Statement-1: If A and B are two events such that P(A)=1, then A and B are independent.
Statement-2: A and B are two independent events iff
$P(A ∩ B) = P(A) P(B)$
Answer & explanation
Correct answer: option 1
Clearly, statement-2 is true. (see Theory)
If $P(A)=1$, then $ P(\overline{A})=0$
Now, $\overline{A} ∩ B ⊂ \overline{A}$
$⇒ P(\overline{A} ∩ B) ≤ P(\overline{A})$
$⇒ P(\overline{A} ∩ B) ≤ 0$
$⇒ P(\overline{A} ∩ B) = 0$
$⇒ P(\overline{A} ∩ B) = P(\overline{A})P(B)$ $[∵ P(\overline{A})=0]$
$⇒(\overline{A})$ and B are independent events.
⇒ A and B are independent events.