If two events are independent, then

If two events A and B are independent, then we know that $P(A \cap B) = P(A) \cdot P(B),P(A) \ne 0,P(B) \ne 0$

Since, $A$ and $B$ have a common outcome.
Further, mutually exclusive events never have a common outcome.

In other words, two independent events having non-zero probabilities of occurrence cannot be mutually exclusive and conversely, i.e., two mutually exclusive events having non-zero probabilities of outcome cannot be independent.