Two events $A$ and $B$ will be independent, if

(A) $A$ and $B$ are mutually exclusive

(B) $P\left( {{A^\prime } \cap {B^\prime }} \right) = [1 - P(A)][1 - P(B)]$

(C) $P(A) = P(B)$

(D) $P(A) + P(B) = 1$

Option B is correct

$A$ and $B$ are independent $\Rightarrow$ $P(A \cap B) = P(A)P(B)$

and
$P\left( {{A^\prime } \cap {B^\prime }} \right) = P{(A \cup B)^\prime } = 1 - P(A \cup B)$

$= 1 - \{ P(A) + P(B) - P(A \cap B)\} = 1 - P(A) - P(B) + P(A)P(B)$

$= (1 - P(A))(1 - P(B))$

Exercise-13.3