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College Mathematics for Business Economics Life Sciences and Social Sciences 12th edition Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen - Solutions
The theoretical probability of an event is less than or equal to its empirical probability. In Problem, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example.
If E and F are mutually exclusive events, then E and F are complementary. In Problem, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example.
A number divisible by 3 in a single roll of a die In Problem, compute the odds in favor of obtaining
1 head when a single coin is tossed twice In Problem, compute the odds in favor of obtaining
2 heads when a single coin is tossed twice In Problem, compute the odds against obtaining
An odd number or a number divisible by 3 in a single roll of a die In Problem, compute the odds against obtaining
(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) The sum is a prime number or is exactly divisible by 4.(B) The sum is an odd number or exactly divisible by 3.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
In Problem, a single card is drawn from a standard 52-card deck. Calculate the probability of and odds for each event. A king or a heart is drawn.
A heart or a number less than 7 (count an ace as 1) is drawn. In Problem, a single card is drawn from a standard 52-card deck. Calculate the probability of and odds for each event.
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 600 positive integers is (exactly) divisible by 6 or 9?
In a group of n people (n ≤ 100), 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?
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
Refer to Problem 71. 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? Problem 71 From a survey involving 1,000 university students, a market research company found that 750
Use the (empirical) probabilities in Problem 73 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
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 trouble shooting if 1 or more in the sample are found to be defective. What is the probability that
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?
If a state resident is selected at random, what is the (empirical) probability that the resident is (A) A Democrat or prefers candidate fl? What are the odds for this event? (B) Not a Democrat and has no preference? What are the odds against this event? Problem refer to the data in the following
In Problem, test each pair of events for independence: A and E
In Problem, test each pair of events for independence: C and F
A fair die is rolled 5 times. (A) What is the probability of getting 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?
Repeat Problem 23 with the following events:E = pointer lands on an odd number F = pointer lands on a prime numberProblem 23A pointer is spun once on the circular spinner shown below. The probability assigned to the pointer landing on a given integer (from 1 to 5) is the ratio of the area of the
Compute the indicated probabilities in Problem by referring to the following probability tree:(A) P(N ˆ© R) (B) P(S)
For each pair of event , discuss whether they are independent and whether they are mutually exclusive. (A) E1 and E3 (B) E3andE4
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 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 M and TV are M = the drawn card is a diamond. N = the drawn card is even (face cards are not valued). (A) Find P(N|M). (B) Test M and N for independence.
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.
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 Problem refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balk Let Ri be the event
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 Problem refer to the following experiment: 2 balls are drawn in succession out of a box containing 2 red and 5 white balk Let Ri be the
If A and B are events, then P(A|B) = P(B|A). In Problem, 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 P(A|B)≤ P(B). In Problem, 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 Problem, 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 (m ≥ 1 and n ≥ l),then P(W1 ∩ R2) = P(R1 ∩ W2) In Problem, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample.
For the experiment in Problem 49, what is the probability that no white balls are drawn? Problem 49 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?
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 \. 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.
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?
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
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 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
Find the probabilities in Problem by referring to the following Venn diagram and using Bayes' formula (assume that the simple events in S are equally likely):P{U2|R')
Find the probabilities in Problem by referring to the following tree diagram and using Bayes' formula. Round answers to three decimal placesP(V|Cʹ)
Find the probabilities in Problem by referring to the following Venn diagram and using Bayes' formula (assume that the simple events in S are equally likely):P(U2|R')
Find the probabilities in Problem by referring to the following Venn diagram and using Bayes' formula (assume that the simple events in S are equally likely):P(U1|R')
Find the probabilities in Problem by referring to the following Venn diagram and using Bayes' formula (assume that the simple events in S are equally likely):P(U3|R')
In Problem, use the probabilities in the first tree diagram to find the probability of each branch of the second tree diagram.
If a white ball is drawn, what is the probability that it came from urn 2? In Problem, 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 Problem, 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.
In Problem, an urn contains 4 red and 5 white balls. Two balls are drawn in succession without replacement. In the second ball is red, what is the probability that the first ball was red?
If the ball drawn from urn 2 is white, what is the probability that the ball drawn from urn 1 was white? In Problem, 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.
(A) If the two balls are drawn with replacement, then P(B1|W2) = P(W2|B1). (B) If the two balls are drawn without replacement, then P{B1|W2)=P(W2|B1). In Problems, two balls are drawn in succession from an urn containing m blue balls and n white balls (m ≥ 2 and n ≥ 2). Discuss the validity of
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?
If the chosen card is a club, what is the probability that exactly two of the cards in the hand are clubs? In Problem, a 3- card hand is dealt from a standard 52- card deck, and then one of the 3 cards is chosen at random.
If the chosen card is not a club, what is the probability that none of the cards in the hand is a club? In Problem, a 3- card hand is dealt from a standard 52- card deck, and then one of the 3 cards is chosen at random.
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
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
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 non pregnant women. If a woman suspects she is pregnant and this
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% of the subjects, was inconclusive for 7%, and indicated no tuberculosis in 3%. For the subjects who did not have
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
Find the probabilities in Problems by referring to the tree diagram below.
Find the probabilities in Problem by referring to the following Venn diagram and using Bayes' formula (assume that the simple events in S are equally likely):P(U2|R)
In Problem 9, for the game to be fair, how much should you lose if a head and a tail turn up? Problem 9 Two coins are flipped. You win $2 if either 2 heads or 2 tails turn up; you lose $3 if a head and a tail turn up. What is the expected value of the game?
