Question

A and B are independent events of a random experiment if and only if:

• A. \( P(A | B) \neq P(A \cap B) \)
• B. \( P(A | B) = P(B | A) \)
• C. \( P(A | B) \neq P(A | B^c) \)
• D. \( P(A | B) = P(A | B^c) \)

Hint:

For two events to be independent, the probability of one event occurring should not change based on the occurrence of the other event. Therefore, conditional probabilities should remain constant regardless of the event.

Answer 1

The Correct Option is D

Two events \(A\) and \(B\) are independent if and only if the occurrence of one event does not affect the probability of the other. This can be mathematically represented as: \[ P(A | B) = P(A) \quad \text{and} \quad P(B | A) = P(B). \] In terms of conditional probabilities, we have the condition for independence of events as: \[ P(A | B) = P(A) \quad \text{which implies that} \quad P(A | B) = P(A | B^c). \] Thus, the correct answer is \(\boxed{(d)}\).