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Questions and Answers of Mathematics Economics Business

(A) Use slack variables s1 and s2 to convert the following i-system (system of inequalities) to an e-system (system of equations).(B) Find the basic solution for which x1 = 0 and s1 = 0.(C) Find the
Form the dual problem: Minimize subject to C = 16x₁ + 9x2 + 21x3 x₁ + x₂ + 3x3 = 12 2x₁ + x₂ + x3 ≥ 16 X1, X2, X30
Refer to the partially completed table of the six basic solutions to the e-systemIn basic solution (B), which variables are basic? (A) (B) (C) (D) (E) (F) 2x1 + 5x2 + S₁ = 32 x₁ + 2x₂ + $₂ =
Construct the table of basic solutions and use it to solve the following linear programming problem: Maximize P = 30x₁ + 40x₂ subject to 2x₁ + 3x₂ = 24 4x₁3x₂36 X1, X₂ = 0
In Problem 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.A 6-card hand that contains exactly two
In Problem 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.A 6-card hand that contains exactly two
In Problem 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.A 4-card hand that contains no acesData
In Problem 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.A 7-card hand that contains exactly 2
In Problem 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.A 7-card hand that contains exactly 1 king
In Problem 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 1 to 6,
In Problem 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 1 to 6,
In Problem a player is dealt two cards from a 52-card 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
An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 and, assuming each simple event is as likely as any other,
In Problem a player is dealt two cards from a 52-card 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
An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 and, assuming each simple event is as likely as any other,
In Problem a player is dealt two cards from a 52-card 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
An insurance company charges an annual premium of $75 for a $200,000 insurance policy against a house burning down. If the (empirical) probability that a house burns down in a given year is .0003,
A card is drawn at random from a standard 52-card deck. Using a graphing calculator to simulate 800 such draws, determine the empirical probability that the card is a black jack and compare with the
An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 and, assuming each simple event is as likely as any other,
Show that P(U1|R) + P(U1′∙R) = 1.
An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 and, assuming each simple event is as likely as any other,
An experiment consists of rolling two fair dice and adding the dots on the two sides facing up. Using the sample space shown in Figure 2 and, assuming each simple event is as likely as any other,
A $2 Powerball lottery ticket has a 1/27.05 probability of winning $4, a 1/317.39 probability of winning $7, a 1/10,376.47 probability of winning $100, a 1/913,129.18 probability of winning $50,000,
Repeat Problem 55, assuming that the Grand Prize is currently $400,000,000.Data from problem 55A $2 Powerball lottery ticket has a 1/27.05 probability of winning $4, a 1/317.39 probability of winning
Show that the solution to the birthday problem in Example 5 can be written in the formFor a calculator that has a nPr function, explain why this form may be better for direct evaluation than the
What is the probability of getting at least 1 diamond in a 5-card hand dealt from a standard 52-card deck?
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
In Problem 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.A 5-card hand that consists entirely of red
In Problem 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.A 5-card hand that consists entirely of
In Problem 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.A 4-card hand that contains no face
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
Use matrix inverse methods to solve the system: 3x1 - x2 + x3 = 1 = -X1 + X2 = 3 X1 + x3 = 2
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
Solve each of the following systems by graphing: (A) x + y = 4 2x - y = 2 - (B) 6x - 3у = 9 2х - y = 3 (C) 2x - y = 4 6х Зу = -18
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
Show that P(A|A) = 1 when P(A'|B) ≠ 0.
An experiment consists of rolling two fair (not weighted) 4-sided dice and adding the dots on the two sides facing up. Each die is numbered 1–4. Compute the probability of obtaining the indicated
Show that P(A|B) + P(A′|B) = 1.
