Two events $A$ and $B$ will be independent, if
(A) $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive
(B) $P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]$
(C) $P(A)=P(B)$
(D) $P(A)+P(B)=1$

Solution:

Answer: (B) $P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]$
Concept: Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other.

Explanation:

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

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

$$
\Rightarrow 1-P(A \cup B)=1-P(A)-P(B)+P(A) P(B)
$$

$=-[P(A)+P(B)-P(A \cap B)]=-P(A)-P(B)+P(A) P(B)$

$=-P(A)-P(B)+P(A \cap B)=-P(A)-P(B)+P(A) P(B)$

$\Rightarrow P(A \cap B)=P(A) \cdot P(B)$

Final Answer: Hence, it shows $A$ and $B$ are Independent events.