Question

If \(A\) and \(B\) are two such events that \(P(A \cup B) = P(A \cap B)\), then which of the following is true?

• A. \(P(A) + P(B) = 0\)
• B. \(P(A) + P(B) = P(A)P(B|A)\)
• C. \(P(A) + P(B) = 2P(A)P(B|A)\)
• D. None of the above

Hint:

Always convert \(P(A \cap B)\) into conditional form when options involve \(P(B|A)\).

Answer 1

The Correct Option is C

Concept: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Step 1:Given condition.
\[ P(A \cup B) = P(A \cap B) \]

Step 2:Substitute.
\[ P(A) + P(B) - P(A \cap B) = P(A \cap B) \] \[ P(A) + P(B) = 2P(A \cap B) \]

Step 3:Use conditional probability.
\[ P(A \cap B) = P(A)P(B|A) \] \[ \Rightarrow P(A) + P(B) = 2P(A)P(B|A) \] Conclusion : (C)