Question

Let \(A\) and \(B\) be two events such that \( P(A) = \frac{5}{11}, \; P(B) = \frac{2}{11} \) and \( P(A \cup B) = \frac{3}{11} \), then \( P(A'|B') = \)_____

• A. \( \frac{8}{9} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{1}{2} \)
• D. \( \frac{2}{9} \)

Hint:

Use complement rule: \( P(A' \cap B') = 1 - P(A \cup B) \).

Answer 1

The Correct Option is A

Concept: \[ P(A'|B') = \frac{P(A' \cap B')}{P(B')} \] Also, \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Step 1: Find \( P(A \cap B) \). \[ \frac{3}{11} = \frac{5}{11} + \frac{2}{11} - P(A \cap B) \] \[ P(A \cap B) = \frac{4}{11} \]
Step 2: Find complements. \[ P(B') = 1 - \frac{2}{11} = \frac{9}{11} \] \[ P(A' \cap B') = 1 - P(A \cup B) = 1 - \frac{3}{11} = \frac{8}{11} \]
Step 3: \[ P(A'|B') = \frac{8/11}{9/11} = \frac{8}{9} \]