Subtract: [2 -3 5] - [3 -2 1]
Use matrix inverse methods to solve the system: X1 - -X2 + x3 = 1 2x2x3 = 2x1 + 3x2 1 1 (1)
An economy is based on three sectors, agriculture (A), energy (E), and manufacturing (M). Production of a dollar’s worth of agriculture requires an input of $0.20 from the agriculture sector and
Refer to the following matrices:How many elements are there in B? In D? -4 0 A = [2 19 6 -5 B = D -19 0 8 7 2 4 0 C = [2 -3 0] || -4 ∞
(A) Suppose that the square matrix M has a row of all zeros. Explain why M has no inverse.(B) Suppose that the square matrix M has a column of all zeros. Explain why M has no inverse.
The solution of Example 3 involved three augmented matrices. Write the linear system that each matrix represents, solve each system graphically, and discuss the relationships among these
Refer to Example 6. The rental company charges $19.95 per day for a 10-foot truck, $29.95 per day for a 14-foot truck, and $39.95 per day for a 24-foot truck. Which of the four possible choices in
In Problem find the additive inverse and the multiplicative inverse, if defined, of each real number.(A) –7 (B) 2 (C) –1
Solve by Gauss–Jordan elimination:2x1 – 2x2 + x3 = 33x1 + x2 – x3 = 7x1 – 3x2 + 2x3 = 0
Find the inverse, if it exists, of the matrix M = 0 2 -1 2 3 -1 0
Ms. Smith and Mr. Jones are salespeople in a new-car agency that sells only two models. August was the last month for this year’s models, and next year’s models were introduced in September.
In Problem solve each equation for x, where x represents a real number4x = 9
In Problem solve each equation for x, where x represents a real number.x = 4x – 5
In Problem write the augmented matrix of the system of linear equations.4x1 + x2 = 83x1 – 5x2 = 6x1 + 9x2 = 4
Let(A) Form the augmented matrix [M | I].(B) Use row operations to transform [M | I] into [I | B].(C) Verify by multiplication that B = M–1 (that is, show that BM = I). M= = 3 −1 1 -1 1 10 0 1
Find a, b, c, and d so that a b [2]-[11 C d -4 2 -3 ]= -2 5 8 2
Solve by Gauss–Jordan elimination:2x1 – 2x2 + x3 = 33x1 + x2 – x3 = 7x1 – 3x2 + 2x3 = 0
Solve by Gauss–Jordan elimination:2x1 – 4x2 + x3 = -44x1 – 8x2 + 7x3 = 2–2x1 + 4x2 – 3x3 = 5
Use matrix inverse methods to solve each of the following systems: (A) x₁ x₂ + x3 = 3 2x2x3 = 1 = 4 2x₁ + 3x₂ (B) x₁ - x₂ + x3 = -5 2x2x3 = 2 2x1 + 3x2 = -3
Refer to the following matrices:What is the size of B? Of D? 2 -4 0 A=8 19 6 -5 B = D -19 0 8 7 2 4 0 C = [2 -3 0] || -4 ∞
Use matrix inverse methods to solve each of the following systems (see Matched Problem 2):Data from Matched Problem 2Use matrix inverse methods to solve the system: (A) 3x₁x2 + x3 = 3 = -3 -x₁ +
Solve to two decimal places using graphical approximation techniques on a graphing calculator: 2x 2х - 5y = -25 5 4x + Зу =
In Problem find the additive inverse and the multiplicative inverse, if defined, of each real number.(A) 2/3(B) –1/7(C) 1.6
Graph(A) y > –3 (B) 2x ≤ 5 (C) x ≤ 3y
Solve the system in Problem 16 by writing the system as a matrix equation and using the inverse of the coefficient matrix (see Problem 15). Also, solve the system if the constants 3 and 5 are
Graph(A) y < 4 (B) 4x ≥ –9 (C) 3x ≥ 2y
Solve the following linear programming problem using the simplex method: Maximize subject to P = 10x₁ + 5x₂ 4x₁ + x₂ = 28 2x+3x₂ 24 = X1, X₂ = 0
Solve the following linear programming problem using the simplex method: Maximize subject to P = 2x₁ + x₂ 5x₁ + x₂ = 9 x₁ + x₂ ≤ 5 X1, X₂0
Is the point (21, 25) in the solution set of 30x – 27y ≤ 1?
