Answer
$\bar E$ and $F$ are independent events
Work Step by Step
Given: E and F are independent.
To Prove: $\overline E$ and $F$ are independent
Proof:
Since E and F are independent.
$$P(E\cap F)=P(E).P(F)$$
Use the complement rule:$$P(\overline E)=1-P(E)$$
Let us determine their product:
$P(\bar E).P(F)=(1-P(E)).P(F)$
$=P(F)-P(E).P(F)$
$=P(F)-P(E\cap F)$
$=P(\bar E \cap F)$
Since $P(\bar E).P(F)=P(\bar E \cap F), $ $\bar E$ and $F$ are independent events.