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CHAPTER FOUR

INTRODUCTION TO PROBABILITY

MULTIPLE CHOICE QUESTIONS

In the following multiple-choice questions, circle the correct answer.

1. The probability of at least one head in two flips of a coin is

a. 0.33

b. 0.50

c. 0.75

d. 1.00

2. Revised probabilities of events based on additional information are

a. joint probabilities

b. posterior probabilities

c. marginal probabilities

d. complementary probabilities

3. Posterior probabilities are computed using

a. the classical method

b. Chebyshev’s theorem

c. the empirical rule

d. Bayes’ theorem

4. The complement of P(A | B) is

a. P(AC | B)

b. P(A | BC)

c. P(B | A)

d. P(A B)

5. An element of the sample space is

a. an event

b. an estimator

c. a sample point

d. an outlier

6. The probability of an intersection of two events is computed using the

a. addition law

b. subtraction law

c. multiplication law

d. division law

7. If A and B are mutually exclusive, then

a. P(A) + P(B) = 0

b. P(A) + P(B) = 1

c. P(A B) = 0

d. P(A B) = 1

8. Posterior probabilities are

a. simple probabilities

b. marginal probabilities

c. joint probabilities

d. conditional probabilities

9. The range of probability is

a. any value larger than zero

b. any value between minus infinity to plus infinity

c. zero to one

d. any value between -1 to 1

10. Since the sun must rise tomorrow, then the probability of the sun rising tomorrow is

a. much larger than one

b. zero

c. infinity

d. None of the other answers is correct.

11. Any process that generates well-defined outcomes is

a. an event

b. an experiment

c. a sample point

d. None of the other answers is correct.

12. In statistical experiments, each time the experiment is repeated

a. the same outcome must occur

b. the same outcome can not occur again

c. a different outcome may occur

d. None of the other answers is correct.

13. Each individual outcome of an experiment is called

a. the sample space

b. a sample point

c. an experiment

d. an individual

14. A sample point refers to a(n)

a. numerical measure of the likelihood of the occurrence of an event

b. set of all possible experimental outcomes

c. individual outcome of an experiment

d. All of these answers are correct.

15. The collection of all possible sample points in an experiment is

a. the sample space

b. a sample point

c. an experiment

d. the population

16. The set of all possible sample points (experimental outcomes) is called

a. a sample

b. an event

c. the sample space

d. a population

17. The sample space refers to

a. any particular experimental outcome

b. the sample size minus one

c. the set of all possible experimental outcomes

d. both any particular experimental outcome and the set of all possible experimental

outcomes are correct

18. An experiment consists of three steps. There are four possible results on the first step,

three possible results on the second step, and two possible results on the third step. The

total number of experimental outcomes is

a. 9

b. 14

c. 24

d. 36

19. An experiment consists of tossing 4 coins successively. The number of sample points in

this experiment is

a. 16

b. 8

c. 4

d. 2

20. A lottery is conducted using three urns. Each urn contains chips numbered from 0 to 9.

One chip is selected at random from each urn. The total number of sample points in the

sample space is

a. 30

b. 100

c. 729

d. 1,000

21. Three applications for admission to a local university are checked to determine whether

each applicant is male or female. The number of sample points in this experiment is

a. 2

b. 4

c. 6

d. 8

22. Assume your favorite football team has 2 games left to finish the season. The outcome of

each game can be win, lose or tie. The number of possible outcomes is

a. 2

b. 4

c. 6

d. None of the other answers is correct.

23. Each customer entering a department store will either buy or not buy some merchandise.

An experiment consists of following 3 customers and determining whether or not they

purchase any merchandise. The number of sample points in this experiment is

a. 2

b. 4

c. 6

d. 8

24. A graphical device used for enumerating sample points in a multiple-step experiment is a

a. bar chart

b. pie chart

c. histogram

d. None of the other answers is correct.

25. Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many

possible selections are there?

a. 20

b. 7

c. 5!

d. 10

26. The “Top Three” at a racetrack consists of picking the correct order of the first three

horses in a race. If there are 10 horses in a particular race, how many “Top Three”

outcomes are there?

a. 302,400

b. 720

c. 1,814,400

d. 10

EMBS4 TB04 - 5

27. When the assumption of equally likely outcomes is used to assign probability values, the

method used to assign probabilities is referred to as the

a. relative frequency method

b. subjective method

c. probability method

d. classical method

28. A method of assigning probabilities that assumes the experimental outcomes are equally

likely is referred to as the

a. objective method

b. classical method

c. subjective method

d. experimental method

29. When the results of experimentation or historical data are used to assign probability

values, the method used to assign probabilities is referred to as the

a. relative frequency method

b. subjective method

c. classical method

d. posterior method

30. A method of assigning probabilities based upon judgment is referred to as the

a. relative method

b. probability method

c. classical method

d. None of the other answers is correct.

31. Of the last 100 customers entering a computer shop, 25 have purchased a computer. If the

classical method for computing probability is used, the probability that the next customer

will purchase a computer is

a. 0.25

b. 0.50

c. 1.00

d. 0.75

32. The probability assigned to each experimental outcome must be

a. any value larger than zero

b. smaller than zero

c. one

d. between zero and one

33. An experiment consists of four outcomes with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4.

The probability of outcome E4 is

a. 0.500

EMBS4 TB04 - 6

b. 0.024

c. 0.100

d. 0.900

34. A graphical method of representing the sample points of a multiple-step experiment is

a. a frequency polygon

b. a histogram

c. an ogive

d. a tree diagram

35. A(n) __________ is a graphical representation in which the sample space is represented

by a rectangle and events are represented as circles.

a. frequency polygon

b. histogram

c. Venn diagram

d. tree diagram

36. A(n) __________ is a collection of sample points.

a. probability

b. permutation

c. experiment

d. event

37. Given that event E has a probability of 0.25, the probability of the complement of event E

a. cannot be determined with the above information

b. can have any value between zero and one

c. must be 0.75

d. is 0.25

38. The symbol shows the

a. union of events

b. intersection of events

c. sum of the probabilities of events

d. sample space

39. The union of events A and B is the event containing

a. all the sample points common to both A and B

b. all the sample points belonging to A or B

c. all the sample points belonging to A or B or both

d. all the sample points belonging to A or B, but not both

40. The probability of the union of two events with nonzero probabilities

a. cannot be less than one

b. cannot be one

EMBS4 TB04 - 7

c. cannot be less than one and cannot be one

d. None of the other answers is correct.

41. The symbol shows the

a. union of events

b. intersection of events

c. sum of the probabilities of events

d. None of the other answers is correct.