The following linear programming problem has only one problem constraint:Solve it by the table method, then solve it by graphing, and compare the two solutions. Maximize subject to P = 2x₁ +
What is the expected value (long-run average) of the number of dots facing up for the roll of a single die?
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 P(R ∩ A) and P(A|R).
Study the probability tree below:(A) Discuss the difference between P(M|U) and P(U|M), and between P(N|V) and P(V|N), in terms of a, b, c, d, e, and f.(B) Show that ac + ad + be + bf = 1. What is the
Suppose that the die in Example 1 is not fair and we obtain (empirically) the following probability distribution for X:What is the expected value of X?Data from Example 1What is the expected value
A pointer is spun once on a circular spinner (Fig. 3). The probability assigned to the pointer landing on a given integer (from 1 to 6) is the ratio of the area of the corresponding circular sector
Suppose that two fair dice are rolled.(A) What is the probability that a sum of 7 or 11 turns up?(B) What is the probability that both dice turn up the same or that a sum less than 5 turns up?
Find the average (mean) of the exam scores 78, 64, 97, 60, 86, and 83.
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
Brittani and Ramon are members of a 15-person ski club. If the president and treasurer are selected by lottery, what is the probability that Brittani will be president and Ramon will be treasurer? (A
A new, inexpensive skin test is devised for detecting tuberculosis. To evaluate the test before it is used, a medical researcher randomly selects 1,000 people. Using precise but more expensive
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?)
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
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
An experiment consists of recording the boy–girl composition of a two-child family. What would be an appropriate sample space(A) If we are interested in the genders of the children in the order of
In college basketball, would it be reasonable to assume that the following events are independent? Explain why or why not.A = the Golden Eagles win in the first round of the NCAA tournament.B = the
Suppose that city records produced the following probability data on a driver being in an accident on the last day of a Memorial Day weekend:(A) Find the probability of an accident, rain or no
A carton of 20 laptop batteries contains 2 defective ones. A random sample of 3 is selected from the 20 and tested. Let X be the random variable associated with the number of defective batteries
Find the average (mean) of the exam scores in Problem 1, if 4 points are added to each score.Data from Problem 1Find the average (mean) of the exam scores 73, 89, 45, 82, and 66.
What is the probability that a number selected at random from the first 500 positive integers is (exactly) divisible by 3 or 4?
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?Data from
A company produces 1,000 refrigerators a week at three plants. Plant A produces 350 refrigerators a week, plant B produces 250 refrigerators a week, and plant C produces 400 refrigerators a week.
In Example 3, what is the probability that a defective refrigerator in the warehouse was produced at plant B? At plant C?Data from Example 3A company produces 1,000 refrigerators a week at three
Consider an experiment of rolling two dice. Figure 2 shows a convenient sample space that will enable us to answer many questions about interesting events. Let S be the set of all ordered pairs in
What is the probability that a number selected at random from the first 140 positive integers is (exactly) divisible by 4 or 6?
Refer to the sample space shown in Figure 2. What is the event that corresponds to each of the following outcomes?(A) A sum of 5 turns up.(B) A sum that is a prime number greater than 7 turns up.
A spinner device is numbered from 0 to 5, and each of the 6 numbers is as likely to come up as any other. A player who bets $1 on any given number wins $4 (and gets the $1 bet back) if the pointer
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?Data from Example 3A spinner device is
Two balls are drawn in succession, without replacement, from a box containing 3 blue and 2 white balls (Fig. 4). What is the probability of drawing a white ball on the second draw? Figure 4
If 60% of a department store’s customers are female and 75% of the female customers have credit cards at the store, what is the probability that a customer selected at random is a female and has a

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