Question

If \( A \) and \( B \) are events such that \( P(A) = 0.42 \), \( P(B) = 0.48 \), and \( P(A \cap B) = 0.16 \), then: I. \( P(\text{not } A) = 0.58 \)
II. \( P(\text{not } B) = 0.52 \)
III. \( P(A \cup B) = 0.47 \)

• A. Only I and III are correct
• B. Only I and II are correct
• C. Only I and III are true
• D. All three statements are correct

Hint:

For any two events, use the addition rule for the union, and the complement rule for finding probabilities of "not" events.

Answer 1

The Correct Option is D

Step 1: Using the complementary rule, \( P(\text{not } A) = 1 - P(A) = 0.58 \).
Step 2: Similarly, \( P(\text{not } B) = 1 - P(B) = 0.52 \).
Step 3: Using the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.42 + 0.48 - 0.16 = 0.74. \]

Final Answer: \[ \boxed{All three statements are correct} \]