Show that if events A and B are independent, then events A' and B are independent.
If events $\displaystyle A,B$ are independent, then we know that
$\displaystyle P(A \cap B) = P(A) P(B)$ by definition. That's our key.
For the proof, we will use the following lemma:
$\displaystyle A^C \cap B = B \setminus (A \cap B)$
Proof of the lemma:
$\displaystyle A^C \cap B = (\Omega \setminus A) \cap B = (\Omega \cap B) \setminus (A \cap B) = B \setminus (A \cap B)$
So now we know that lemma is true, we can use it to conclude that:
$\displaystyle P(A^C \cap B) = P(B) - P(A \cap B)$
Looking at the right hand side, we have:
$\displaystyle =P(B) - P(A \cap B)$
$\displaystyle =P(B) - P(A)P(B)$ (using our key assumption)
$\displaystyle =(1-P(A))P(B)$
$\displaystyle =P(A^C)P(B)$
So we conclude that
$\displaystyle P(A^C \cap B) = P(A^C)P(B)$ as desired.
$\displaystyle QED$
$\displaystyle \begin{array}{*{20}c}
{P\left( {A^c \cap B} \right)} & = & {P\left( B \right) + P\left( {A \cap B} \right)} \\
{} & = & {P\left( B \right) + P\left( A \right)P\left( B \right)} \\
{} & = & {P(B)\left( {1 - P(A)} \right)} \\
{} & = & {P(B)P\left( {A^c } \right)} \\
\end{array} $