CHAPTER 6: PROBABILITY
TRUE/FALSE
1. The relative frequency approach to probability uses long term relative frequencies, often based on past
data.
2. Predicting the outcome of a football game is using the subjective approach to probability.
3. You think you have a 90% chance of passing your next advanced financial accounting exam. This is
an example of subjective approach to probability.
4. P(A) + P(B) = 1 for any events A and B that are mutually exclusive.
5. The collection of all the possible outcomes of a random experiment is called a sample space.
6. If events A and B cannot occur at the same time, they are called mutually exclusive.
7. If either event A or event B must occur, they are called mutually exclusive.
8. If either event A or event B must occur, then A and B are mutually exclusive and collectively
exhaustive events.
9. If P(A) = 0.4 and P(B) = 0.6, then A and B must be collectively exhaustive.
10. If P(A) = 0.4 and P(B) = 0.6, then A and B must be mutually exclusive.
MULTIPLE CHOICE
1. Of the last 500 customers entering a supermarket, 50 have purchased a wireless phone. If the relative
frequency approach for assigning probabilities is used, the probability that the next customer will
purchase a wireless phone is
a.
0.10
c.
0.50
b.
0.90
d.
None of these choices.
2. If A and B are mutually exclusive events with P(A) = 0.75, then P(B):
a.
can be any value between 0 and 1.
c.
cannot be larger than 0.25.
b.
can be any value between 0 and 0.75.
d.
equals 0.25.
3. If you roll a balanced die 50 times, you should expect an even number to appear:
a.
on every other roll.
c.
25 times on average, over the long term.
b.
exactly 50 times out of 100 rolls.
d.
All of these choices are true.
4. An approach of assigning probabilities which assumes that all outcomes of the experiment are equally
likely is referred to as the:
a.
subjective approach
c.
classical approach
b.
objective approach
d.
relative frequency approach
5. The collection of all possible outcomes of an experiment is called:
a.
a simple event
c.
a sample
b.
a sample space
d.
a population
6. Which of the following is an approach to assigning probabilities?
a.
Classical approach
c.
Subjective approach
b.
Relative frequency approach
d.
All of these choices are true.
7. A sample space of an experiment consists of the following outcomes: 1, 2, 3, 4, and 5. Which of the
following is a simple event?
a.
At least 3
c.
3
b.
At most 2
d.
15
8. Which of the following is a requirement of the probabilities assigned to outcome Oi?
a.
P(Oi) 0 for each i
c.
0 P(Oi) 1 for each i
b.
P(Oi) 1 for each i
d.
P(Oi) = 1 for each i
9. If an experiment consists of five outcomes with P(O1) = 0.10, P(O2) = 0.20, P(O3) = 0.30, P(O4) =
0.25, then P(O5) is
a.
0.75
b.
0.15
c.
0.50
d.
Cannot be determined from the information given.
10. If two events are collectively exhaustive, what is the probability that one or the other occurs?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
11. If two events are collectively exhaustive, what is the probability that both occur at the same time?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
12. If two events are mutually exclusive, what is the probability that one or the other occurs?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
13. If two events are mutually exclusive, what is the probability that both occur at the same time?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
14. If two events are mutually exclusive and collectively exhaustive, what is the probability that both
occur?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
15. If the two events are mutually exclusive and collectively exhaustive, what is the probability that one or
the other occurs?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
16. If events A and B are mutually exclusive and collectively exhaustive, what is the probability that event
A occurs?
a.
0.25
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
17. If two equally likely events A and B are mutually exclusive and collectively exhaustive, what is the
probability that event A occurs?
a.
0.00
b.
0.50
c.
1.00
d.
Cannot be determined from the information given.
18. If event A and event B cannot occur at the same time, then A and B are said to be
a.
mutually exclusive
c.
collectively exhaustive
b.
independent
d.
None of these choices.
19. The collection of all possible events is called
a.
an outcome
c.
an event
b.
a sample space
d.
None of these choices.
COMPLETION
1. A random experiment is an action or process that leads to one of several possible
____________________.
2. The outcomes of a sample space must be ____________________, which means that all possible
outcomes must be included.
3. The outcomes of a sample space must be ____________________, which means that no two outcomes
can occur at the same time.
4. A(n) ____________________ of a random experiment is a list of all possible outcomes of the
experiment.
5. The outcomes of a sample space must be ____________________ and ____________________.
6. There are ____________________ requirements of probabilities for the outcomes of a sample space.
7. An individual outcome of a sample space is called a(n) ____________________ event.
8. A(n) ____________________ is a collection or set of one or more simple events in a sample space.
9. The probability of an event is the ____________________ of the probabilities of the simple events
that constitute the event.
10. No matter which approach was used to assign probability (classical, relative frequency, or subjective)
the one that is always used to interpret a probability is the ____________________ approach.
