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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Finite Mathematics for Business, Economics, Life Sciences and Social Sciences

In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A6-card hand that contains exactly two clubs In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A4-card hand that contains no aces In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A4-card hand that contains no face cards In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A7-card hand that contains exactly 2 diamonds and exactly 2 spades In Problems 79-86, find the probability of being dealt the given hand from a standard 52 -card deck. Refer to the description of a standard 52 -card deck on page 384. A7-card hand that contains exactly 1 king and exactly 2 jacks In Problems 87-90, several experiments are simulated using the random number feature on a graphing calculator. For example, the roll of a fair die can be simulated by selecting a random integer from, and 50 rolls of a fair die by selecting 50 random integers from 1to6 (see Fig. A for Problem 87 and your user's manual). From the statistical plot of the outcomes of rolling a fair die 50 times (see Fig. B), we see, for example, that the number 4 was rolled exactly 5 times. (A) What is the empirical probability that the number 6 was rolled? (B) What is the probability that a 6 is rolled under the equally likely assumption? (C) Use a graphing calculator to simulate 100 rolls of a fair die and determine the empirical probabilities of the six outcomes.In Problems 87-90, several experiments are simulated using the random number feature on a graphing calculator. For example, the roll of a fair die can be simulated by selecting a random integer from, and 50 rolls of a fair die by selecting 50 random integers from 1to6 (see Fig. A for Problem 87 and your user's manual). Use a graphing calculator to simulate 200 tosses of a nickel and dime, representing the outcomes HH,HT,TH,TTby1,2,3,and4, respectively. (A) Find the empirical probabilities of the four outcomes. (B) What is the probability of each outcome under the equally likely assumption?In Problems 87-90, several experiments are simulated using the random number feature on a graphing calculator. For example, the roll of a fair die can be simulated by selecting a random integer from, and 50 rolls of a fair die by selecting 50 random integers from 1to6 (see Fig. A for Problem 87 and your user's manual). A) Explain how a graphing calculator can be used to simulate 500 tosses of a coin. (B) Carry out the simulation and find the empirical probabilities of the two outcomes. (C) What is the probability of each outcome under the equally likely assumption?In Problems 87-90, several experiments are simulated using the random number feature on a graphing calculator. For example, the roll of a fair die can be simulated by selecting a random integer from, and 50 rolls of a fair die by selecting 50 random integers from 1to6 (see Fig. A for Problem 87 and your user's manual). From a box containing 12 balls numbered 1 through 12, one ball is drawn at random. (A) Explain how a graphing calculator can be used to simulate 400 repetitions of this experiment. (B) Carry out the simulation and find the empirical probability of drawing the 8 ball. (C) What is the probability of drawing the 8 ball under the equally likely assumption?Consumer testing. Twelve popular brands of beer are used in a blind taste study for consumer recognition. (A) If 4 distinct brands are chosen at random from the 12 and if a consumer is not allowed to repeat any answers, what is the probability that all 4 brands could be identified by just guessing? (B) If repeats are allowed in the 4 brands chosen at random from the 12 and if a consumer is allowed to repeat answers, what is the probability that all 4 brands are identified correctly by just guessing?Consumer testing. Six popular brands of cola are to be used in a blind taste study for consumer recognition. (A) If 3 distinct brands are chosen at random from the 6 and if a consumer is not allowed to repeat any answers, what is the probability that all 3 brands could be identified by just guessing? (B) If repeats are allowed in the 3 brands chosen at random from the 6 and if a consumer is allowed to repeat answers, what is the probability that all 3 brands are identified correctly by just guessing?Personnel selection. Suppose that 6 female and 5 male applicants have been successfully screened for 5 positions if the 5 positions are filled at random from the 11 finalists what is the probability of selecting (A) 3 females and 2 males? (B) 4 females and 1 male? (C) 5 females? (D) At least 4 females?Committee selection. A 4-person grievance committee is to include employees in 2 departments, A and B, with 15 and 20 employees, respectively. If the 4-person are selected at random from the 35 employees, what is the probability of selecting (A) 3 from A and 1 from B ? (B) 2 from A and 2 from B ? (C) All 4 from A ? (D) At least 3 from A ?Medicine. A laboratory technician is to be tested on identifying blood types from 8 standard classifications. (A) If 3 distinct samples are chosen at random from the 8 types and if the technician is not allowed to repeat any answers, what is the probability that all 3 could be correctly identified by just guessing? (B) If repeats are allowed in the 3 blood types chosen at random from the 8 and if the technician is allowed to repeat answers, what is the probability that all 3 are identified correctly by just guessing?Medical research. Because of limited funds, 5 research centers are to be chosen out of 8 suitable ones for a study on heart disease. If the selection is made at random, what is the probability that 5 particular research centers will be chosen?Politics. A town council has 9 members: 5 Democrats and 4 Republicans. A 3-person zoning committee is selected at random. (A) What is the probability that all zoning committee members are Democrats. (B) What is the probability that a majority of zoning committee members are Democrats?Politics. There are 10 senators (half Democrats, half Republicans) and 16 representatives (half Democrats, half Republicans) who wish to serve on a joint congressional committee on tax reform. An 8-person committee is chosen at random from those who wish to serve. (A) What is the probability that the joint committee contains equal numbers of senators and representatives? (B) What is the probability that the joint committee contains equal numbers of Democrats and Republicans?