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
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.