Independent Event Example 1 - Solution.

Example 1.) Mathematical Statistics by K.Knight. Chapter 1, 1.2
Suppose that A and B are independent events. Determine which of the following pairs of events are always independent and which are always disjoint.
(a) A and $B^c$
(c) $A^c$ and $B^c$  


Solution
(a) A and $B^c$
We need to check whether P(A \cap B^c)=P(A)\cdot P(B^c) or not.
$P(A)=P(A \cap B) \cup (A \cap B^c) \Rightarrow P(A \cap B^c)=P(A)-P(A \cap B)$
         $=P(A)\left \{ 1-P(B^c) \right \} = P(A) \cdot P(B^c)$
Therefore, it's true! They are independent.

(c) $A^c$ and $B^c$  
We need to check whether $P(A^c \cap B^c)=P(A^c) \cdot P(B^c)$ or not.
We know $(A^c \cap B^c)=(A \cup B)^c$, $\Rightarrow P( (A \cup B)^c) = 1 - P(A\cup B)=1 - P(A)-P(B)+P(A)\cdot P(B)$
     $= 1 - P(A)+P(B)(P(A)-1)=(1-P(A))(1-P(B))=P(A^c) \cdot P(B^c)$
Therefore, it's true! They are independent.