(A) Suppose that E and F are complementary events. Are E and F necessarily mutually exclusive? Explain why or why not. (B) Suppose that E and F are mutually exclusive events. Are E and F necessarily complementary? Explain why or why not.Determine the smallest number n such that in a group of n people, the probability that 2 or more have a birthday in the same month is greater than. 5. Discuss the assumptions underlying your computation.Use the sample space in Example 1 to answer the following: (A) What is the probability of rolling an odd number and a prime number? (B) What is the probability of rolling an odd number or a prime number?Use the sample space in Example 2 to answer the following: (A) What is the probability that a sum of 2 or 3 turns up? (B) What is the probability that both dice turn up the same or that a sum greater than 8 turns up?What is the probability that a number selected at random from the first 140 positive integers is (exactly) divisible by 4 or 6 ?A shipment of 40 precision parts, including 8 that are defective, is sent to an assembly plant. The quality control division selects 10 at random for testing and rejects the entire shipment if 1 or more in the sample are found to be defective. What is the probability that the shipment will be rejected?Use equation 3 to evaluate PE for n=4.(A) What are the odds for rolling a sum of 8 in a single roll of two fair dice? (B) If you bet $5 that a sum of 8 will turn up, what should the house pay (plus returning your $5 bet) if a sum of 8 does turn up in order for the game to be fair?If in repeated rolls of two fair dice, the odds against rolling a 6 before rolling a 7 are 6 to 5, then what is the probability of rolling a 6 before rolling a 7 ?Refer to Example 8. If a person from Springfield is selected at random, what is the (empirical) probability that (A) He or she has not tried either cola? What are the (empirical) odds for this event? (B) He or she has tried the diet cola or has not tried the regular cola? What are the (empirical) odds against this event?In Problems 1-6, write the expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1). 310910 2E In Problems 1-6, write the expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1). 1837 In Problems 1-6, write the expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1). 4556 In Problems 1-6, write the expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1). 29129 In Problems 1-6, write the expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1). 3161316 Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB Problems 7-12 refer to the Venn diagram below for events A and B in an equally likely sample space S. Find each of the indicated probabilities. PAB A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PD A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PD A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF A single card is drawn from a standard 52-card deck. Let D be the event that the card drawn is a diamond, and let F be the event that the card drawn is a face card. In Problems 13-24, find the indicated probabilities. PDF In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Odd or a multiple of 4 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Even or a multiple of 7 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Prime or greater than 20 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Less than 10 or greater than 10 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is A multiple of 2 or a multiple of 5 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is A multiple of 3 or a multiple of 4 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Less than 5 or greater than 20 In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. In Problems 25-32, use equation ( 1 ) to compute the probability that the number drawn is Prime or less than 14 If the probability is .51 that a candidate wins the election, what is the probability that he loses?If the probability is .03 that an automobile tire fails in less than 50,000 miles, what is the probability that the tire does not fail in 50,000 miles?In Problems 35-38, use the equally likely sample space in Example 2 to compute the probability of the following events: A sum that is less than or equal to 5 In Problems 35-38, use the equally likely sample space in Example 2 to compute the probability of the following events: A sum that is greater than 9 In Problems 35-38, use the equally likely sample space in Example 2 to compute the probability of the following events: The number on the first die is a 6 or the number on the second die is a 3.In Problems 35-38, use the equally likely sample space in Example 2 to compute the probability of the following events: The number on the first die is even or the number on the second die is even.Given the following probabilities for an event E. find the odds for and against E : A38B14C.4D.55 Given the following probabilities for an event E. find the odds for and against E : A35B17C.6D.35 Compute the probability of event E if the odds in favor of E are A38B117C41D4951 Compute the probabilities of event E if the odds in favor of E are A59B43C37D2377 In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If the odds for E equal the odds against E,thenPE=12.In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If the odds for E are a:b, then the odds against E are b:a.In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If PE+PF=PEF+PEF,thenEandF are mutually exclusive events.In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. The theoretical probability of an event is less than or equal to its empirical probability.In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If EandF are complementary events, then EandF are mutually exclusive.In Problems 43-48, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. If EandF are mutually exclusive events, then EandF are complementary.In Problems 49-52, compute the odds in favor of obtaining A head in a single toss of a coin In Problems 49-52, compute the odds in favor of obtaining A number divisible by 3 in a single roll of a die In Problems 49-52, compute the odds in favor of obtaining At least 1 head when a single coin is tossed 3 times In Problems 49-52, compute the odds in favor of obtaining 1 head when a single coin is tossed twice In Problems 53-56, compute the odds against obtaining A number greater than 4 in a single roll of a die In Problems 53-56, compute the odds against obtaining 2 heads when a single coin is tossed twice In Problems 53-56, compute the odds against obtaining A 3 or an even number in a single roll of a die In Problems 53-56, compute the odds against obtaining An odd number or a number divisible by 3 in a single roll of a die (A) What are the odds for rolling a sum of 5 in a single roll of two fair dice? (B) If you bet $1 that a sum of 5 will turn up, what should the house pay (plus returning your $1 bet) if a sum of 5 turns up in order for the game to be fair?