If A and B are independent events, then which of the following statements are TRUE?
(A) $P(A ∩ B) = P(A).P(B)$
(B) $P(A ∩ B) = P(A) - P(B)$
(C) $P(A ∪ B) = P(A) + P(B) - P(A).P(B)$
(D) $P(A ∩ B) = P(A).P(B|A)$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (3) → (A), (C) and (D) only
(A) $P(A ∩ B) = P(A).P(B)$ (True)
(B) $P(A ∩ B) = P(A) - P(B)$ (False)
(C) $P(A ∪ B) = P(A) + P(B) - P(A).P(B)$ (True)
(D) $P(A ∩ B) = P(A).P(B|A)$ (True)
Given that $A$ and $B$ are independent events, evaluate the truth of each statement:
(A) $P(A \cap B) = P(A) \cdot P(B)$
True. By definition of independence.
(B) $P(A \cap B) = P(A) - P(B)$
False. This is not generally true.
(C) $P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B)$
True. Since $P(A \cap B) = P(A) P(B)$ for independent events, this follows from the general formula:
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
(D) $P(A \cap B) = P(A) \cdot P(B|A)$
True. This is the definition of conditional probability, always true.