Independence and Mutual Exclusivity

More examples:

• Two dice are rolled. Let event $A$ be "first die shows even," and $B$ be "second die shows a 5." Are these events independent?
  Solution: Yes, because $P(\text{first: even} \cap \text{second: 5}) \;$$= P(\text{first: even}) \cdot P(\text{second: 5}) = \frac{3}{6} \cdot \frac{1}{6} = \frac{1}{12}$.
• A single die is rolled. Let $A$ be "roll a 2," and $B$ be "roll a 5." Are these events independent?
  Solution: No, because $P(A \cap B) = 0$, but $P(A) \cdot P(B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$, so $P(A \cap B) \neq P(A) \cdot P(B)$. In fact, these events are mutually exclusive.

Bottom line: Two independent events are never mutual exclusive, and two mutual exclusive events are never independent.