(A) What are the odds for rolling a sum of 10 in a single roll of two fair dice? (B) If you bet $1 that a sum of 10 will turn up, what should the house pay (plus returning your $1 bet) if a sum of 10 turns up in order for the game to be fair?A pair of dice are rolled 1,000 times with the following frequencies of outcomes: Use these frequencies to calculate the approximate empirical probabilities and odds for the events in Problems 59 and 60. (A) The sum is less than 4 or greater than 9. (B) The sum is even or exactly divisible by 5.A pair of dice are rolled 1,000 times with the following frequencies of outcomes: Use these frequencies to calculate the approximate empirical probabilities and odds for the events in Problems 59 and 60. (A) The sum is a prime number or is exactly divisible by 4. (B) The sum is an odd number or exactly divisible by 3.In Problems 61-64, a single card is drawn from a standard 52-card deck. Calculate the probability of each event. A face card or a club is drawn.In Problems 61-64, a single card is drawn from a standard 52-card deck. Calculate the probability of each event. A king or a heart is drawn.In Problems 61-64, a single card is drawn from a standard 52-card deck. Calculate the probability of each event. A black card or an ace is drawn.In Problems 61-64, a single card is drawn from a standard 52-card deck. Calculate the probability of each event. A heart or a number less than 7 (count an ace as 1 ) is drawn.What is the probability of getting at least 1 diamond in a 5-card hand dealt from a standard 52-card deck?What is the probability of getting at least 1 black card in a 7-card hand dealt from a standard 52-card deck?What is the probability that a number selected at random from the first 100 positive integers is (exactly) divisible by 6or8 ?What is the probability that a number selected at random from the first 60 positive integers is (exactly) divisible by 6or9 ?Explain how the three events A,B,andC from a sample space S are related to each other in order for the following equation to hold true: PABC=PA+PB+PCPAB Explain how the three events A,B,andC from a sample space S are related to each other in order for the following equation to hold true: PABC=PA+PB+PC Show that the solution to the birthday problem in Example 5 can be written in the form PE=1P365n365n For a calculator that has a Pnr function, explain why this form may be better for direct evaluation than the other form used in the solution to Example 5. Try direct evaluation of both forms on a calculator for n=25.In a group of n people n12, what is the probability that at least 2 of them have the same birth month? (Assume that any birth month is as likely as any other.)In a group of n people n100, each person is asked to select a number between 1 and 100, write the number on a slip of paper and place the slip in a hat. What is the probability that at least 2 of the slips in the hat have the same number written on them?If the odds in favor of an event E occurring are atob, show that PE=aa+b [Hint: Solve the equation PE/PE=a/bforPE. ]If PE=c/d, show that odds in favor of E occurring are ctodc.The command in Figure A was used on a graphing calculator to simulate 50 repetitions of rolling a pair of dice and recording their sum. A statistical plot of the results is shown in Figure B. (A) Use Figure B to find the empirical probability of rolling a 7or8. (B) What is the theoretical probability of rolling a 7or8 ? (C) Use a graphing calculator to simulate 200 repetitions of rolling a pair of dice and recording their sum, and find the empirical probability of rolling a 7or8.Consider the command in Figure A and the associated statistical plot in Figure B. (A) Explain why the command does not simulate 50 repetitions of rolling a pair of dice and recording their sum. (B) Describe an experiment that is simulated by this command. (C) Simulate 200 repetitions of the experiment you described in part (B). Find the empirical probability of recording a 7or8, and the theoretical probability of recording a 7or8.Market research. From a survey involving 1,000 university students, a market research company found that 750 students owned laptops, 450 owned cars, and 350 owned cars and laptops. If a university student is selected at random, what is the (empirical) probability that (A) The student owns either a car or a laptop? (B) The student owns neither a car nor a laptop?Market research. Refer to Problem 79. If a university student is selected at random, what is the (empirical) probability that (A) The student does not own a car? (B) The student owns a car but not a laptop?Insurance. By examining the past driving records of city drivers, an insurance company has determined the following (empirical) probabilities: If a city driver is selected at random, what is the probability that (A) He or she drives less than 10,000 miles per year or has an accident? (B) He or she drives 10,000 or more miles per year and has no accidents?Insurance. Use the (empirical) probabilities in Problem 81 to find the probability that a city driver selected at random (A) Drives more than 15,000 miles per year or has an accident (B) Drives 15,000 or fewer miles per year and has an accident Quality control. A shipment of 60 game systems, including 9 that are defective, is sent to a retail store. The receiving department selects 10 at random for testing and rejects the whole shipment if 1 or more in the sample are found to be defective. What is the probability that the shipment will be rejected?Quality control. An assembly plant produces 40 outboard motors, including 7 that are defective. The quality control department selects 10 at random (from the 40 produced) for testing and will shut down the plant for troubleshooting if 1 or more in the sample are found to be defective. What is the probability that the plant will be shut down?Medicine. In order to test a new drug for adverse reactions, the drug was administered to 1,000 test subjects with the following results: 60 subjects reported that their only adverse reaction was a loss of appetite, 90 subjects reported that their only adverse reaction was a loss of sleep, and 800 subjects reported no adverse reactions at all. If this drug is released for general use, what is the (empirical) probability that a person using the drug will suffer both a loss