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Math-132

Fall 2017 Final Exam Review

Section 1.1

#1) Find the equation of a line with slope 1

2 that passes through the point (3, 1), and write the equation

in slope-intercept form.

[answer: 𝒚 =𝟏

𝟐𝒙 −

𝟏

𝟐]

#2) Find the equation of the line that connects the points (−6, 2) 𝑎𝑛𝑑 (−1,−2), and write the

equation in slope-intercept form.

[answer: 𝒚 = −𝟒

𝟓𝒙 −

𝟏𝟖

𝟓]

#3) Find the equation of the line that passes through the points (2, −1) 𝑎𝑛𝑑 (0, 5), and write the equation in slope-intercept form. [𝒂𝒏𝒔𝒘𝒆𝒓: 𝒚 = −𝟑𝒙 + 𝟓 ] Section 1.2 #4) Determine the solution of each of the following systems and put your solution in point form (𝑥, 𝑦).

a) {4𝑥 − 2𝑦 = 164𝑥 + 2𝑦 = 0

[answer= (𝟐,−𝟒)]

b) {4𝑥 + 3𝑦 = 22𝑥 − 𝑦 = −1

[answer= (−𝟏

𝟏𝟎,𝟒

𝟓)]

#5) A retiree needs $10,000 per year in supplementary income. He has $150,000 to invest and can invest

and can invest in AA bonds at 10% annual interest or in Savings and Loans certificates of 5% interest per

year. How much money should be invested in each so that he realizes exactly $10,000 in extra income

per year?

[answer= $𝟏𝟎𝟎, 𝟎𝟎𝟎 𝒊𝒏 𝑺𝒂𝒗𝒊𝒏𝒈𝒔 &𝒍𝒐𝒂𝒏𝒔 𝒂𝒏𝒅 $𝟓𝟎, 𝟎𝟎𝟎 𝒊𝒏 𝒃𝒐𝒏𝒅𝒔]

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Section 1.3 #6) A manufacturer produces gameday pennants at a cost of $0.85 per item and sells them for $0.95 per

item. The daily operational overhead is $350.

a) What is the daily cost function?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑪(𝒙) = 𝟎. 𝟖𝟓𝒙 + 𝟑𝟓𝟎]

b) What is the daily revenue function?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑹(𝒙) = 𝟎. 𝟗𝟓𝒙]

c) How many pennants should the manufacturer sell per day in order to break-even?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑𝟓𝟎𝟎]

#7) The supply and demand equations for sugar have been estimated to be given by the equations:

𝑆 = 0.6𝑝 + 0.5

𝐷 = −0.5𝑝 + 2.7

Where 𝑝 is the price in dollars per pound and 𝑆 and 𝐷 are in millions of pounds.

a) Find the market price.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒑 = $𝟐 ]

b) What quantity of supply is required at this market price?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑺 ≅ 𝟏. 𝟕 𝒖𝒏𝒊𝒕𝒔]

#8) It costs a cellphone parts manufacturer $12 per item to produce a certain electronic component. There is a fixed cost of $594 per month for the manufacturer to produce the component. The company sells the components for $18 a piece.

a) What is the total cost function per month for producing the electronic component? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝑪(𝒙) = 𝟏𝟐𝒙 + 𝟓𝟗𝟒 ]

b) What is the total revenue function? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝑹(𝒙) = 𝟏𝟖𝒙 ]

c) After selling how many of the electronic components per month will the manufacturer break-even? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙 = 𝟗𝟗 ]

Section 2.1 #9) Using the method of elimination, solve each of the following systems and put your solution in the form (𝑥, 𝑦, 𝑧).

a) {

2𝑥 + 𝑦 − 3𝑧 = −2−2𝑥 + 2𝑦 + 𝑧 = −93𝑥 − 4𝑦 − 3𝑧 = 15

[answer= (𝟐,−𝟑, 𝟏)

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b) {

3𝑥 − 2𝑦 + 2𝑧 = 67𝑥 − 3𝑦 + 2𝑧 = −12𝑥 − 3𝑦 + 4𝑧 = 0

[answer= 𝑵𝒐 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏]

c) {

𝑥 + 4𝑦 − 3𝑧 = −83𝑥 − 𝑦 + 3𝑧 = 12𝑥 + 𝑦 + 6𝑧 = 1

[answer= (𝟑,−𝟖

𝟑,𝟏

𝟗)]

Section 2.2 & 2.3 #10) The following augmented matrix is already in Row Echelon Form (REF). Find the solution of the system and give the answer in the form (𝑥1, 𝑥2, 𝑥3).

