Let's analyze the question step by step: ## Given: Two events \(A\) and \(B) are **independent**. Definition of Independent Events: Two events \(A\) and \(B\) are independent if: \[ P(A \cap B) = P(A) \cdot P(B) \] --- ## Option Analysis: **A) \(P(A \cap B) = P(A) + P(B)\)** - This is incorrect. This formula is not true for independent events. **B) \(P(A \cap B) = P(A)P(B)\)** - This is the **definition of independence**. **Correct!** **C) \(P(A|B) = P(B|A)\)** - This is not generally true for independent events. **D) \(P(A \cup B) = P(A)P(B)\)** - This is incorrect. The correct formula is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] --- ## **Final Answer:** \[ \boxed{B) \; P(A \cap B) = P(A)P(B)} \] This option matches the definition of independence.

# Solution to the Problem ## Given: Two events \( A \) and \( B \) are **independent**. ### Definition of Independent Events: Two events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) \cdot P(B) \] --- ## Option Analysis: ### **A)** \( P(A \cap B) = P(A) + P(B) \) - **Incorrect**: This formula does not hold for independent events. ### **B)** \( P(A \cap B) = P(A)P(B) \) - **Correct**: This is the definition of independence. ### **C)** \( P(A|B) = P(B|A) \) - **Incorrect**: This statement is not necessarily true for independent events. ### **D)** \( P(A \cup B) = P(A)P(B) \) - **Incorrect**: The correct formula for the union of two events is: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] --- ## **Final Answer:** \[ \boxed{B) \; P(A \cap B) = P(A)P(B)} \] This option correctly reflects the definition of independence for events \( A \) and \( B \).