42. The addition law is potentially helpful when we are interested in computing the

probability of

a. independent events

b. the intersection of two events

c. the union of two events

d. conditional events

43. If P(A) = 0.38, P(B) = 0.83, and P(A B) = 0.57; then P(A B) =

a. 1.21

b. 0.64

c. 0.78

d. 1.78

44. If P(A) = 0.62, P(B) = 0.47, and P(A B) = 0.88; then P(A B) =

a. 0.2914

b. 1.9700

c. 0.6700

d. 0.2100

45. If P(A) = 0.85, P(A B) = 0.72, and P(A B) = 0.66, then P(B) =

a. 0.15

b. 0.53

c. 0.28

d. 0.15

46. Two events are mutually exclusive if

a. the probability of their intersection is 1

b. they have no sample points in common

c. the probability of their intersection is 0.5

d. the probability of their intersection is 1 and they have no sample points in

common

47. Events that have no sample points in common are

a. independent events

b. posterior events

c. mutually exclusive events

d. complements

48. The probability of the intersection of two mutually exclusive events

a. can be any value between 0 to 1

b. must always be equal to 1

c. must always be equal to 0

d. can be any positive value

49. If two events are mutually exclusive, then the probability of their intersection

a. will be equal to zero

b. can have any value larger than zero

c. must be larger than zero, but less than one

d. will be one

50. Two events, A and B, are mutually exclusive and each has a nonzero probability. If event

A is known to occur, the probability of the occurrence of event B is

a. one

b. any positive value

c. zero

d. any value between 0 to 1

51. If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then

P(A B) =

a. 0.30

b. 0.15

c. 0.00

d. 0.20

52. If A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.5, then

P(A B) =

a. 0.00

b. 0.15

c. 0.8

d. 0.2

53. In an experiment, events A and B are mutually exclusive. If P(A) = 0.6, then the

probability of B

a. cannot be larger than 0.4

b. can be any value greater than 0.6

c. can be any value between 0 to 1

d. cannot be determined with the information given

EMBS4 TB04 - 9

54. Which of the following statements is(are) always true?

a. -1 P(Ei) 1

b. P(A) = 1 − P(Ac)

c. P(A) + P(B) = 1

d. both P(A) = 1 − P(Ac) and P(A) + P(B) = 1

55. One of the basic requirements of probability is

a. for each experimental outcome Ei, we must have P(Ei) 1

b. P(A) = P(Ac) − 1

c. if there are k experimental outcomes, then

P(E1) + P(E2) + … + P(Ek) = 1

d. both P(A) = P(Ac) − 1 and if there are k experimental outcomes, then

P(E1) + P(E2) + … + P(Ek) = 1

56. Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.2. The probability

of the complement of Event B equals

a. 0.00

b. 0.06

c. 0.7

d. None of the other answers is correct.

57. The multiplication law is potentially helpful when we are interested in computing the

probability of

a. mutually exclusive events

b. the intersection of two events

c. the union of two events

d. None of the other answers is correct.

58. If P(A) = 0.80, P(B) = 0.65, and P(A B) = 0.78, then P(BA) =

a. 0.6700

b. 0.8375

c. 0.9750

d. Not enough information is given to answer this question.

59. If two events are independent, then

a. they must be mutually exclusive

b. the sum of their probabilities must be equal to one

c. the probability of their intersection must be zero

d. None of the other answers is correct.

60. If A and B are independent events with P(A) = 0.38 and P(B) = 0.55, then P(AB) =

a. 0.209

b. 0.000

c. 0.550

d. None of the other answers is correct.

61. If X and Y are mutually exclusive events with P(X) = 0.295, P(Y) = 0.32, then P(XY) =

a. 0.0944

b. 0.6150

c. 1.0000

d. 0.0000

62. Two events with nonzero probabilities

a. can be both mutually exclusive and independent

b. cannot be both mutually exclusive and independent

c. are always mutually exclusive

d. cannot be both mutually exclusive and independent and are always mutually

exclusive

63. If P(A) = 0.50, P(B) = 0.60, and P(A B) = 0.30; then events A and B are

a. mutually exclusive events

b. not independent events

c. independent events

d. Not enough information is given to answer this question.

64. On a December day, the probability of snow is .30. The probability of a "cold" day is

.50. The probability of snow and a "cold" day is .15. Are snow and "cold" weather

independent events?

a. only if given that it snowed

b. no

c. yes

d. only when they are also mutually exclusive

65. If P(A) = 0.5 and P(B) = 0.5, then P(A B) is

a. 0.00

b. 0.25

c. 1.00

d. cannot be determined from the information given

66. If A and B are independent events with P(A) = 0.4 and P(B) = 0.6, then P(A B) =

a. 0.76

b. 1.00

c. 0.24

d. 0.2

67. If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A B) =

a. 0.62

b. 0.12

c. 0.60

d. 0.68

68. If A and B are independent events with P(A) = 0.4 and P(B) = 0.25, then P(A B) =

a. 0.65

b. 0.55

c. 0.10

d. Not enough information is given to answer this question.