SHORT ANSWER
1. Alana, Eva, and Stephanie, three candidates for the presidency of a college’s student body, are to
address a student forum. The forum’s organizer is to select the order in which the candidates will give
their speeches, and must do so in such a way that each possible order is equally likely to be selected.
a.
What is the random experiment?
b.
List the outcomes in the sample space.
c.
Assign probabilities to the outcomes.
d.
What is the probability that Stephanie will speak first?
e.
What is the probability that Alana will speak before Stephanie does?
b.
c.
The probability assigned to each outcome is 1/6.
d.
e.
2. There are three approaches to determining the probability that an outcome will occur: classical,
relative frequency, and subjective. For each situation that follows, determine which approach is most
appropriate.
a.
A Russian will win the French Open Tennis Tournament next year.
b.
The probability of getting any single number on a balanced die is 1/6.
c.
Based on the past, it’s reasonable to assume the average book sales for a certain textbook
is 6,500 copies per month.
c.
relative frequency
Hobby Shop Sales
Sales records of a hobby shop showed the following number of radio controlled trucks sold weekly for
each of the last 50 weeks.
Number of Weeks
20
15
10
4
1
3. {Hobby Shop Sales Narrative} Define the random experiment of interest to the store.
4. {Hobby Shop Sales Narrative} List the outcomes in the sample space.
5. {Hobby Shop Sales Narrative} What approach would you use in determining the probabilities for next
week’s sales? Assign probabilities to the outcomes.
ANS:
6. {Hobby Shop Sales Narrative} What is the probability of selling at least two trucks in any given week?
7. {Hobby Shop Sales Narrative} What is the probability of selling between 1 and 3 (inclusive) trucks in
any given week?
Mutual Fund Price
An investor estimates that there is a 75% chance that a particular mutual fund’s price will increase to
$100 per share over the next three weeks, based on past data.
8. {Mutual Fund Price Narrative} Which approach was used to produce this figure?
9. {Mutual Fund Price Narrative} Interpret the 75% probability.
10. The sample space of the toss of a balanced die is S = {1, 2, 3, 4, 5, 6}. If the die is balanced, each
simple event (outcome) has the same probability. Find the probability of the following events:
a.
Rolling an odd number
b.
Rolling a number less than or equal to 3
c.
Rolling a number greater than or equal to 5
d.
Rolling a number between 2 and 5, inclusive.
ANS:
3/6
2/6
Equity Loan Rates
A survey of banks estimated the following probabilities for the interest rate being charged on a equity
loan based on a 30-year loan, based on past records.
Interest Rate
6.0%
6.5%
7.0%
7.5%
>7.5%
Probability
0.20
0.23
0.25
0.28
.04
11. {Equity Loan Rates Narrative} If a bank is selected at random from this distribution, what is the
probability that the interest rate charged on a home loan exceeds 7.0%?
ANS:
12. {Equity Loan Rates Narrative} What is the most common interest rate?
ANS:
13. {Equity Loan Rates Narrative} What approach was used in estimating the probabilities for the
interest rates?
14. The probability of the intersection is called a joint probability.
15. Two or more events are said to be independent when the occurrence of one event has no effect on the
probability that another will occur.
16. The union of events A and B is the event that occurs when either A or B or both occur. It is denoted as
‘A or B’.
17. If A and B are independent events with P(A) = 0.35 and P(B) = 0.55, then P(A|B) is 0.35/0.55 = .64.
18. Two events A and B are said to be independent if P(A|B) = P(B).
19. The conditional probability of event B given event A is denoted by P(A|B).
20. If A and B are independent events with P(A) = .40 and P(B) = .50, then P(A and B) = .20.
21. The intersection of two events A and B is the event that occurs when both A and B occur.
22. Two events A and B are independent if P(A and B) = 0.
23. The union of events A and B is the event that occurs when either A or B occurs but not both.
24. If A and B are independent, then P(A|B) = P(A) or P(B|A) = P(B).
25. If P(A) = .30, P(B) = .60, and P(A and B) = .20, then P(A|B) = .40.
26. Suppose the probability that a person owns both a cat and a dog is 0.10. Also suppose the probability
that a person owns a cat but not a dog is 0.20. The marginal probability that someone owns a cat is
0.30.
ANS:
27. The probability of the intersection of two events A and B is denoted by P(A and B) and is called the:
a.
marginal probability
b.
joint probability
c.
conditional probability of A given B
d.
conditional probability of B given A
28. The intersection of events A and B is the event that occurs when:
a.
either A or B occurs but not both
b.
neither A nor B occur
c.
both A and B occur
d.
All of these choices are true.
29. The probability of event A given event B is denoted by
a.
P(A and B)
b.
P(A or B)
c.
P(A|B)
d.
P(B|A)
30. Which of the following is equivalent to P(A|B)?
a.
P(A and B)
b.
P(B|A)
c.
P(A)/P(B)
d.
None of these choices.
31. Which of the following best describes the concept of marginal probability?
a.
