chapter 6 practice test problems (fa06)

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MTH 110 Chapter 6 Practice Test Problems Name___________________________________

1) Probability

A) assigns realistic numbers to random events.

B) is the branch of mathematics that studies long-term patterns of random events by repeated observations.

C) can be applied to events that we see in both our personal and professional lives.

D) all of the above

2) The probability of an event is

A) is a number that expresses te long-run liklihood that an event will occur.

B) is the branch of mathematics that studies long-term patterns of random events by repeated observations.

C) an event that we see in both our personal and professional lives.

D) all of the above

3) The number of events associated with a sample space having n outcomes is

A) n!.

B) n.C) 2n.

D) n2.

E) none of the above

4) Which of the following events are mutually exclusive?

A) being a college student and being a high school graduate

B) living in Baltimore and working in Washington, D.C.

C) being a mother and being an uncle

D) being a steelworker and being a stamp collector


5) Two events A and B are mutually exclusive if

A) A∩B=U.

B) A∩B=∅.

C) A∪B=U.

D) A∪B=∅.


Solve the problem.

6) There are five finalists, A, B, C, D, E, in a lottery drawing. Three of them are to be selected to win $10,000 prizes.(a) What is the sample space for this experiment?

(b) Describe the event ʺC and E win $10,000 prizesʺ as a subset of the sample space.

(c) Describe the event ʺneither A nor B wins a $10,000 prizeʺ.

1



7) An experiment consists of tossing a coin three times and recording the sequence of heads and tails.

(a) What is the sample space?

(b) Determine the event E = ʺMore heads than tails occur.ʺ

(c) Determine the event F = ʺThe number of heads equals the number of tails.ʺ

8) A quality control process consists of selecting two transistors at random, testing them, and recording whether

each is defective (D) or nondefective (N). What is the sample space for this experiment?

9) Let S = {a, b, c, d, e} be a sample space, E = {a, b, e}, and F = {b, c}.

(a) Determine the events E ∪ F and E′ ∩ F′ .

(b) Are E ∪ F and E′ ∩ F′ mutually exclusive?

10) A letter is selected at random from the word ʺTEXTBOOK.ʺ

(a) What is the sample space for this experiment?

(b) Describe the event ʺthe letter chosen is a vowelʺ as a subset of the sample space.

11) If E and F are mutually exclusive and Pr(E) = 0.2 and Pr(F) = 0.6, then Pr(E ∪ F) is

A) 0.B) 0.12.

C) 1.

D) 0.8.


12) Which of the following is a valid probability distribution for a sample space S = {a, b, c, d}?

A) Pr(a) = 0.3, Pr(b) = 0.1, Pr(c) = 0.2, Pr(d) = 0.5

B) Pr(a) = -0.2, Pr(b) = 0.5, Pr(c) = 0.4, Pr(d) = 0.3

C) Pr(a) = 0.5, Pr(b) = 0.2, Pr(c) = 0.1, Pr(d) = 0.3

D) Pr(a) = 0.6, Pr(b) = 0, Pr(c) = 0.3, Pr(d) = 0.1E) none of the above

13) Two fair die are rolled. The probability that the numbers that appear add to 4 is

A)1

5

B)1

36

C)1

12

D)1

6


2



14) Two fair die are rolled. The probability that the numbers that appear are both three is

A)1

6

B)1

2

C)1

3

D) 136


15) If the odds against an event are 2 to 5, then the probability that the event will occur is

A)2

5.

B)2

7.

C)3

5

.

D)3

7.


16) The probability of getting either a black card or an ace in one draw from an ordinary deck of 52 cards is

A)29

52.

B)28

52.

C)

30

52 .

D)26

52.


17) A fair coin is tossed six times. The probability of obtaining no heads is

A)6

64=.

3

32

B)1

64.

C) 132

.

D)0

64=0.


3



18) A fair coin is tossed six times. The probability of obtaining at most five heads is

A)1

64

B)59

64

C)5

64

D) 6364

E) none of the above.

Solve the problem.

19) If the odds against an event are 2 to 3, what is the probability that the event will occur?