Find the probabilities in Problem by referring to the following tree diagram and using Bayes' formula. Round answers to three decimal placesP(V|Cʹ)
A single die is rolled once. You lose $12 if a number divisible by 3 turns up. How much should you win if a number not divisible by 3 turns up in order for the game to be fair? Describe the process and reasoning used to arrive at your answer.
A coin is tossed three times. Suppose you lose $3 if 3 heads appear, lose $2 if 2 heads appear, and win $3 if 0 heads appear. How much should you win or lose if 1 head appears in order for the game to be fair?
A card is drawn from a standard 52-card deck. If the card is a diamond, you win $10; otherwise, you lose $4. What is the expected value of the game?
A 5-card hand is dealt from a standard 52-card deck. If the hand contains at least one diamond, you win $10; otherwise, you lose $4. What is the expected value of the game?
The payoff table for three possible courses of action is given below. Which of the three actions will produce the largest expected value? What is it?
In roulette, the numbers from 1 to 36 are evenly divided between red and black. A player who bets $1 on black wins $1 (and gets the $1 bet back) if the ball comes to rest on black; otherwise (if the ball lands on red, 0 or 00), the $1 bet is lost. What is the expected value of the game?
A game has an expected value to you of -$0.50. It costs $2 to play, but if you win, you receive $20 (including your $2 bet) for a net gain of $18. What is the probability of winning? Would you play this game? Discuss the factors that would influence your decision.
Ten thousand raffle tickets are sold at $2 each for a local library benefit. Prizes are awarded as follows: 2 prizes of $1,000, 4 prizes of $500, and 10 prizes of $100. What is the expected value of this raffle if you purchase 1 ticket?
A box of 8 flashbulbs contains 3 defective bulbs. A random sample of 2 is selected and tested. Let X be the random variable associated with the number of defective bulbs in a sample. (A) Find the probability distribution of X. (B) Find the expected number of defective bulbs in a sample.
Repeat Problem 31 with the purchase of 10 tickets. Problem 31 One thousand raffle tickets are sold at $1 each. Three tickets will be drawn at random (without replacement), and each will pay $200. Suppose you buy 5 tickets. (A) Create a payoff table for 0,1,2, and 3 winning tickets among the 5
A 3-card hand is dealt from a standard deck. You win $100 for each king in the hand. If the game is fair, how much should you lose if the hand contains no kings?
Repeat Problem 37 from the point of view of the insurance company. Problem 37 The annual premium for a $5,000 insurance policy against the theft of a painting is $150. If the (empirical) probability that the painting will be stolen during the year is .01, what is your expected return from the
In a family with 2 children, excluding multiple births and assuming that a boy is as likely as a girl at each birth, what is the expected number of boys?
Repeat Problem 39, assuming that additional analysis caused the estimated probability of success in field B to be changed from .1 to .11. Problem 39 After careful testing and analysis, an oil company is considering drilling in two different sites. It is estimated that site A will net $30 million if
A pink-flowering plant is of genotype RW. If two such plants are crossed, we obtain a red plant (RR) with probability .25, a pink plant (RW or WR) with probability .50, and a white plant (WW) with probability .25, as shown in the table. What is the expected number of W genes present in a crossing
Repeat Problem 5, assuming an unfair coin with the probability of a head being .55 and a tail being .45. Problem 5 A fair coin is flipped. If a head turns up, you win $ 1. II a tail turns up, you lose $1. What is the expected value of the game? Is the game fair?
On three rolls of a single die, you will lose $10 if a 5 turns i at least once, and you will win $7 otherwise. What is the expected value of the game?
In problem find S1 for the indicated initial- state matrix S0 and interpret with a tree diagram.S0 = [0 1]Problem refer to the following transition matrix:
In Problem could the given matrix be the transition matrix of a Markov chain?
In Problem, could the given matrix be the transition matrix of Markov chain?
In Problem, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.
In Problem, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.
In Problem, is there a unique way of filling in the missing probabilities in the transition diagram? If so, complete the transition diagram and write the corresponding transition matrix. If not, explain why.
In Problem, are there unique values of a, b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why.
In Problem, are there unique values of a, b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why.
In Problem, are there unique values of a, b, and c that make P a transition matrix? If so, complete the transition matrix and draw the corresponding transition diagram. If not, explain why.
A Markov chain has two states, A and B. The probability of going from state A to state A in one trial is .6, and the probability of going from state B to state B in one trial is .2. In Problem, use the given information to draw the transition diagram and find the transition matrix.
In problem find S1 for the indicated initial- state matrix S0 and interpret with a tree diagram.S0 = [.3 .7 ]Problem refer to the following transition matrix:
A Markov chain has three states, A, B, and C. The probability of going from state A to state B in one trial is 1. The probability of going from state B to state A in one trial is .5, and the probability of going from state B to state C in one trial is .5. The probability of going from state C to
Problem refer to the following transition matrix P and its powers:Find S2 for S0 = [0 1 0] and explain what it represents.
Problem refer to the following transition matrix P and its powers:Find S3 for S0 = [1 0 0 ] and explain what it represents.
In Problem, given the transition matrix P and initial - state matrix S0, find P4 to find S4.
In Problem, given the transition matrix P and initial - state matrix S0, find P4 to find S4.
In Problem find S2 for the indicated initial - state matrix S0, and explain what it represents.S0 = [0 1]Problem refer to the following transition matrix:
Repeat Problem 63 for transition matrixProblem 63 Given the transition matrix (A) Find P4. (B) Find the probability of going from state A to state D in four trials. (C) Find the probability of going from state C to state B in four trials. (D) Find the probability of going from state B to state A in
Show that ifAnd S = [ c 1 -c ] Are probability matrices, then SP is a probability matrix.
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