of appetite and a loss of sleep?Product testing. To test a new car, an automobile manufacturer wants to select 4 employees to test-drive the car for 1 year. If 12 management and 8 union employees volunteer to be test drivers and the selection is made at random, what is the probability that at least 1 union employee is selected?Problems 87 and 88 refer to the data in the following table, obtained from a random survey of 1,000 residents of a state. The participants were asked their political affiliations and their preferences in an upcoming election. (In the table, D=Democrat,R=Republican,andU=Unaffliliated. ) Politics. If a state resident is selected at random, what is the (empirical) probability that the resident is (A) Not affiliated with a political party or has no preference? What are the odds for this event? (B) Affiliated with a political party and prefers candidate A ? What are the odds against this event?Problems 87 and 88 refer to the data in the following table, obtained from a random survey of 1,000 residents of a state. The participants were asked their political affiliations and their preferences in an upcoming election. (In the table, D=Democrat,R=Republican,andU=Unaffliliated. ) Politics. If a state resident is selected at random, what is the (empirical) probability that the resident is (A) A Democrat or prefers candidate B ? What are the odds for this event? (B) Not a Democrat and has no preference? What are the odds against this event?Refer to the table on rain and accidents in Example 2 and use formula (1), where appropriate, to complete the following probability tree: Discuss the difference between PRAandPAR.In college basketball, would it be reasonable to assume that the following events are independent? Explain why or why not. A=theGoldenEagleswininthefirstroundoftheNCAAtournament.B=theGoldenEagleswininthesecondroundoftheNCAAtournament.Refer to Example 1. (A) What is the probability of the pointer landing on a number greater than 4 ? (B) What is the probability of the pointer landing on a number greater than 4, given that it landed on an even number?Referring to the table in Example 2, determine the following: (A) Probability of no rain (B) Probability of an accident and no rain (C) Probability of an accident, given no rain [Use formula (1) and the results of parts (A) and (B).]If 80 of the male customers of the department store in Example 3 have store credit cards, what is the probability that a customer selected at random is a male and has a store credit card?Two balls are drawn in succession without replacement from a box containing 4 red and 2 white balls. What is the probability of drawing a red ball on the second draw?In Example 5, what is the probability that a given on board computer came from company EorC ?In Example 6, compute PBA and compare with PB.A single card is drawn from a standard 52-card deck. Test the following events for independence: (A) E= the drawn card is a red card F=thedrawncardsnumberisdivisibleby5(facecardsarenotassignedvalues) (B) G= the drawn card is a king H= the drawn card is a queen A single die is rolled 6 times. What is the probability of getting the sequence 1,2,3,4,5,6 ?1E 2E 3E 4E 5E 6E A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is an ace, given that it is a heart.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is red, given that it is a face card.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is a heart, given that it is an ace.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is a face card, given that it is red.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is black, given that it is a club.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is a jack, given that it is red.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is a club, given that it is black.A single card is drawn from a standard 52-card deck. In Problems 7-14 find the conditional probability that The card is red, given that it is a jack.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The sum is less than 6, given that the sum is even.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The sum is 10, given that the roll is doubles.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The sum is even, given that the sum is less than 6.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The roll is doubles, given that the sum is 10.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The sum is greater than 7, given that neither die is a six.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that The sum is odd, given that at least one die is a six.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that Neither die is a six, given that the sum is greater than 7.In Problems 15-22, find the conditional probability, in a single roll of two fair dice, that At least one die is a six, given that the sum is odd.In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 23-26, find each probability directly from the table. PB In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 23-26, find each probability directly from the table. PE In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 23-26, find each probability directly from the table. PBD In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 23-26, find each probability directly from the table. PCE In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PDB In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PCE In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PBD In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PEC In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PDC In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PEA In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PAC In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 27-34, compute each probability using formula ( 1 ) on page 423 and appropriate table values. PBB In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence AandD In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence AandE In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence BandD In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence BandE In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence BandF In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence CandF In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence AandB In Problems 23-42, use the table below. Events A,B, and C mutually exclusive; so are D,E, and F. In Problems 35-42, test each pair of events for independence DandF A Fair coin is tossed 8 times. (A) What is the probability of tossing a head on the 8th toss, given that the preceding 7 tosses were heads? (B) What