(1 7 60 1 00 0 1

|7−11)

[answer= (𝟖,−𝟏, 𝟏)]

#11) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form (𝑥1, 𝑥2).

(3 21 − 2

|142)

[answer= (𝟒, 𝟏)]

#12) Given the following augmented matrix, use row operations (i.e., the Gaussian elimination) to put the matrix into Row Echelon Form (REF). Then, determine the solution to the system and put your answer in the form (𝑥1, 𝑥2, 𝑥3).

(1 1 13 2 − 13 1 2

|6411)

[answer= (𝟏, 𝟐, 𝟑)]

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Section 3.2

#13) Find the product of the following matrices:

a) [4 − 2 3] [033]

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑 ]

b) [1 9] [5 09 − 5

]

[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟖𝟔 − 𝟒𝟓] ]

c) [1 − 4 52 0 10

] [0 − 51 04 − 10

]

[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟏𝟔 − 𝟓𝟓𝟒𝟎 − 𝟏𝟏𝟎

]]

d) [8 010 − 811 − 1

] [8 19 − 8

]

[𝒂𝒏𝒔𝒘𝒆𝒓: [𝟔𝟒 𝟖𝟖 𝟕𝟒𝟕𝟗 𝟏𝟗

]]

Section 3.3

#14) Using elementary row operations (not the determinant method) to find the inverse of the

following matrices:

a) 𝐴 = [8 164 9

]

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [𝟗

𝟖 −

𝟏

𝟖

−𝟖 𝟏] ]

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b) 𝐴 = [1 − 1 0−3 4 02 0 1

]

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [4 1 03 1 0−8 − 2 1

]]

c) #8) 𝐴 = [1 − 1 0−6 7 08 0 1

]

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨−𝟏 = [7 1 06 1 0−56 − 8 1

]]

Sections 4.1 & 4.2

#15) The given figure illustrates the graph of the set of feasible points of a linear system of inequalities.

Find the minimum and maximum values of the objective function 𝑧 = 3𝑥 + 11𝑦.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒎𝒊𝒏𝒊𝒎𝒖𝒎 = 𝟏. 𝟐 ]

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 = 𝟐𝟏 ]

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#16) Using the geometrical method, find a graphical solution to each of the following linear systems of

inequalities. Then, find the coordinates of the corner points for each solution region.

a)

{

𝑥 ≤ 10𝑦 ≤ 8

4𝑥 + 3𝑦 ≥ 12𝑥 ≥ 0𝑦 ≥ 0

[𝒂𝒏𝒔𝒘𝒆𝒓: (𝟑, 𝟎) (𝟏𝟎, 𝟎) (𝟏𝟎, 𝟖) (𝟎, 𝟖) (𝟎, 𝟒) ]

#17) Use the Geometric method to maximize the objective function 𝑧 = 7𝑥 + 5𝑦 subject to the

constraints: {

𝑥 + 𝑦 ≥ 22𝑥 + 3𝑦 ≤ 6

𝑥 ≥ 0𝑦 ≥ 0

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒂𝒕 (𝟑, 𝟎) 𝒕𝒉𝒆 𝒐𝒃𝒋𝒆𝒄𝒕𝒊𝒗𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒉𝒂𝒔 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒐𝒇 𝟐𝟏 ]

Section 5.2

#18) For each of the following tableaus perform the pivot operations on the circled element and classify

the next tableau that results according to the resulting solution type as “solved,” “no solution,” or

“ready for another set of pivot operations.”

a) (0 2 16 1 00 3 1 0 11 − 1 − 4 0 0

|10180)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒔𝒐𝒍𝒗𝒆𝒅 ]

b)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒏𝒐 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 ]

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c)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒓𝒆𝒂𝒅𝒚 𝒇𝒐𝒓 𝒂𝒏𝒐𝒕𝒉𝒆𝒓 𝒔𝒆𝒕 𝒐𝒇 𝒑𝒊𝒗𝒐𝒕 𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏𝒔 ]

