6 - Prob. 45RECh. 6 - Prob. 46RECh. 6 - Left-Handedness According to a geneticist at...Ch. 6 - Tax Audits An auditing procedure for income tax...Ch. 6 - 49. Weighing Produce A supermarket has three...Ch. 6 - 50. Dragons An island contains an equal number of...Ch. 6 - Prob. 51RECh. 6 - Prob. 52RECh. 6 - Prob. 53RECh. 6 - Prob. 54RECh. 6 - Prob. 55RECh. 6 - Prob. 56RECh. 6 - First Paradox: Under certain circumstances, you...Ch. 6 - First Paradox: Under certain circumstances, you...Ch. 6 - First Paradox: Under certain circumstances, you...Ch. 6 - Prob. 4PCh. 6 - First Paradox: Under certain circumstances, you...Ch. 6 - Second Paradox: The probability of a male...Ch. 6 - Prob. 7PCh. 6 - Prob. 8PCh. 6 - Prob. 9PCh. 6 - Prob. 10PCh. 6 - Prob. 11P
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- Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games.To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage.It turns out that with a five-point advantage Bob wins 60% of the time; he wins 60% of the time with a ten-point advantage and 60% of the time with a fifteen-point advantage.Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in the…arrow_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 70% of the time; he wins 80% of the time with a ten-point advantage and 90% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive game won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in…arrow_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 60% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 40% of the time; he wins 70% of the time with a ten-point advantage and 70% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in…arrow_forward
- Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 80% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 40% of the time; he wins 70% of the time with a ten-point advantage and 90% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for system, and determine the proportion games that Doug will win the long…arrow_forwardBob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 80% of their games. To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game, and if he still wins the next game he will spot him fifteen points, and continue to spot him fifteen points as long as he keeps winning. Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a five-point advantage Bob wins 20% of the time; he wins 50% of the time with a ten-point advantage and 80% of the time with a fifteen-point advantage. Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three or more consecutive games won by Doug. Find the transition matrix of this system, the steady-state vector for the system, and determine the proportion of games that Doug will win in…arrow_forwardSuppose Janice has a 25% chance of totaling her car (worth $13,500) this year and Sam has a 19% chance of totaling his car ($23,700) this year. If you have a car insurance company and you want to offer an insurance policy to these two customers and offer it for the same price, what price should you charge (without any profit mark-up)?arrow_forward
- A ticket between Corellia and Dantooine (located in a galaxy far far away) sells for $150. The plane can hold 100 people. It costs $8000 to fly an empty plane. Each person on the plane incurs variable costs of $30 (for food and fuel). If the flight is overbooked, anyone who cannot get a seat receives $300 in compensation. On average, 95% of all people who have a reservation show up for the flight. To maximize profit, how many reservations for the flight should you book? Hint: You can assume that the number of people that show up for a flight follows a binomial random variable.arrow_forwardSuppose Kim is deciding whether to open a gym or a diner. Her friend Tim, who is a finance expert, advised that there is a 45% chance that the gym will be successful and return a payoff of $55000 and a 55% chance that it will fail and lose $9500. Tim also advised that there is an 80% chance that the diner will be successful with a payoff of $30000 and a 20% chance of failure with a loss of $6000. 13) What is the expected payoff for the gym? a) 18000 b) 16525 c) 19555 d) 19525arrow_forwardJanus Seagull had a car accident and was out of work for a year. Janus believes that the accident was caused by a vehicle defect. He consulted some lawyers and planned to sue the vehicle manufacturer. During negotiations, the legal team of the vehicle manufacturer offered a $700K settlement. However, Janus needs to settle the $100K in legal fees. Janus asked his lawyer for advice and the lawyer told him that he has a 50% chance of winning the case. If Janus loses, he will incur legal fees of $75K. If he wins, the full requested settlement is also not guaranteed. The lawyer believes that there is a 50% chance that Janus will receive full settlement of $3 million, of which Janus needs to settle $1 million in legal fees. There is a 50% chance that the jury will award Janus $1 million, of which 50% will be taken up by legal fees.arrow_forward
- . The table below states the payoffs in political points (measured in billions of rubles) to two nations that are rivals in world politics, Russia and Ukraine. Each country can take one of two courses: peace; or war. In each cell, the first payoff is for Russia, and the second payoff is for Ukraine. (a) Assume that neither country observes the military strategy of its rival, and solve the game (if it can be solved). Explain your solution step-by-step. Does this outcome maximize total political points? (b) In general, what is a Nash equilibrium? Is the solution to this game a Nash equilibrium? (c) Suppose that each country deposits a fund of two billion rubles with the United Nations. Either country would forfeit this fund if it wages war. What is the solution now to the game? Is this a Nash equilibrium? Russia, Ukraine Peace War Peace 4,4 1, 5 War 5, 1 2,2arrow_forwardA friend who lives in Los Angeles makes frequent consultingtrips to Washington, D.C.; 50% of the time shetravels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on airline #3. For airline#1, flights are late into D.C. 30% of the time and late intoL.A. 10% of the time. For airline #2, these percentagesare 25% and 20%, whereas for airline #3 the percentagesare 40% and 25%. If we learn that on a particular trip shearrived late at exactly one of the two destinations, whatare the posterior probabilities of having flown on airlines#1, #2, and #3? Assume that the chance of a late arrival inL.A. is unaffected by what happens on the flight to D.C.[Hint: From the tip of each first-generation branch on atree diagram, draw three second-generation brancheslabeled, respectively, 0 late, 1 late, and 2 late.]arrow_forwardWhile walking around you see a carnival game! The game has a 1% chance of you winning $150, a 3% chance of you winning $20, a 6% chance of you winning $6, and an 90% chance of you winning nothing. If the game costs $3 to play, how much should you expect to win (or lose) if you play this game once? For the same situation, how much should you expect to win (or lose) if you play the game 300 times?arrow_forward
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