Question

If \(A\) and \(B\) are any two events such that \( P(A) + P(B) - P(A and B) = P(A) \), then _____

• A. \( P(B|A') = 1 \)
• B. \( P(B|A) = 0 \)
• C. \( P(A|B) = 1 \)
• D. \( P(A|B) = 0 \)

Hint:

If \( P(A \cap B) = P(B) \), then event \(B\) is completely inside \(A\).

Answer 1

The Correct Option is C

Concept: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Step 1: Given condition. \[ P(A) + P(B) - P(A \cap B) = P(A) \] \[ \Rightarrow P(B) = P(A \cap B) \]
Step 2: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = 1 \]