69. Events A and B are mutually exclusive. Which of the following statements is also true?

a. A and B are also independent.

b. P(A B) = P(A)P(B)

c. P(A B) = P(A) + P(B)

d. P(A B) = P(A) + P(B)

70. If A and B are independent events with P(A) = 0.05 and P(B) = 0.65, then P(AB) =

a. 0.05

b. 0.0325

c. 0.65

d. 0.8

71. A six-sided die is tossed 3 times. The probability of observing three ones in a row is

a. 1/3

b. 1/6

c. 1/27

d. 1/216

72. If a coin is tossed three times, the likelihood of obtaining three heads in a row is

a. zero

b. 0.500

c. 0.875

d. 0.125

73. If a penny is tossed four times and comes up heads all four times, the probability of heads

on the fifth trial is

a. zero

b. 1/32

c. 0.5

d. larger than the probability of tails

74. If a penny is tossed three times and comes up heads all three times, the probability of

heads on the fourth trial is

a. smaller than the probability of tails

b. larger than the probability of tails

c. 1/16

d. None of the other answers is correct.

75. A perfectly balanced coin is tossed 6 times and tails appears on all six tosses. Then, on

the seventh trial

a. tails cannot appear

b. heads has a larger chance of appearing than tails

c. tails has a better chance of appearing than heads

d. None of the other answers is correct.

76. The probability of the occurrence of event A in an experiment is 1/3. If the experiment is

performed 2 times and event A did not occur, then on the third trial event A

a. must occur

b. may occur

c. could not occur

d. has a 2/3 probability of occurring

77. Bayes’ theorem is used to compute

a. the prior probabilities

b. the union of events

c. both the prior probabilities and the union of events

d. the posterior probabilities

78. Initial estimates of the probabilities of events are known as

a. sets

b. posterior probabilities

c. conditional probabilities

d. prior probabilities

79.

A

B

C

D

E

1

Prior

Conditional

Joint

2

Event

Probability

Probability

Probability

3

A1

0.25

0.31

For the Excel worksheet above, which of the following formulas would correctly

calculate the joint probability for cell D3?

a. =SUM(B3:C3)

b. B3+C3

c. B3/C3

d. =B3*C3

80.

A

B

C

D

E

1

Prior

Conditional

Joint

Posterior

2

Event

Probability

Probability

Probability

Probability

3

A1

0.45

0.22

0.099

4

A2

0.55

0.16

0.088

5

0.187

For the Excel worksheet above, which of the following formulas would correctly

calculate the posterior probability for cell E3?

a. =SUM(B3:D3)

b. =D3/$D$5

c. =D5/$D$3

d. B3/C3+D3

PROBLEMS

1. All the employees of ABC Company are assigned ID numbers. The ID number consists

of the first letter of an employee’s last name, followed by four numbers.

a. How many possible different ID numbers are there?

b. How many possible different ID numbers are there for employees whose last

name starts with an “A”?

2. A company plans to interview 10 recent graduates for possible employment. The

company has three positions open. How many groups of three can the company select?

3. A student has to take 7 more courses before she can graduate. If none of the courses are

prerequisites to others, how many groups of three courses can she select for the next

semester?

4. A committee of 4 is to be selected from a group of 12 people. How many possible

committees can be selected?

5. The sales records of a real estate agency show the following sales over the past 200 days:

Number of

Houses Sold

Number

of Days

0

60

1

80

2

40

3

16

4

4

a. How many sample points are there?

b. Assign probabilities to the sample points and show their values.

c. What is the probability that the agency will not sell any houses in a given day?

d. What is the probability of selling at least 2 houses?

e. What is the probability of selling 1 or 2 houses?

f. What is the probability of selling less than 3 houses?

6. The results of a survey of 800 married couples and the number of children they had is

shown below.

Number

of Children

Probability

0

0.050

1

0.125

2

0.600

3

0.150

4

0.050

5

0.025

If a couple is selected at random, what is the probability that the couple will have

a. Less than 4 children?

b. More than 2 children?

c. Either 2 or 3 children?

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