We are told that events A and B are conditionally independent, given a third event C, and that P(B∣C)>0. For each one of the following statements, decide whether the statement is “Always true", or “Not always true."

Question

We are told that events A and B are conditionally independent, given a third event C, and that P(B∣C)>0. For each one of the following statements, decide whether the statement is “Always true", or “Not always true."

A and B are conditionally independent, given the event Cc.

- unanswered

A and Bc are conditionally independent, given the event C.
- unanswered

P(A∣B∩C)=P(A∣B)
- unanswered

P(A∣B∩C)=P(A∣C)

Answer 1

1. Not always true

2. Always true

3. Not always true

4. Always true

(I am taking this exam, too)

Answer 2

Not always true.

Answer 3

To determine whether each statement is "Always true" or "Not always true," we need to understand what it means for events to be conditionally independent given another event.

Two events A and B are conditionally independent given event C if and only if the conditional probability of A given B and C is equal to the conditional probability of A given C. In other words, if P(A|B∩C) = P(A|C), then events A and B are considered conditionally independent given event C.

Now let's analyze each statement:

1. A and B are conditionally independent, given the event Cc.
This statement is "Not always true." The complement of event C, denoted as Cc, refers to the event that C does not occur. For A and B to be conditionally independent given Cc, we need to have P(A|B∩Cc) = P(A|Cc). However, there is no guarantee that this equality holds true in every scenario. Therefore, the statement is not always true.

2. A and Bc are conditionally independent, given the event C.
This statement is "Not always true." For A and Bc to be conditionally independent given C, we require P(A|Bc∩C) = P(A|C). Similarly to the previous statement, there is no guarantee that this equality always holds true.

3. P(A|B∩C) = P(A|B)
This statement is "Not always true." If P(A|B∩C) = P(A|B), it implies that events A and B are conditionally independent given C. However, there is no guarantee that this equality holds true in every case.

4. P(A|B∩C) = P(A|C)
This statement is "Always true." If P(A|B∩C) = P(A|C), it indicates that events A and B are conditionally independent given C. This equality is a definition of conditional independence and holds true by its definition.

In summary, statements 1, 2, and 3 are "Not always true," while statement 4 is "Always true."