#19) Use the Simplex Method to solve the maximum problem:

Maximize: 𝑝 = 4𝑥1 + 2𝑥2 + 3𝑥3

Subject to the constraints:

𝑥1 + 3𝑥2 + 3𝑥3 ≤ 40

2𝑥1 + 𝑥2 + 3𝑥3 ≤ 14

𝑥1 ≥ 0 𝑥2 ≥ 0 𝑥3 ≥ 0

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙𝟏 = 𝟕 𝒙𝟐 = 𝟎 𝒙𝟑 = 𝟎 𝒑 = 𝟐𝟖 ]

#20) Use the Simplex Method to solve the maximum problem:

Maximize: 𝑝 = 7𝑥1 + 3𝑥2

Subject to the constraints:

𝑥1 + 𝑥2 ≤ 2

2𝑥1 + 4𝑥2 ≤ 5

𝑥1 ≥ 0 𝑥2 ≥ 0

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙𝟏 = 𝟐 𝒙𝟐 = 𝟎 𝒑 = 𝟏𝟒 ]

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Section 5.3 #21) Write the dual problem of the following minimum problem. Do not solve the dual problem.

Minimize 𝐶 = 3𝑥1 + 2𝑥2

Subject to the constraints:

2𝑥1 + 𝑥2 ≥ 6

𝑥1 + 𝑥2 ≥ 4

𝑥1 ≥ 0 𝑥2 ≥ 0

𝒂𝒏𝒔𝒘𝒆𝒓: 𝑴𝒂𝒙𝒊𝒎𝒊𝒛𝒆 𝑷 = 𝟔𝒚𝟏 + 𝟒𝒚𝟐

Subject to the constraints:

𝟐𝒚𝟏 + 𝒚𝟐 ≤ 𝟑

𝒚𝟏 + 𝒚𝟐 ≤ 𝟐

𝒚𝟏 ≥ 𝟎 𝒚𝟐 ≥ 𝟎

#22) Write the dual problem of the following minimum problem. Do not solve the dual problem.

Minimize 𝐶 = 3𝑥1 + 4𝑥2

Subject to the constraints:

8𝑥1 + 9𝑥2 ≥ 5

11𝑥1 + 12𝑥2 ≥ 7

𝑥1 ≥ 0 𝑥2 ≥ 0

𝒂𝒏𝒔𝒘𝒆𝒓: 𝑴𝒂𝒙𝒊𝒎𝒊𝒛𝒆 𝑷 = 𝟓𝒚𝟏 + 𝟕𝒚𝟐

Subject to the constraints:

𝟖𝒚𝟏 + 𝟏𝟏𝒚𝟐 ≤ 𝟑

𝟗𝒚𝟏 + 𝟏𝟐𝒚𝟐 ≤ 𝟒

𝒚𝟏 ≥ 𝟎 𝒚𝟐 ≥ 𝟎

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#23) The following tableau was obtained using the simplex method for optimizing a Minimum problem.

Perform pivot operations on the circled element to obtain the final tableau. Then, using the final

tableau, state the solution of the Minimum problem in terms of variables x1, x2, and the value of the

objective function, C.

a) What is the final tableau? Write it down.

b) What is the solution to the minimum problem?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙𝟏 = 𝟎 𝒙𝟐 = 𝟐 𝑪 = 𝟔 ]

#24) The following tableau was obtained using the simplex method for optimizing a Minimum problem.

Perform pivot operations on the circled element to obtain the final tableau. Then, using the final

tableau, state the solution of the Minimum problem in terms of variables x1, x2, and the value of the

objective function, C.

a) What is the final tableau? Write it down.

b) What is the solution to the minimum problem?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒙𝟏 = 𝟓 𝒙𝟐 = 𝟎 𝑪 = 𝟏𝟎 ]

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Section 7.3

#25) A man has 13 shirts and 5 ties. How many different shirts and tie arrangements can he wear?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟔𝟓 ]

#26) A restaurant offers 3 different salads, 6 different main courses, 13 different desserts, and 7

different drinks. How many different launches are possible?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏𝟔𝟑𝟖 ]

#18) How many different ways can 4 people be seated in a row of 4 seats?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟐𝟒 ]

#27) How many 4-letter code words are possible using the first 5 letters of the alphabet with no letters

repeated? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏𝟐𝟎 ] How many codes are possible when letters are allowed to repeat?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟔𝟐𝟓 ]

Section 7.4

#28) Three letters are picked from the alphabet (repetitions are allowed, assume order is important).