20) In the game Clue there are six equally likely suspects and six equally likely murder weapons. What are the

odds that Professor Plum killed the victim with the rope?

21) If the odds against an event are 3 to 5 , what is the probability of the event occurring?

22) If the odds against an event are a to b, what is the probability of the event occurring?

23) Data was collected about four crimes: robbery, assault, rape and murder. The number of times each crime was

reported is displayed below.

Crime Frequency

Robbery 30

Assault 25

Rape 7

Murder 5

What is the probability that one of the four reported crimes is robbery?

24) Let E and F be events such that Pr(E) = 0.3, Pr(F) = 0.6, Pr(E ∩ F) = 0.2. Find Pr(E ∪ F).

25) The probability that a student will pass mathematics is 0.8, that she will pass physics is 0.65 and that she will

pass both courses is 0.6. Find the probability that

(a) she will pass at least one of the two courses.

(b) she will pass physics but not mathematics.

(c) she will fail both courses.

4



An experiment with outcomes s1, s2, s3, and s4 is described by the probability table below.

Outcome Probability

s1 0.5

s2 0.1

s3 0.2

s4 0.2

26) For the experiment above what is Pr({ s1 , s3 , s4})?

An experiment with outcomes s1, s2, s3, s4, and s5 is described by the probability table below.

Outcome Probability

s1 0.05

s2 0.10

s3 0.20

s4 0.50

s5 0.15

Let E = {s1, s4} and F = {s1, s2, s3, s5}.

27) For the experiment above, compute Pr(F).

28) For the experiment above, compute Pr(E ∪ F).

An experiment consists of tossing a coin three times and observing the sequence of heads and tails. Each of the eight

outcomes has the same probability of occurrence.

29) For the experiment above, compute the probability the number of heads is larger than the number of tails.

An experiment consists of selecting a letter at random from the letters of the word ʺTEXTBOOK.ʺ

30) For the experiment above, compute the probability that the letter selected is an ʺX.ʺ

31) For the experiment above, compute the probability that the letter selected is a vowel.

An analyst develops a partial prediction of the return on a $1000 investment after one year. Note that a return of less than

$1000 is a loss and a return of more than $1000 is a profit.

Return Probability

Less than $1000 0.10

$1000 0.10

More than $1000, less than $1100 0.35

At least $1100, less than $1500 0.25

At least $1500, less than $2000 0.15

32) Based on the table above, what is the probability that the investment at least doubles?

5



In a study of the ages of its employees, over a period of several years a university finds the following:

Age (years) Probability

18-30 0.20

18-45 0.65

18-60 0.90

18-80 1.00

33) Find the probability associated with each of the events: 18-30 years, 31-45 years, 46-60 years, and 61-80 years.

34) Let E and F be events such that Pr(F) = 0.4 and Pr(E ∩ F) = 0.3.

Compute Pr(E′ ∩ F).

A) 0.05

B) 0.55

C) 0.006

D) 0.10


35) Five horses are running at a race track. Being an inexperienced bettor, you assume that every order of finish is

equally likely. You bet that Son-of-a-Gun will win and that Gentle Lady will come in second. The probability

that you will win both bets is

A)1

25.

B)1

20.

C)9

20.

D)2

5.


36) The probability that a family with 6 children has exactly two girls is

A)3

8.

B)1

64.

C)1

3.

D)15

64.


6



A basket contains five red balls, four white balls and three blue balls. Two balls are drawn, one after the other, with the

first ball replaced before the second is drawn. Find the probablility of drawing

37) two white balls.

A)4

12 =

1

3.

B)2

12 =

1

6.

C) 29

.

D)1

9.


A student is studying mathematics and chemistry. The probability that he passes mathematics is 0.75, the probability

that he fails chemistry is 0.2, and the probability that he passes mathematics but fails chemistry is 0.05.

38) The probability that he passes both courses is

A) 0.60.

B) 0.70.C) 0.

D) 0.75.


39) The probability that he either passes mathematics or fails chemistry is

A) 0.90.

B) 0.95.

C) 0.15.

D) 1.0.


Solve the problem.

