4
Probability
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4.6
Are Mutual Exclusiveness
and Independence Related?
Copyright © Cengage Learning. All rights reserved.
Are Mutual Exclusiveness and Independence Related?
Mutually exclusive events and independent events are two
very different concepts, based on definitions that start from
very different orientations. The two concepts can easily be
confused because they interact with each other and are
intertwined by the probability statements we use in
describing these concepts.
To describe these two concepts and eventually understand
the distinction between them as well as the relationship
between them, we need to agree that the events being
considered are two nonempty events defined on the same
sample space and therefore each has nonzero
probabilities.
3
Calculating Probabilities and
the Addition Rule
4
Calculating Probabilities and the Addition Rule
A pair of dice is rolled. Event T is defined as the occurrence
of “a total of 10 or 11,” and event D is the occurrence of
“doubles.”
To find the probability
P(T or D), you need to
look at the sample space
of 36 ordered pairs for
the rolling of two dice in
Figure 4.5.
Sample Space for the Roll of Two Dice
Figure 4.5
5
Calculating Probabilities and the Addition Rule
Event T occurs if any one of 5 ordered pairs occurs: (4, 6),
(5, 5), (6, 4), (5, 6), (6, 5). Therefore, P(T) = .
Event D occurs if any one of 6 ordered pairs occurs: (1, 1),
(2, 2), (3, 3), (4, 4), (5, 5), (6, 6). Therefore, P(D) = .
Notice, however, that these two events are not mutually
exclusive. The two events “share” the ordered pair (5, 5).
Thus, the probability P(T and D) =
.
6
Calculating Probabilities and the Addition Rule
As a result, the probability P(T or D) will be found using
formula (4.4).
P(T or D) = P(T) + P(D) – P(T and D)
7
Calculating Probabilities and the Addition Rule
Look at the sample space in Figure 4.5 and verify
P(T or D) = .
Sample Space for the Roll of Two Dice
Figure 4.5
8
Using Conditional Probabilities
to Determine Independence
9
Using Conditional Probabilities to Determine Independence
You can also use conditional probabilities to determine
independence. In a sample of 150 residents, each person
was asked if he or she favored the concept of having a
single, countywide police agency.
The county is composed of one large city and many
suburban townships. The residences (city or outside the
city) and the responses of the residents are summarized in
Table 4.4.
Sample Results
Table 4.4
10
Using Conditional Probabilities to Determine Independence
If one of these residents was to be selected at random,
what is the probability that the person would
(a) favor the concept?
(b) favor the concept if the person selected is a city
resident?
(c) favor the concept if the person selected resides outside
the city?
(d) Are the events F (favors the concept) and C (resides in
city) independent?
11
Using Conditional Probabilities to Determine Independence
A series of simple calculations provides the solution:
(a) P(F) is the proportion of the total sample that favors the
concept. Therefore,
12
Using Conditional Probabilities to Determine Independence
(b) P(F | C) is the probability that the person selected
favors the concept, given that he or she lives in the city.
The condition “is a city resident” reduces the sample
space to the 120 city residents in the sample.
Of these, 80 favored the concept; therefore,
13
Using Conditional Probabilities to Determine Independence
(c) P(F | ) is the probability that the person selected
favors the concept, given that the person lives outside
the city.
The condition “lives outside the city” reduces the sample
space to the 30 non-city residents; therefore,
14
Using Conditional Probabilities to Determine Independence
(d) All three probabilities have the same value, .
Therefore, we can say that the events F (favors) and C
(resides in city) are independent.
The location of residence did not affect P(F).
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