Determine whether each of the following statements about events X, Y, Z is always true or not.
1. Suppose that X, Y, and Z are mutually exclusive events; then X^c (X complement) and Y U Z^c are mutually exclusive. (T/F)
2. From now on, we do not assume that X, Y, Z are mutually exclusive. Suppose that X is independent of Y, given Z.
(a) X^c is independent of Y, given Z. (T/F)
(b) X∩Z is independent of Y∩Z, given Z. (T/F)
(c) X is independent of Y,given Z^c. (T/F)
To determine whether each of the statements is always true or not, we need to understand the concepts involved and apply logical reasoning. Let's break down each statement and determine their truth values.
1. Suppose that X, Y, and Z are mutually exclusive events; then X^c (X complement) and Y U Z^c are mutually exclusive. (T/F)
To determine the truth value of this statement, we need to understand what it means for events to be mutually exclusive and what the complement of an event is.
Mutually exclusive events are events that cannot occur at the same time or have no common outcomes. If X, Y, and Z are mutually exclusive, it means that they cannot occur simultaneously.
The complement of an event X, denoted as X^c, includes all outcomes that are not in X. In other words, X^c consists of all events that are not part of X.
Given these definitions, we can analyze the statement. If X, Y, and Z are mutually exclusive, it means that their outcomes do not overlap. In this case, X^c would contain all outcomes that are not part of X, and Y U Z^c would contain outcomes that are part of Y or are not part of Z.
Since X^c and Y U Z^c both contain outcomes that are not part of X, they can overlap. Therefore, X^c and Y U Z^c are not necessarily mutually exclusive. Hence, the statement is not always true. (F)
2. From now on, we do not assume that X, Y, Z are mutually exclusive. Suppose that X is independent of Y, given Z.
(a) X^c is independent of Y, given Z. (T/F)
To determine the truth value of this statement, we need to understand what it means for events to be independent.
If events X and Y are independent given event Z, it means that the occurrence of event Z does not affect the probability of events X and Y occurring together. In other words, knowing whether Z occurred or not does not change the relationship between X and Y.
In this case, if X is independent of Y given Z, it implies that X^c (the complement of X) is also independent of Y given Z because the complement of an event only consists of outcomes that are not part of X.
Therefore, X^c would have the same independence from Y, given Z as X does. Thus, the statement is always true. (T)
(b) X∩Z is independent of Y∩Z, given Z. (T/F)
To determine the truth value of this statement, we need to consider the concept of independence and the intersection of events.
Independence between events means that the occurrence of one event does not affect the occurrence or probability of the other event. In this case, X∩Z represents the intersection of X and Z, which includes outcomes that are both in X and Z.
Given that X is independent of Y given Z, it means that knowing whether Z occurred or not does not change the relationship between X and Y. However, the intersection of X and Z (X∩Z) is not independent of Y∩Z because the occurrence of Z is common to both events.
Therefore, X∩Z is dependent on Y∩Z, given Z. Hence, the statement is not always true. (F)
(c) X is independent of Y, given Z^c. (T/F)
To determine the truth value of this statement, we need to understand the concept of independence and the complement of an event.
If X is independent of Y given Z, it means that the occurrence of event Z does not affect the probability of events X and Y occurring together.
In this case, Z^c represents the complement of Z, which includes all outcomes that are not part of Z. If X is independent of Y given Z, it does not automatically mean that X is independent of Y given Z^c. The complement of an event can have different relationships with other events compared to the original event.
Therefore, X is not necessarily independent of Y given Z^c. Hence, the statement is not always true. (F)
In summary:
1. False
2. (a) True
(b) False
(c) False