40) An coin is to be tossed 10 times. What is the probability of obtaining 7 heads and 3 tails?

41) If the odds in favor of an events are 4 to 7, what is the probability that the event will NOT occur?

A basket contains five red balls, four white balls and three blue balls. Two balls are drawn, one after the other, with the

first ball replaced before the second is drawn. Find the probablility of drawing

42) at most one white ball.

43) at least one white ball.

44) a white ball and a red ball.

The letters of the word ʺSCRAMBLEʺ are scrambled and arranged in a random order.

45) In the situation above, how many arrangements start with a vowel?

7



46) In the situation above, what is the probability that the resulting arrangement reads ʺSCRAMBLEʺ?

Solve the problem.

47) Suppose that a pair of dice is tossed and the number on the uppermost faces are observed.

(a) What is the probability that the sum is 11?

(b) What is the probability that the sum is less than 5?

An urn contains six red balls and four green balls. A sample of seven balls is selected at random.

48) Find the probability that five red and two green balls are selected.

49) Find the probability that at least four red balls are selected.

Solve the problem.

50) An exam contains six ʺtrue or falseʺ questions. What is the probability that a student guessing at the answers

will get exactly four correct?

51) A factory produces screws, which are packaged in boxes of 30. Four screws are selected from each box for

inspection. A

box

fails

inspection

if

two

or

more

of

these

four

screws

are

defective.

What

is

the

probability

that

a box containing two defective screws will pass inspection?

52) What is the probability that in a group of seven people two or more of them have the same birth month?

(Assume that each month is equally likely.)

53) An urn contains three white balls and four red balls. Two balls are chosen at random. What is the probability

that at least one of the balls is red?

54) Compute the probability of obtaining three face cards when five cards are dealt from a standard 52-card deck.

55) If E and F are independent and Pr(E) = 0.3 and Pr(F) = 0.6, then Pr(E ∪ F) is

A) 0.

B) 0.18.

C) 0.90.

D) 0.72.


56) Suppose that Pr(E) = 0.85, Pr(F) = 0.4, and Pr(E ∩ F) = 0.3. Then Pr(F|E) =

A)6

11.

B)3

4.

C)6

17.

D)3

10.


8



57) Suppose that Pr(E) = 0.85, Pr(F) = 0.4, and Pr(E ∩ F) = 0.3. Then Pr(F|E′) =

A)5

11.

B)1

5.

C)11

7.

D) 23


58) Two cards are drawn (without replacement) from an ordinary deck of 52 cards. The probability that the second

card is black if the first card is the ace of hearts is

A)2

51.

B)1

104.

C) 12 .

D)26

51.


59) A biased coin with Pr(H) = 1

4 and Pr(T) =

3

4 is thrown twice, then the probablility of getting two tails is

A)3

2.

B)9

16.

C)2

3.

D)1

3.


60) Two cards are drawn (without replacement) from an ordinary deck of 52 cards. The probability that both cards

are aces is

A)1

17.

B)1

221.

C)4

663.

D)2

13.


9



The probability that person A will pass Finite Mathematics is 5

8 and the probability that person B will pass is

6

7. Assume

the events are independent.

61) In the situation above, the probability that neither will pass is

A)29

56.

B)53

56.

C)3

56.

D) 1.


62) In the situation above, the probability that both will pass is

A)15

28.

B)53

56

.

C)30

54.

D) 0.


63) In the situation above, the probability that at least one will pass is

A)53

56.

B) 0.

C)30

56 .

D)13

28.


10



The table below gives crime statistics relating to the location of the crime and the type of crime.

Robbery Murder Assault

Residential 130 40 30

Commercial 102 28 20

64) Based on the table above, the probability that a randomly-selected crime committed in a residential area is a

murder is

A)34

175.

B)10

17.

C)1

5.

D)4

35.


65) Based on the table above, the probability that a randomly-selected crime was committed in a commercial areagiven that it was an assault is

A)1

7.

B)1

3.

C)2

5.

D)2

15.


A shipment of twenty radios contains six defective radios. Two radios are randomly selected from the shipment.

66) For the situation above, the probability that both radios selected are defective is

A)3

10.

