If A and B are independent events, then which of the following statements are TRUE?
(A) \(P(A \cap B) = P(A).P(B)\)
(B) \(P(A \cap B) = P(A) - P(B)\)
(C) \(P(A \cup B) = P(A) + P(B) - P(A).P(B)\)
(D) \(P(A \cap B) = P(A).P(B|A)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
For independent events, \(P(A \cap B) = P(A)P(B)\).
Checking the statements
(A) \(P(A \cap B) = P(A).P(B)\) — Correct. This is the definition of independent events.
(B) \(P(A \cap B) = P(A) - P(B)\) — Incorrect. With \(P(A) = P(B) = \frac{1}{2}\), \(P(A \cap B) = \frac{1}{4}\), while \(P(A) - P(B) = 0\).
(C) \(P(A \cup B) = P(A) + P(B) - P(A).P(B)\) — Correct. By the addition rule, with \(P(A \cap B) = P(A)P(B)\):
\( \displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
\( \displaystyle = P(A) + P(B) - P(A)P(B) \)
(D) \(P(A \cap B) = P(A).P(B|A)\) — Correct. This is the multiplication rule, true for any events with \(P(A) \ne 0\). For independent events \(P(B|A) = P(B)\), so it agrees with (A).