is the probability of getting 8 heads or 8 tails?A fair die is rolled 5 times. (A) What is the probability of gelling a 6 on the 5th roll, given that a 6 turned up on the preceding 4 rolls? (B) What is the probability that the same number turns up every time?A pointer is spun once on the circular spinner shown below. The probability assigned to the pointer landing on a given integer (from 1to5 ) is the ratio of the area of the corresponding circular sector to the area of the whole circle, as given in the table: Given the events E= pointer lands on an even number F= pointer lands on a number less than 4 (A) Find PFE. (B) Test events E and F for independence.Repeat Problem 45 with the following events: E= pointer lands on an odd number F= pointer lands on a prime number Compute the indicated probabilities in Problems 47 and 48 by referring to the following probability tree: (A) PMS (B) PR Compute the indicated probabilities in Problems 47 and 48 by referring to the following probability tree: (A) PNR (B) PS A fair coin is tossed twice. Consider the sample space S=HH,HT,TH,TT of equally likely simple events. We are interested in the following events: E1= a head on the first toss E2= a tail on the first toss E3= a tail on the second toss E4= a head on the second toss For each pair of events, discuss whether they are independent and whether they are mutually exclusive. (A) E1 and E4 (B) E1 and E2 For each pair of events (see Problem 49 ), discuss whether they are independent and whether they are mutually exclusive. (A) E1 and E3 (B) E3 and E4 In 2 throws of a fair die, what is the probability that you will get an even number on each throw? An even number on the first or second throw?In 2 throws of a fair die, what is the probability that you will get at least 5 on each throw? At least 5 on the first or second throw?Two cards are drawn in succession from a standard 52-card deck. What is the probability that the first card is a club and the second card is a heart? (A) If the cards are drawn without replacement? (B) If the cards are drawn with replacement?Two cards are drawn in succession from a standard 52-card deck. What is the probability that both cards are red (A) If the cards are drawn without replacement? (B) If the cards are drawn with replacement?A card is drawn at random from a standard 52-card deck. Events G and H are G= the drawn card is black. H= the drawn card is divisible by 3 (face cards are not valued). (A) Find PHG. (B) Test H and G for independence.A card is drawn at random from a standard 52-card deck. Events M and N are M= the drawn card is a diamond. N= the drawn card is even (face cards are not valued). (A) Find PNM. (B) Test M and N for independence.Let A be the event that all of a family’s children are the same gender, and let B be the event that the family has at most 1 boy. Assuming the probability of having a girl is the same as the probability of having a boy ( both .5 ), test events A and B for independence if (A) The family has 2 children. (B) The family has 3 children.An experiment consists of tossing n coins. Let A be the event that at least 2 heads turn up, and let B be the event that all the coins turn up the same. Test A and B for independence if (A) 2 coins are tossed. (B) 3 coins are tossed.Problems 59-62 refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balls. Let Ri, be the event that the ith ball is red, and let Wi, be the event that the ith ball is white. Construct a probability tree for this experiment and find the probability of each of the events R1R2,R1W2,W1R2,W1W2,, given that the first ball drawn was (A) Replaced before the second draw (B) Not replaced before the second draw Problems 59-62 refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balls. Let Ri, be the event that the ith ball is red, and let Wi, be the event that the ith ball is white. Find the probability that the second ball was red, given that the first ball was (A) Replaced before the second draw (B) Not replaced before the second draw Problems 59-62 refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balls. Let Ri, be the event that the ith ball is red, and let Wi, be the event that the ith ball is white. Find the probability that at least 1 ball was red, given that the first ball was (A) Replaced before the second draw (B) Not replaced before the second draw Problems 59-62 refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balls. Let Ri, be the event that the ith ball is red, and let Wi, be the event that the ith ball is white. Find the probability that both balls were the same color, given that the first ball was (A) Replaced before the second draw (B) Not replaced before the second draw In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If PAB=PB, then A and B are independent.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If A and B are independent, then PAB=PBA.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If A is nonempty and AB, then PABPA.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If A and B are events, then PABPB.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If A and B are mutually exclusive, then A and B are independent.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If A and B are independent, then A and B are mutually exclusive.In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If two balls are drawn in succession, with replacement, from a box containing m red and n white balls m1andn1, then PW1R2=PR1W2 In Problems 63-70, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If two balls are drawn in succession, without replacement, from a box containing m red and n white balls m1andn1, then PW1R2=PR1W2 A box contains 2 red, 3 white, and 4 green balls. Two balls are drawn out of the box in succession without replacement. What is the probability that both balls are the same color?For the experiment in Problem 71, what is the probability that no white balls are drawn?An urn contains 2 one-dollar bills, 1 five-dollar bill, and 1 ten-dollar bill. A player draws bills one at a time without replacement from the urn until a ten-dollar bill is drawn. Then the game stops. All bills are kept by the player. (A) What is the probability of winning $16 ? (B) What is the probability of winning all bills in the urn? (C) What is the probability of the game stopping at the second draw?Ann and Barbara are playing a tennis match. The first player to win 2 sets wins the match. For any given set, the probability that Ann wins that set is 23. Find the probability that (A) Ann wins the match. (B) 3 sets are played. (C) The player who wins the first set goes on to win the