Copyright © Cengage Learning. All rights reserved. 4 Probability

Copyright © Cengage Learning. All rights reserved. Are Mutual Exclusiveness and Independence Related? 4.6

3 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.

4 Calculating Probabilities and the Addition Rule

5 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

6 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) =.

7 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)

8 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

9 Using Conditional Probabilities to Determine Independence

10 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

11 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?

12 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,

13 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,

14 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,

15 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).