Two events $A$ and $B$ are independent if and only if $P(A \cap B) = P(A)P(B)$.
Now, let's prove that $A$ and $B^{c}$ are independent. We need to show that $P(A \cap B^{c}) = P(A)P(B^{c})$. Using the complement rule, we have $B^{c} = \Omega \setminus B$, where
Show more…