Question

Let \( A \) and \( B \) be two events in a probability space with \( P(A) = 0.3 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.1 \). Which of the following statements is/are TRUE?

• A. The two events \( A \) and \( B \) are independent
• B. \( P(A \cup B) = 0.7 \)
• C. \( P(A \cap B^c) = 0.2 \), where \( B^c \) is the complement of the event \( B \)
• D. \( P(A^c \cap B^c) = 0.4 \), where \( A^c \) and \( B^c \) are the complements of the events \( A \) and \( B \), respectively

Hint:

When solving probability problems, always verify independence using ( P(A cap B) = P(A) cdot P(B) ) and use complement and union rules to simplify calculations.

Answer 1

The Correct Option is B

Step 1: Check independence of events \( A \) and \( B \). Two events \( A \) and \( B \) are independent if: \[ P(A \cap B) = P(A) \cdot P(B). \] Here: \[ P(A \cap B) = 0.1, \quad P(A) \cdot P(B) = 0.3 \cdot 0.5 = 0.15. \] Since \( P(A \cap B) \neq P(A) \cdot P(B) \), the events are NOT independent. Option (1) is FALSE. Step 2: Compute \( P(A \cup B) \). Using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B), \] substitute the values: \[ P(A \cup B) = 0.3 + 0.5 - 0.1 = 0.7. \] Thus, Option (2) is TRUE. Step 3: Compute \( P(A \cap B^c) \). Using the complement rule: \[ P(A \cap B^c) = P(A) - P(A \cap B). \] Substitute the values: \[ P(A \cap B^c) = 0.3 - 0.1 = 0.2. \] Thus, Option (3) is TRUE. Step 4: Compute \( P(A^c \cap B^c) \). Using the complement rule: \[ P(A^c \cap B^c) = 1 - P(A \cup B). \] Substitute the value of \( P(A \cup B) \): \[ P(A^c \cap B^c) = 1 - 0.7 = 0.3. \] Since the option states \( P(A^c \cap B^c) = 0.4 \), Option (4) is FALSE. Final Answer: \[ \boxed{\text{(2) } P(A \cup B) = 0.7, \text{(3) } P(A \cap B^c) = 0.2} \]