B)15

19.

C)14

17.

D)1

3.


11



67) For the situation above, the probability that neither radio selected is defective is

A)1

2.

B)91

190.

C)3

17.

D) 17

.


68) A shipment contains 25 defective items and 100 nondefective items. Two items are randomly chosen in

succession without replacement. The probability that both items are defective is

A)6

155.

B)1

16.

C) 125 .

D)6

99.


69) Of the 1000 freshmen enrolled at a certain college, 100 have verbal SAT scores above 650. Thirty of these 100

students earned an A in freshman composition. The probability that a freshman has a verbal SAT score above

650 and earned an A in freshman composition is

A)3

100.

B) 13100

.

C)1

10.

D) impossible to determine.


Suppose that Pr(E) = 0.3, Pr( F ) = 0.5, and Pr(E ∩ F ) = 0.2.

70) Calculate Pr(E′).

71) Calculate Pr(E′ ∩

F

′).

72) Calculate Pr(F′|E′).

Let E and F be events with Pr(E) = 0.4, Pr( F ) = 0.6, and Pr(E ∪ F ) = 0.8.

73) Find Pr(E|F).

12



74) Find Pr(E′ ∩ F).

Let A and B be events such that Pr( ) = 0.6, Pr(B) = 0.5, and Pr( |B) = 0.4.

75) Compute Pr(B| A).

A coin has Pr(T)=1

4. If it is tossed six times in succession, find the probablility of getting

76) at least five tails

Two cards are drawn in succession (without replacement) from an ordinary deck of 52 cards.

77) Find the probability that the second card is red if the first card is the king of hearts.

78) Find the probability that both cards are black.

The probability that a person passes organic chemistry the first time he enrolls is 0.8. The probability that a person

passes organic chemistry the second time he enrolls is 0.9.

79) Find the probability that a person fails the first time but passes the second time.

Enrollment statistics at a certain college show that 45% of all students are men, 10% of the student body consists of

women majoring in business administration, and 35% of all students major in business administration. A student is

selected at random.

80) What is the probability that the selected student majors in business administration if the selected student is a

women?

81) What is the probability that the selected student is a woman if the selected student is a business administration

major?

Two people X and Y enter a supermarket. The probabilities that person X and Y will make a purchase are 0.3 and 0.4

respectively. Assume that whether each makes a purchase is independent of whether the other makes a purchase.

82) What is the probability that neither of the two people described above makes a purchase?

Solve the problem.

83) There are three children in a family. Are the events ʺthere are more boys than girlsʺ and ʺthe first child is a girlʺ

(a) mutually exclusive?

(b) independent?

The table below gives the distribution of blood type by sex in a group of 1000 individuals.

Blood Type Male Female Total

O 80 370 450A 150 250 400

B 50 50 100

AB 20 30 50

Total 400 600

A person is selected at random from this group.

84) Based on the table above, what is the probability that the person selected has blood type B?

13



85) Based on the table above, what is the probability that the person selected is female if the personʹs blood type is

O?

Solve the problem.

86) An urn contains an equal number of red and blue balls. One third of the red balls have a white dot on them.

What is the probability that a randomly selected ball is red with a white dot?

87) The probabilities that two species will become extinct in five years are 0.3 and 0.2 respectively. Given that these

probabilities are independent, what is the probability that at least one group will become extinct in the next five

years?

88) A shipment contains 25 defective and 100 nondefective items. Two items are randomly chosen in succession

without replacement and the shipment is rejected if at least one of these is defective. What is the probability

that the shipment will be rejected?

89) A pair of dice is tossed twice and the numbers on the uppermost faces are observed. What is the probability of

getting a sum of 9 in each of the two successive tosses?

90) A basketball player makes 60% of all foul shots that she tries. What is the probability that, in two foul shots, she

makes at least one?

A certain soccer goalkeeper catches 30% of all penalty kicks against her team.

91) What is the probability that out of five penalty kicks she catches at least two?

