If A and B are independent events then which of the following are TRU…

The correct answer is Option (2) → (B) and (D) only

Independent events satisfy \( P(A\cap B)=P(A)P(B) \). For the complements,

    \( \displaystyle P(A'\cap B') \)

    \( \displaystyle =1-P(A)-P(B)+P(A)P(B) \)

    \( \displaystyle =(1-P(A))(1-P(B)) \)

    \( \displaystyle =P(A')P(B') \)

So \( P(A\cap B)=P(A)P(B) \) and \( P(A'\cap B')=P(A')P(B') \), while \( P(A) \) and \( P(B) \) can take any values.

Checking the statements

(A) A and B are mutually exclusive — Incorrect. That needs \( P(A\cap B)=0 \), but \( P(A)P(B)\ne 0 \) when both probabilities are non-zero.

(B) A' and B' are independent — Correct. \( P(A'\cap B')=(1-P(A))(1-P(B))=P(A')P(B') \).

(C) P (A) + P (B) = 1 — Incorrect. For example, \( P(A)=P(B)=\frac{1}{4} \) gives independent events with sum \( \frac{1}{2} \).

(D) P (A ∩ B) = P(A) P(B) — Correct. This is the definition of independence.