Question

If events \( A \) and \( B \) are independent, then:

• A. \( P(A \cap B) = P(A)P(B) \)
• B. \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
• C. \( P(A \cup B) = 0 \)
• D. \( P(A \cap B) = P(A) + P(B) \)

Hint:

If two events \( A \) and \( B \) are independent: - \( P(A \cap B) = P(A)P(B) \) - Independence is different from mutual exclusivity. - Always verify definitions carefully in probability problems.

Answer 1

The Correct Option is A

Step 1: By definition, two events \( A \) and \( B \) are independent if and only if: \[ P(A \cap B) = P(A) \cdot P(B) \] Step 2: This means the occurrence of one event does not affect the probability of the other. Step 3: Let's briefly check the other options: - (B) is always true for any two events (not just independent). - (C) implies both events are impossible, which is not implied by independence. - (D) is incorrect unless \( A \) and \( B \) are mutually exclusive and at least one of them has probability 0. Final Answer: \( \boxed{P(A \cap B) = P(A)P(B)} \)