Fifty percent of students enrolled in an astronomy class have previously taken physics. Thirty percent of these studens

received an A for the astronomy class, whereas twenty percent of the other students received an A for astronomy. Find

the probablility that a student selected at random

92) previously took a physics course and did not receive an A in the astronomy course.

A) .40B) .10

C) .15

D) .35


93) received an A in the astronomy course.

A) .10

B) .45

C) .50

D) .25


14



94) previously took a physics course, given that they received an A in the astronomy course.

A) .15

B) .40

C) .6

D) .25


Suppose that 30% of all small businesses are undercapitalized, 40% of all undercapitalized small business fail, and 20%

of all small businesses that are not undercapitalized fail.

95) A small business is chosen at random. Based on the statistics above, the probability that the small business

succeeds if it is undercapitalized is

A) 0.56.

B) 0.60.

C) 0.80.

D) 0.18.


96) A small business is chosen at random. Based on the statistics above, the probability that the small business is

not undercapitalized and yet fails is

A) 0.20.

B) 0.40.

C) 0.12.

D) 0.14.


Solve the problem.

97) A box contains four good light bulbs and three defective ones. Bulbs are selected one at a time (without

replacement). Find the probability that the second defective bulb is found on the third selection.

98) Urn I contains three red balls and one white ball. Urn II contains two red and two white balls. An urn is

selected at random, and a ball is chosen. If the ball is red, what is the probability that Urn I was chosen?

99) Every day Aaron, Bebe, and Cindy eat a piece of fruit for lunch. Aaron will eat an apple, banana, peach, or

orange with equal likelihood. Bebe will eat an apple, banana, or mango with a 50% likelihood for an apple, 30%

for a banana, and 20% for a mango. Cindy will eat either an apple or a banana with equal likelihood. What is

the probability that they all eat the same kind of fruit for lunch?

A

local

store

orders

lightbulbs

from

two

suppliers,

AAA

Electronics

and

ZZZ

Electronics.

The

local

store

purchases

30%

ofthe bulbs from AAA and 70% of the bulbs from ZZZ. Two percent of the bulbs from AAA are defective while 3% of the

bulbs from ZZZ are defective. Find the probability that

100) was purchased from ZZZ and is not defective.

15



Solve the problem.

101) An election between two candidates is held in two districts. The first district, which has 60% of the voters, votes

40% for candidate I and 60% for candidate II. The second district, with 40% of the voters, votes 60% for

candidate I and 40% for candidate II. Who wins?

Data maintained by a university records the distribution of the student population by college and by the proportion of

each collegeʹs population who are honors students.

College Proportion

of university Proportion who

are honors students

Arts and Sciences 0.35 0.10

Education 0.10 0.08

Engineering 0.30 0.15

Journalism 0.05 0.12

Nursing 0.20 0.16

102) A student is chosen at random from the university. If the student is an engineering student, what is the

probability that he/she is an honors student?

103) A student is chosen at random from the university. If the student is an honors student, what is the probability

that he/she is not a nursing student?

In a factory, assembly lines I, II, and III produce 60%, 30%, and 10% of the total output, respectively. One percent of line

Iʹs output is defective, 2% of line IIʹs output is defective, and 3% of line IIIʹs output is defective.

104) An item is chosen at random. Based on the data above, if the selected item is defective, what is the probability

that it was produced by line III?

Balls of different colors are placed in two urns as follows:

Red Green Blue

Urn I 3 4 3Urn II 5 3 4

105) Based on the data above, given that a ball chosen is blue, what is the probability that it came from urn I?

In the current first-year class of a community college, all the students come from three local high schools. Schools I, II,

and III supply respectively 40%, 50%, and 10% of the students. The failure rate of students is 4%, 2%, and 6%,

respectively.

106) In the situation above, given that a student fails, what is the probability that he or she came from school I?

Three boxesI, II, and IIIcontain three red and two green chips, two red and four green chips, and four red and five

green chips, respectively. A box is selected at random and a chip is drawn at random from the box .

107) In the situation above, given the the chip is green, what is the probability that it came from box II?