It is a measure of the likelihood that a particular event will occur, regardless of whether
another event occurs.
b.
It is a measure of the likelihood that a particular event will occur, if another event has
already occurred.
c.
It is a measure of the likelihood of the simultaneous occurrence of two or more events.
d.
None of these choices.
32. If two events are independent, what is the probability that they both occur?
a.
0
b.
0.50
c.
1.00
d.
Cannot be determined from the information given
33. If the outcome of event A is not affected by event B, then events A and B are said to be
a.
mutually exclusive
b.
independent
c.
collectively exhaustive
d.
None of these choices.
34. If A and B are disjoint events with P(A) = 0.70, then P(B):
a.
can be any value between 0 and 1
b.
can be any value between 0 and 0.70
c.
cannot be larger than 0.30
d.
cannot be determined with the information given
ANS:
35. If P(A) = 0.65, P(B) = 0.58, and P(A and B) = 0.76, then P(A or B) is:
a.
1.23
b.
0.47
c.
0.18
d.
0.11
36. Suppose P(A) = 0.60, P(B) = 0.85, and A and B are independent. The probability of the complement of
the event (A and B) is:
a.
.4 .15 = .060
b.
0.40 + .15 = .55
c.
1 − (.40 + .15) = .45
d.
1 − (.6 .85) = .490
37. Which of the following statements is correct if the events A and B have nonzero probabilities?
a.
A and B cannot be both independent and disjoint
b.
A and B can be both independent and disjoint
c.
A and B are always independent
d.
A and B are always disjoint
38. A and B are disjoint events, with P(A) = 0.20 and P(B) = 0.30. Then P(A and B) is:
a.
0.50
b.
0.10
c.
0.00
d.
0.06
ANS:
39. If P(A) = 0.35, P(B) = 0.45, and P(A and B) = 0.25, then P(A|B) is:
a.
1.4
b.
1.8
c.
0.714
d.
0.556
40. If A and B are independent events with P(A) = 0.60 and P(A|B) = 0.60, then P(B) is:
a.
1.20
b.
0.60
c.
0.36
d.
cannot be determined with the information given
41. If A and B are independent events with P(A) = 0.20 and P(B) = 0.60, then P(A|B) is:
a.
0.20
b.
0.60
c.
0.40
d.
0.80
42. If P(A) = 0.25 and P(B) = 0.65, then P(A and B) is:
a.
0.25
b.
0.40
c.
0.90
d.
cannot be determined from the information given
Cars
Suppose X = the number of cars owned by a family in the U.S. The probability distribution of X is
shown in the table below.
X
0
1
2
3
Probability
0.56
0.23
0.12
0.09
43. {Car Narrative}What is the chance that a family owns more than one car?
a.
0.23
b.
0.21
c.
0.44
d.
None of these choices.
44. {Cars Narrative} Suppose you choose two families at random. What is the chance that they each own
one car? (That means family A owns a car and family B owns a car.)
a.
0.23
b.
0.23 + 0.23 = 0.46
c.
0.23 + 0.23 − (0.23)*(0.23) = .4071
d.
(0.23)*(0.23) = 0.0529
45. The ____________________ of events A and B is the event that occurs when both A and B occur.
46. The probability of an intersection of two events is called a(n) ____________________ probability.
47. Suppose two events A and B are related. The ____________________ probability of A is the
probability that A occurs, regardless of whether event B occurred or not.
48. If two events are mutually exclusive, their joint probability is ____________________.
49. A conditional probability of A given B is written in probability notation as ____________________.
50. If A and B are independent, then P(A|B) = ____________________.
51. The ____________________ of two events A and B is the event that occurs when either A or B or both
occur.
52. If A and B are mutually exclusive, their joint probability is ____________________.
53. P(A|B) is the conditional probability of ____________________ given ____________________.
54. If P(A|B) = P(A) then events A and B are ____________________.
Tea and Seltzer
Suppose 55 percent of adults drink tea, 45 percent drink seltzer, and 10 percent drink both.
55. {Tea and Seltzer Narrative} What is the probability that a randomly chosen adult does not drink
seltzer?
56. {Tea and Seltzer Narrative} What is the probability that a randomly chosen adult drinks seltzer or tea
or both?
57. {Tea and Seltzer Narrative} What is the probability that a randomly chosen adult doesn’t drink tea or
seltzer?
Club Members
A survey of a club’s members indicates that 50% own a home, 80% own a car, and 90% of the
homeowners who subscribe also own a car.
58. {Club Members Narrative} What is the probability that a subscriber owns both a car and a house?
59. {Club Members Narrative} What is the probability that a club member owns a car or a house, or both?
60. {Club Members Narrative} What is the probability that a club member owns neither a car nor a house?
Business Majors
Suppose 30% of business majors major in accounting. You take a random sample of 3 business
majors.
61. {Business Majors Narrative} What is the chance that they all major in accounting?
62. {Business Majors Narrative} What is the chance that at least one majors in accounting?