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}\;=$$\;\frac{1}{12}$.
• A single die is rolled once. For that single rolling, let $A$ be "roll a 2," and $B$ be "roll a 5." Are these events independent?
  Solution: No, because $P(A \cap B) = 0$, but $P(A) \cdot P(B)\;=$$\;\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.