A local store orders lightbulbs from two suppliers, AAA Electronics and ZZZ Electronics. The local store purchases 30% of

the bulbs from AAA and 70% of the bulbs from ZZZ. Two percent of the bulbs from AAA are defective while 3% of the

bulbs from ZZZ are defective. Find the probability that

108) a randomly selected defective light bulb was purchased from AAA Electronics.

16



109) is defective.

A test for a certain drug produces a false negative 5% of the time and a false positive 8% of the time. Suppose 12% of the

employees at a certain company use the drug.

110) What is the probability that a nondrug user at the company tests positive twice in a row?

17



Answer KeyTestname: MTH 110 CHAPTER 6 PRACTICE TEST PROBLEMS (FA06)

1) D

2) A

3) C

4) C

5) B6) (a) {( A, B, C), ( A, B, D), ( A, B, E), ( A, C, D), ( A, C, E), ( A, D, E), (B, C, D), (B, C, E), (B, D, E), (C, D, E)}

(b) {( A, C, E), (B, C, E), (C, D, E)}

(c) {(C, D, E)}

7) (a) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

(b) {HHH, HHT, HTH, THH}

(c) ∅

8) {(D, D), (D, N), (N, D), (N, N)}

9) (a) E ∪ F = {a, b, c, e}, E′ ∩ F′ = {d}.

(b) yes

10) (a) {T, E, X, B, O, K}

(b) {E, O}

11) D12) D

13) C

14) D

15) E

16) B

17) B

18) D

19)3

5

20) 1 to 35 (or 35 to 1 that he did not kill the victim with the rope)

21) 1 -

3

3+5 =

5

8

22) 1 - a

a + b =

b

a + b

23)30

67

24) 0.7

25) (a) 0.85

(b) 0.05

(c) 0.15

26) 0.9

27) 0.5

28) 1

29) 0.5 = 1

2

30) 0.125 = 1

8

31) 0.375 = 3

8

32) 0.15

18




33) Pr(18-30 years) = 0.20

Pr(31-45 years) = 0.45

Pr(46-60 years) = 0.25

Pr(61-80 years) = 0.10

34) D35) B

36) D

37) D

38) B

39) A

40)

10

7

210 =

15

128

41)7

11

42)

1 - 1

9 =

8

9

43) 1 - 8*8

12*12 =

5

9

44)2*5*4

12*12 =

5

18

45) 2 · 7! = 10,080

46)1

8! =

1

40320 ≈ 0.0000248

47) (a) 118

(b)1

6

48)

6

5

4

2

10

7

= 3

10

49)

6

4

4

3 +

6

5

4

2 +

6

6

4

1

107

= 5

6

50)15

64

51)

1 -

2

2

28

2

30

4

= 143

145 ≈ 0.986

19




52) 1 - 12 × 11 × 10 × 9 × 8 × 7 × 6

127 =

3071

3456 ≈ 0.889

53)

1 -

3

2

7

2

= 6

7

54)

12

3 ·

40

2

52

5

= 55

833 ≈ 0.066

55) D

56) C

57) D

58) D59) B

60) B

61) C

62) A

63) A

64) C

65) C

66) E

67) B

68) A

69) A

70) 0.771) 0.4

72)4

7

73)1

3

74) 0.4

75)1

3

76)6

5

1

4

5 3

4+

1

4

6=

19

4096

77) 2551

78) 1

2 ·

25

51 =

25

102

79) 0.18

80)2

11

81)2

7

20




82) 0.42

83) (a) no

(b) no

84) 0.1

85) 3745

86)1

6

87) 0.44

88)56

155

89)1

81

90) 1 - (0.40)2 = 0.84

91) 1 - (0.7)5 - [ C(5, 1) · (0.3) · (0.7)4 ] = 0.47178

92) D93) D

94) C

95) B

96) D

97)8

35

98) 0.6

99) 10%

100) .679

101) Candidate II

102) 0.15

103)45

61 ≈ 0.738

104)1

5

105)9

19

106) 0.5

107)30

73

108) 0.222

109) .027

110) 0.64%

21

chapter 6 practice test problems (fa06)

Documents

b describetheeventcandewin

b determinetheevente

c determinetheeventf

determinetheeventsefande

thenpre fisa