## Definition ==Independent Events== are two or more events whose occurrence does not affect each other's probability. Events $A$ and $B$ are independent if and only if: $ P(A \cap B) = P(A) \cdot P(B) $ Equivalently, $P(A \mid B) = P(A)$ and $P(B \mid A) = P(B)$. > [!example]- A fair coin is tossed twice. Are the events $A$ = "heads on first toss" and $B$ = "heads on second toss" independent? > $P(A) = 0.5$, $P(B) = 0.5$, and $P(A \cap B) = P(\text{HH}) = 0.25$. > Since $P(A) \cdot P(B) = 0.5 \times 0.5 = 0.25 = P(A \cap B)$, the events are **independent**. Knowing the first toss was heads gives no information about the second toss.