match.Show that if A and B are independent events with nonzero probabilities in a sample space S, then PAB=PAandPBA=PB Show that if A and B are events with nonzero probabilities in a sample space S and either PAB=PAorPBAPB, then events A and B are independent.Show that PAA=1 when PA0.Show that PAB+PAB=1.Show that A and B are dependent if A and B are mutually exclusive and PA0,PB0.80E Labor relations. In a study to determine employee voting patterns in a recent strike election, 1,000 employees were selected at random and the following tabulation was made: (A) Convert this table to a probability table by dividing each entry by 1,000. (B) What is the probability of an employee voting to strike? Of voting to strike given that the person is paid hourly? (C) What is the probability of an employee being on salary S ? Of being on salary given that he or she voted in favor of striking?Quality control. An automobile manufacturer produces 37 of its cars at plant A. If 5 of the cars manufactured at plant A have defective emission control devices, what is the probability that one of this manufacturer’s cars was manufactured at plant A and has a defective emission control device?Bonus incentives. If a salesperson has gross sales of over $600,000 in a year, then he or she is eligible to play the company’s bonus game: A black box contains 1 twenty-dollar bill, 2 five dollar bills, and 1 one-dollar bill. Bills are drawn out of the box one at a time without replacement until a twenty-dollar bill is drawn. Then the game stops. The sales person’s bonus is 1,000 times the value of the bills drawn. (A) What is the probability of winning a $26,000 bonus? (B) What is the probability of winning the maximum bonus, $31,000, by drawing out all bills in the box? (C) What is the probability of the game stopping at the third draw?Personnel selection. To transfer into a particular technical department, a company requires an employee to pass a screening test. A maximum of 3 attempts are allowed at 6 -month intervals between trials. From past records it is found that 40 pass on the first trial; of those that fail the first trial and take the test a second time, 60 pass; and of those that fail on the second trial and take the test a third time, 20 pass. For an employee wishing to transfer: (A) What is the probability of passing the test on the first or second try? (B) What is the probability of failing on the first 2 trials and passing on the third? (C) What is the probability of failing on all 3 attempts?U.S. Food and Drug Administration. An ice cream company wishes to use a new red dye to enhance the color in its strawberry ice cream. The U.S. Food and Drug Administration (FDA) requires the dye to be tested for cancer-producing potential using laboratory rats. The results of one test on 1,000 rats are summarized in the following table (A) Convert the table into a probability table by dividing each entry by 1,000. (B) Are “developing cancer” and “eating red dye” independent events? (C) Should the FDA approve or ban the use of the dye? Explain why or why not using PCRandPC.Genetics. In a study to determine frequency and dependency of color-blindness relative to females and males, 1,000 people were chosen at random, and the following results were recorded: (A) Convert this table to a probability table by dividing each entry by 1,000. (B) What is the probability that a person is a woman, given that the person is color-blind? Are the events color blindness and female independent? (C) What is the probability that a person is color-blind, given that the person is a male? Are the events color-blindness and male independent?Problems 87 and 88 refer to the data in the following table, obtained in a study to determine the frequency and dependency of IQ ranges relative to males and females. 1,000 people were chosen at random and the following results were recorded: Psychology. (A) What is the probability of a person having an IQ below 90, given that the person is a female? A male? (B) What is the probability of a person having an IQ below 90 ? (C) Are events AandF dependent? AandF ?Problems 87 and 88 refer to the data in the following table, obtained in a study to determine the frequency and dependency of IQ ranges relative to males and females. 1,000 people were chosen at random and the following results were recorded: (A) What is the probability of a person having an IQ above 120, given that the person is a female? A male? (B) What is the probability of a person being female and having an IQ above 120 ? (C) Are events CandF dependent? CandF?Voting patterns. A survey of a precinct’s residents revealed that 55 of the residents were members of the Democratic party and 60 of the Democratic party members voted in the last election. What is the probability that a person selected at random from this precinct is a member of the Democratic party and voted in the last election?Study the probability tree below: (A) Discuss the difference between PMUandPUM, and between PNVandPVN in terms of a,b,c,d,eandf. (B) Show that ac+ad+be+bf=1.What is the significance of this result?Repeat Example 1, but find PU1WandPU2W.What is the probability that a person has tuberculosis given that the test indicates no tuberculosis is present? (That is, what is the probability of the skin test giving a false negative result?) What is the probability that a person does not have tuberculosis given that the test indicates no tuberculosis is present?In Example 3, what is the probability that a defective refrigerator in the warehouse was produced at plant B ? At plant C ?In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1.) 1313+12 In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1.) 2714+27 In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1.) 1313+12 In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms. (If necessary, review Section A.1.) 2714+27 In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms.(If necessary, review Section A.1.) 45341513+4534 In Problems 1-6, write each expression as a quotient of integers, reduced to lowest terms.(If necessary, review Section A.1.) 