Find the number of outcomes in the sample space.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏𝟕𝟓𝟕𝟔 ]

#29) Alice is going through a pile of applications for admission to the Bachelor’s program, the Master’s

program, and the PhD program in business. She is determined to select one candidate for each program

before she leaves for the day. In how many ways she may select one candidate for each program, given

the applicant’s gender? List the sample space.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟖]

𝑺 = {𝑭𝑭𝑭,𝑭𝑭𝑴,𝑭𝑴𝑭,𝑴𝑭𝑭,𝑴𝑴𝑴,𝑴𝑴𝑭,𝑴𝑭𝑴,𝑭𝑴𝑴}

#30) Assume your favorite football team has 2 games left to finish the lesson. The outcome of each

game can be win, lose, or tie. What is the total number of possible outcomes? List the sample space.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟗]

𝑺 = {𝑾𝑾,𝑳𝑳, 𝑻𝑻,𝑾𝑳,𝑾𝑻, 𝑳𝑾,𝑻𝑾, 𝑳𝑻, 𝑻𝑳}

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#31) Each customer entering a department store will either buy or not buy some merchandise. A

researcher shadows three customers to monitor their behavior in terms of whether or not each

individual buys or does not buy something. How many outcomes are possible? List the sample space.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟖]

𝑺 = {𝑩𝑩𝑩,𝑩𝑩𝑵,𝑩𝑵𝑩,𝑵𝑩𝑩,𝑵𝑵𝑵,𝑵𝑵𝑩,𝑵𝑩𝑵,𝑩𝑵𝑵}

Section 7.5

#32) If events E and F belong to the same sample space, and 𝑃(𝐸) = 0.2, 𝑃(𝐹) = 0.6,

𝑃(𝐸 ∩ 𝐹)=0.40 find 𝑃(𝐸 ∪ 𝐹) =?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨𝒑𝒑𝒍𝒚 𝒕𝒉𝒆 𝒂𝒅𝒅𝒊𝒕𝒊𝒗𝒆 𝒓𝒖𝒍𝒆 ]

#33) If events E and F belong to the same sample space, and 𝑃(𝐸) = 0.7, 𝑃(𝐹) = 0.5,

𝑃(𝐸 ∪ 𝐹)=0.80 find 𝑃(𝐸 ∩ 𝐹) =?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑨𝒑𝒑𝒍𝒚 𝒕𝒉𝒆 𝒂𝒅𝒅𝒊𝒕𝒊𝒗𝒆 𝒓𝒖𝒍𝒆 ]

#34) Anne is taking courses in both mathematics and English. She estimates her probability of passing

mathematics at 0.3 and English at 0.4, and she estimates her probability of passing at least one of

them at 0.53. What is her probability of passing both courses?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟏𝟕 ]

#35) A financial consultant estimates that there is an 11% chance a mutual fund will outperform the

market during any given year. She also estimates that there is a 10% chance that the mutual fund will

outperform the market for the next two years. What is the probability that the mutual fund will

outperform the market in at least one of the next two years?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟏𝟐 ]

#36) Three letters, with repetition allowed, are selected from the alphabet. What is the probability that

none is repeated?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟖𝟖𝟕𝟔 ]

12

#37) Determine the probability of E if the odds in favor of E are 3 to 1.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑬) =𝟑

𝟒 ]

#38) Determine the probability of E if the odds in favor of E are 1 to 1.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑬) =𝟏

𝟐 ]

#39) Determine the odds for and against the event E if 𝑃(𝐸) = 0.6.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝒕𝒉𝒆 𝒐𝒅𝒅𝒔 𝒇𝒐𝒓 𝑬 𝒂𝒓𝒆 𝟑 𝒕𝒐 𝟐 𝒂𝒏𝒅 𝒂𝒈𝒂𝒊𝒏𝒔𝒕 𝑬 𝒂𝒓𝒆 𝟐 𝒕𝒐 𝟑 ]

