If p(A | B) = p(A), what can be concluded?

If p(A | B) = p(A), what can be concluded?

• A. Two events are independent
• B. Two events are mutually exclusive
• C. p(A) = p(B)
• D. None of the above apply

Answer: (A)

When analyzing the statement that the probability of event A given event B (p(A | B)) is equal to the probability of event A (p(A)), it indicates that the occurrence of event B does not affect the likelihood of event A occurring. This is the defining characteristic of independent events in probability theory. In formal terms, two events A and B are said to be independent if the probability of A occurring does not change when B occurs. The mathematical condition for independence is precisely expressed as p(A | B) = p(A). Therefore, when this condition is met, it conclusively shows that events A and B are independent. The other choices do not correctly interpret the implied relationship from the equality stated in the question. For example, mutual exclusivity would mean that the occurrence of one event precludes the occurrence of another, which is not addressed by the equality of p(A | B) and p(A). Similarly, stating that p(A) = p(B) does not follow from the provided condition and is unrelated to the independence of events A and B. Thus, the correct conclusion based on the given information is that the two events are independent.

When analyzing the statement that the probability of event A given event B (p(A | B)) is equal to the probability of event A (p(A)), it indicates that the occurrence of event B does not affect the likelihood of event A occurring. This is the defining characteristic of independent events in probability theory.

In formal terms, two events A and B are said to be independent if the probability of A occurring does not change when B occurs. The mathematical condition for independence is precisely expressed as p(A | B) = p(A). Therefore, when this condition is met, it conclusively shows that events A and B are independent.

The other choices do not correctly interpret the implied relationship from the equality stated in the question. For example, mutual exclusivity would mean that the occurrence of one event precludes the occurrence of another, which is not addressed by the equality of p(A | B) and p(A). Similarly, stating that p(A) = p(B) does not follow from the provided condition and is unrelated to the independence of events A and B. Thus, the correct conclusion based on the given information is that the two events are independent.