15231523+4514 7E 8E 9E 10E 11E 12E Find the probabilities in Problems 13-16 by referring to the following Venn diagram and using Bayes’ formula (assume that the simple events in S are equally likely): PU1R Find the probabilities in Problems 13-16 by referring to the following Venn diagram and using Bayes’ formula (assume that the simple events in S are equally likely): PU2R Find the probabilities in Problems 13-16 by referring to the following Venn diagram and using Bayes’ formula (assume that the simple events in S are equally likely): PU1R Find the probabilities in Problems 13-16 by referring to the following Venn diagram and using Bayes’ formula (assume that the simple events in S are equally likely): PU2R 17E Find the probabilities in Problems 17-22 by referring to the following tree diagram and using Bayes’ formula. Round answers to three decimal places. PVC 19E Find the probabilities in Problems 17-22 by referring to the following tree diagram and using Bayes’ formula. Round answers to three decimal places. PUC Find the probabilities in Problems 17-22 by referring to the following tree diagram and using Bayes’ formula. Round answers to three decimal places. PVC Find the probabilities in Problems 17-22 by referring to the following tree diagram and using Bayes’ formula. Round answers to three decimal places. PWC 23E 24E 25E 26E 27E 28E In Problems 29 and 30, use the probabilities in the first tree diagram to find the probability of each branch of the second tree diagram.In Problems 29 and 30, use the probabilities in the first tree diagram to find the probability of each branch of the second tree diagram.In Problems 31-34, one of two urns is chosen at random, with one as likely to be chosen as the other. Then a ball is withdrawn from the chosen urn. Urn 1 contains 1 white and 4 red balls, and urn 2 has 3 white and 2 red balls. If a white ball is drawn, what is the probability that it came from urn 1 ?In Problems 31-34, one of two urns is chosen at random, with one as likely to be chosen as the other. Then a ball is withdrawn from the chosen urn. Urn 1 contains 1 white and 4 red balls, and urn 2 has 3 white and 2 red balls. If a white ball is drawn, what is the probability that it came from urn 2 ?In Problems 31-34, one of two urns is chosen at random, with one as likely to be chosen as the other. Then a ball is withdrawn from the chosen urn. Urn 1 contains 1 white and 4 red balls, and urn 2 has 3 white and 2 red balls. If a red ball is drawn, what is the probability that it came from urn 2 ?In Problems 31-34, one of two urns is chosen at random, with one as likely to be chosen as the other. Then a ball is withdrawn from the chosen urn. Urn 1 contains 1 white and 4 red balls, and urn 2 has 3 white and 2 red balls. If a red ball is drawn, what is the probability that it came from urn 1 ?In Problems 35 and 36, an urn contains 4 red and 5 white balls. Two balls are drawn in succession without replacement. If the second ball is white, what is the probability that the first ball was white?In Problems 35 and 36, an urn contains 4 red and 5 white balls. Two balls are drawn in succession without replacement. If the second ball is red, what is the probability that the first ball was red?In Problems 37 and 38, urn 1 contains 7 red and 3 white balls. Urn 2 contains 4 red and 5 white balls. A ball is drawn from urn 1 and placed in urn 2. Then a ball is drawn from urn 2. If the ball drawn from urn 2 is red, what is the probability that the ball draw n from urn 1 was red?In Problems 37 and 38, urn 1 contains 7 red and 3 white balls. Urn 2 contains 4 red and 5 white balls. A ball is drawn from urn 1 and placed in urn 2. Then a ball is drawn from urn 2. If the ball drawn from urn 2 is white, what is the probability that the ball drawn from urn 1 was white?In Problems 39 and 40 refer to the following probability tree: c+d=1a+b=1a,b,c,d,e,f0e+f=1 Suppose that c=e. Discuss the dependence or independence of events U and M.In Problems 39 and 40 refer to the following probability tree: c+d=1a+b=1a,b,c,d,e,f0e+f=1 Suppose that c=d=e=f. Discuss the dependence or independence of events MandN.In Problems 41 and 42, two halls are drawn in succession front an urn containing m blue balls and n white balls ( m2andn2 ). Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. (A) If the two balls are drawn with replacement, then PB1B2=PB2B1. (B) If the two balls are drawn without replacement, then PB1B2=PB2B1.In Problems 41 and 42, two halls are drawn in succession front an urn containing m blue balls and n white balls ( m2andn2 ). Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. (A) If the two balls are drawn with replacement, then PB1W2=PW2B1. (B) If the two balls are drawn without replacement, then PB1W2=PW2B1.If 2 cards are drawn in succession from a standard 52-card deck without replacement and the second card is a heart, what is the probability that the first card is a heart?A box contains 10 balls numbered 1 through 10. Two balls are drawn in succession without replacement. If the second ball drawn has the number 4 on it, what is the probability that the first ball had a smaller number on it? An even number on it?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. What is the probability of drawing a red card on the second draw?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. What is the probability of drawing a black card on the second draw?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. If the second card drawn is red, what is the probability that the first card drawn was red?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. If the second card drawn is black, what is the probability that the first card drawn was red?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. If the second card drawn is red, what is the probability that the first card drawn was black?In Problems 45-50, a player is dealt two cards from a 52- deck. If the first card is black, the player returns it to the deck before drawing the second card. If the first card is red, the player sets it aside and then draws the second card. If the second card drawn is black, what is the probability that the first card drawn was black?Show that PU1R+PU1R=1.If U1andU2 are two mutually exclusive events whose union is the equally likely sample space S and if E is an arbitrary event in S such that PE0, show that PU1E=nU1EnU1E+nU2E Employee screening. The management of a company finds that 30 of the administrative assistants hired are unsatisfactory. The personnel director is instructed to devise a test that will improve the situation. One hundred employed administrative assistants are chosen at random and are given the newly constructed test. Out of these, 90 of the satisfactory administrative assistants pass the test and 20 of the unsatisfactory administrative assistants pass. Based on these results, if a person applies for a job, takes the test, and passes it, what is the probability that he or she is a satisfactory administrative assistant? If the applicant fails the test, what is the probability that he or she is a satisfactory administrative assistant?Employee rating. A company