#40) Determine the odds for and against the event F if 𝑃(𝐹) =3

4

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑶𝒅𝒅𝒔 𝒇𝒐𝒓 𝑭: 𝟑 𝒕𝒐 𝟏 𝒂𝒏𝒅 𝑶𝒅𝒅𝒔 𝒂𝒈𝒂𝒊𝒏𝒔𝒕 𝑭: 𝟏 𝒕𝒐 𝟑 ]

#41) Suppose events A and B are independent, with 𝑃(𝐵) = 0.64 and 𝑃(𝐴 ∩ 𝐵) = 0.16.

a) Find the odds for A.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏 𝒕𝒐 𝟑 ]

b) Find the odds for �̅� (i.e., the odds against A)

c) [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑 𝒕𝒐 𝟏 ]

#42) Events A and B are mutually exclusive, with 𝑃(𝐵) = 0.10 and 𝑃(𝐴 ∪ 𝐵) = 0.82.

a) Find the odds for A

b) [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟒 𝒕𝒐 𝟏 ]

c) Find the odds for �̅� (i.e., the odds against A)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏 𝒕𝒐 𝟒 ]

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Section 7.6

#43)The information about attendance at a football game in a certain city is given in the table below.

How many fans are expected to attend each game?

Extremely Cold Cold Moderate Warm

Attendance 40,000 50,000 70,000 90,000

Probability 0.09 0.41 0.41 0.09

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟔𝟎, 𝟗𝟎𝟎 ]

#44) In a lottery, 1000 tickets are sold at $0.32 each. There are 3 cash prizes: one for $130, one for $80,

and one for $10. Alice buys 7 tickets.

A) Considering the expected value of each ticket, what would have been a fair price for a ticket?

[𝒂𝒏𝒔𝒘𝒆𝒓: $𝟎. 𝟐𝟐 ]

B) In total, how much extra did Alice pay?

[𝒂𝒏𝒔𝒘𝒆𝒓: $𝟎. 𝟕𝟎 ]

#45) A multiple choice test has 65 questions and each question has 6 possible answer choices, only 1

of which is correct. If a student guesses on every question, how many questions should he expect to get

wrong?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟓𝟒 ]

Section 8.1

#46) If E and F are events with P(E)=0.2 P(F)=0.3 𝑃(𝐸 ∩ 𝐹) = 0.1

a) Find P(E|F)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟑𝟑 ]

b) Find P(F|E)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟓𝟎 ]

c) Find 𝑃(𝐸 ∪ 𝐹)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟒𝟎 ]

#47) If E and F are events with 𝑃(𝐸 ∩ 𝐹) = 0.2 and P(E|F)=0.5 what is P(F)=?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑭) = 𝟎. 𝟒 ]

14

#48) Given the following tree diagram probabilities, find the probability of obtaining D in any final

outcome.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟔𝟓𝟓𝟏 ]

#49) A jar contains 6 white marbles, 2 yellow marbles, 2 red marbles, and 5 blue marbles. Two marbles

are picked at random one after another and without replacement.

a) Draw the tree diagram for this experiment.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑫𝒐 𝒚𝒐𝒖 𝒌𝒏𝒐𝒘 𝒉𝒐𝒘 𝒕𝒐 𝒅𝒐 𝒊𝒕? ]

b) What is the probability that both are blue?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟎𝟗𝟓 ]

c) What is the probability that exactly one is blue?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟒𝟕𝟔 ]

d) What is the probability that at least one is blue?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟓𝟕𝟏 ]

#50) If P(E)=0.71 P(F)=0.79 𝑃(𝐸 ∪ 𝐹) = 0.99 find 𝑃(𝐸 ∩ 𝐹)

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑬 ∩ 𝑭) = 𝟎. 𝟓𝟏 ]

#51) If P(E)=0.20 P(F)=0.65 𝑃(𝐸 ∪ 𝐹) = 0.77 P(E|F)=?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑬|𝑭) = 𝟎. 𝟏𝟐𝟑𝟏 ]

A

B

D

C

D

0.11

C

0.89

0.87

0.13

0.28

0.72

15

#52) Assume you have applied to two graduate schools: school A and school B. Suppose it is very

competitive to get into school A and only 8 percent of all applicants are accepted, while the probability

of getting accepted by school B is 0.25.

