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

Step 1: Understanding the Concept:
The expression $P(A) + P(B) - P(A \cap B)$ represents the probability of the union $P(A \cup B)$. We simplify the given equation to find the relationship between $A$ and $B$.
Step 2: Formula Derivation:
Given: $P(A) + P(B) - P(A \cap B) = P(A)$
Subtracting $P(A)$ from both sides: $$P(B) - P(A \cap B) = 0 \implies P(A \cap B) = P(B)$$
Step 3: Explanation:
Check option (c): $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B)}{P(B)} = 1$. Therefore, if $P(A \cap B) = P(B)$, then $P(A|B) = 1$ is the mathematically correct identity. Note: If the original equation was $P(A) + P(B) - P(A \cap B) = P(A) + P(B)$, then $P(A \cap B) = 0$, leading to option (b). Based on the exact text provided ($= P(A)$), (c) is the result.
Step 4: Final Answer:
The correct option is (c).