has rated 75 of its employees as satisfactory and 25 as unsatisfactory. Personnel records indicate that 80 of the satisfactory workers had previous work experience, while only 40 of the unsatisfactory workers had any previous work experience. If a person with previous work experience is hired, what is the probability that this person will be a satisfactory employee? If a person with no previous work experience is hired, what is the probability that this person will be a satisfactory employee?Product defects. A manufacturer obtains GPS systems from three different subcontractors: 20 from A. 40 from B. and 40 from C. The defective rates for these subcontractors are 1, 3, and 2, respectively. If a defective GPS system is returned by a customer, what is the probability that it came from subcontractor A ? From B ? From C ?Product defects. A store sells three types of flash drives: brand A, brand B, and brand C. Of the flash drives it sells, 60 are brand A. 25 are brand B. and 15 are brand C. The store has found that 20 of the brand A flash drives, 15 of the brand B flash drives, and 5 of the brand C flash drives are returned as defective. If a flash drive is returned as defective, what is the probability that it is a brand A flash drive? A brand B flash drive? A brand C flash drive?Cancer screening. A new, simple test has been developed to detect a particular type of cancer. The test must be evaluated before it is used. A medical researcher selects a random sample of 1,000 adults and finds (by other means) that 2 have this type of cancer. Each of the 1,000 adults is given the test, and it is found that the test indicates cancer in 98 of those who have it and in 1 of those who do not. Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer? Of a person having cancer given that the test does not indicate cancer?Pregnancy testing. In a random sample of 200 women who suspect that they are pregnant, 100 turn out to be pregnant. A new pregnancy test given to these women indicated pregnancy in 92 of the 100 pregnant women and in 12 of the 100 nonpregnant women. If a woman suspects she is pregnant and this test indicates that she is pregnant, what is the probability that she is pregnant? If the test indicates that she is not pregnant, what is the probability that she is not pregnant?Medical research. In a random sample of 1,000 people, it is found that 7 have a liver ailment. Of those who have a liver ailment, 40 are heavy drinkers, 50 are moderate drinkers, and 10 are nondrinkers. Of those who do not have a liver ailment, 10 are heavy drinkers. 70 are moderate drinkers, and 20 are nondrinkers. If a person is chosen at random and he or she is a heavy drinker, what is the probability of that person having a liver ailment? What is the probability for a nondrinker?Tuberculosis screening. A test for tuberculosis was given to 1,000 subjects, 8 of whom were known to have tuberculosis. For the subjects who had tuberculosis, the test indicated tuberculosis in 90#37; of the subjects, was inconclusive for 7, and indicated no tuberculosis in 3. For the subjects who did not have tuberculosis, the test indicated tuberculosis in 5 of the subjects, was inconclusive for 10, and indicated no tuberculosis in the remaining 85. What is the probability of a randomly selected person having tuberculosis given that the test indicates tuberculosis? Of not having tuberculosis given that the test was inconclusive?Police science. A new lie-detector test has been devised and must be tested before it is used. One hundred people are selected at random, and each person draws a card from a box of 100 cards. Half the cards instruct the person to lie, and the others instruct the person to tell the truth. Of those who lied, 80 fail the new lie-detector test (that is, the test indicates lying). Of those who told the truth. 5 failed the test. What is the probability that a randomly chosen subject will have lied given that the subject failed the test? That the subject will not have lied given that the subject failed the test?Politics. In a given county, records show that of the registered voters, 45 are Democrats, 35 are Republicans, and 20 are independents. In an election, 70 of the Democrats, 40 of the Republicans, and 80 of the independents voted in favor of a parks and recreation bond proposal. If a registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Republican? An independent? A Democrat?From Example 1 we can conclude that the probability is 0 that a single roll of a fair die will equal the expected value for a roll of a die (the number of dots facing up is never 3.5 ). What is the probability that the sum for a single roll of a pair of dice will equal the expected value of the sum for a roll of a pair of dice?Suppose that the die in Example 1 is not fair and we obtain (empirically) the following probability distribution for X : [Note: sum = 1]. What is the expected value of X ?Repeat Example 2 using a random sample of 4.Repeat Example 3 with the player winning $5 instead of $4 if the chosen number turns up. The loss is still $1 if any other number turns up. Is this a fair game?Find the expected value in Example 4 from the insurance company’s point of view.In Example 5, what is the insurance company’s expected value if it writes the policy?In Problems 1-8, if necessary', review Section B.1. Find the average (mean) of the exam scores 73,89,45,82and66.2E In Problems 1-8, if necessary', review Section B.1. Find the average (mean) of the exam scores in Problem 1, if 4 points are added to each score.4E 5E 6E In Problems 1-8, if necessary', review Section B.1. If the probability distribution for the random variable X is given in the table, what is the expected value of X ?In Problems 1-8, if necessary', review Section B.1. If the probability distribution for the random variable X is given in the table, what is the expected value of X ?You draw and keep a single bill from a hat that contains a $5,$20,$50,and$100 bill. What is the expected value of the game to you?You draw and keep a single bill from a hat that contains a $1,$10,$20,$50,and$100 bill. What is the expected value of the game to you?You draw and keep a single coin from a bowl that contain 15pennies,10dimes,and25quarters What is the expected value of the game to you?You draw and keep a single coin from a bowl that contains 120 nickels and 80 quarters. What is the expected value of the game to you?13E You draw a single card from a standard 52-card deck. If it is an ace, you win $104. Otherwise you get nothing. What is the expected value of the game to you?15E

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