a) What is the probability of your application being accepted by both schools? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟎𝟐 ]

b) What is the probability of receiving at least one letter of acceptance? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟑𝟏 ]

c) What is the probability that both schools decline your application? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟔𝟗]

#53) If you are fishing in South Puget Sound in the state of Washington in October, there is a 30%

chance that you’ll catch Salmon, and a 5% chance that you’ll catch Flat Fish. You are there to catch two

fish before heading back home. Assuming that catching the fish is mutually exclusive:

a) What is the probability that you’ll catch either a Salmon or a Flat Fish?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟑𝟓 ]

b) What is the probability that you won’t be able to catch either kind of fish?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟔𝟓 ]

#54) A survey of magazine subscribers showed that 63.2% rented a car during the past 12 months for

business reasons, 36% rented a car during the past 12 months for personal reasons, and 23% rented a

car during the past 12 months for both business and personal reasons.

a) What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟕𝟔𝟐]

b) What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟐𝟑𝟖 ]

16

#55) The following table of data is the result of a survey of 574 individuals at a shopping mall:

Likes the deodorant Does not like the deodorant No opinion

Group I 167 65 24

Group II 100 85 13

Group III 48 63 9

a) what is the probability that a customer likes the deodorant given he/she is from group I?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟔𝟓𝟐 ]

b) What is the probability that a customer is from group II and does not like the deodorant?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟏𝟒𝟖 ]

c) What is the probability that a customer has no opinion, if he/she is from group III?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟎𝟕𝟓 ]

#56) The two-way table below shows the favorite leisure activities for 50 adults (30 women and 20

men).

Dance Sports TV Total

Women 16 6 8 30

Men 2 10 8 20

Total 18 16 16 50

a) Calculate the probability of an individual whose favorite leisure activity is sports, given that the individual is a woman. [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟐 ]

b) (4 points) Calculate the probability of an individual whose favorite leisure activity is dance, given that the individual is a man. [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟏 ]

c) What is the probability that someone’s favorite activity is watching TV? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟑𝟐 ]

d) What is the probability of a woman who likes watching TV? [𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟏𝟔 ]

17

Section 8.2

#57) Let E and F be independent events, P(E)=0.11 P(F)=0.35. What is 𝑃(𝐸 ∩ 𝐹) =?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟎𝟒 ]

#58) If E and F are independent, and if P(E)=0.3 and 𝑃(𝐸 ∩ 𝐹) = 0.15, find P(F)=?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝑷(𝑭) = 𝟎. 𝟓 ]

#59) If E and F are independent and P(E)=0.40, 𝑃(𝐸 ∪ 𝐹) = 0.6 , find P(F)=?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟑𝟑𝟑 ]

Section 8.4

#60) Nine horses are competing in a race. Assume you have purchased a Trifecta ticket (In Trifecta, the

player selects three horses as first, second, and third place winners, and to win, those three horses must

finish the race in the precise order the player has selected). How many possibilities of a Trifecta exist?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟓𝟎𝟒 ]

#61) From 1200 lottery tickets that are sold, three are to be selected for first, second, and third prizes.

How many possible outcomes are there?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏𝟕𝟐𝟑𝟔𝟖𝟐𝟒𝟎𝟎 ]

#62) Five girls and six boys are sitting in a row. How many ways can they sit if all girls are sitting together

and all boys are sitting together in the same row?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟏𝟕𝟐𝟖𝟎𝟎 ]

#63) Ten individuals are candidates for positions of president, vice president of an organization. How

many possibilities exist for a pair of president and vice president?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟗𝟎 ]

Section 8.5

#64) Three individuals are to be selected from a pool of nine English professors to form a panel of judges

in a poetry contest. How many different groups of three may be selected at random from the entire

pool?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟖𝟒 ]

18

#65) Eleven people are waiting to get on a bus. When the bus arrives, the driver tells the awaiting

customers that there are only four seats available on the bus and no one is allowed to stand up when

the bus is moving. In how many different ways can any four customers get on the bus?

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟑𝟑𝟎 ]

#66) From a stack of 12 cards of 7 spades and 5 hearts, 3 cards are randomly selected. Find the

probability that there are 2 spades and 1 heart.

[𝒂𝒏𝒔𝒘𝒆𝒓: 𝟎. 𝟒𝟕𝟕 ]