Statistical Independence

Two events $A$ and $B$ are statistically independent if the occurrence of one does not affect the probability of the other.

Definition

Events $A$ and $B$ are independent if:

$$ Equivalently, using conditional probability: $$ P(A|B) = P(A) \quad \text{and} \quad P(B|A) = P(B) $$ ## Quick Example **Scenario:** Flipping a fair coin and rolling a fair six-sided die simultaneously. - Event $A$: Coin shows heads → $P(A) = 0.5$ - Event $B$: Die shows a 3 → $P(B) = \frac{1}{6}$ Since the coin flip and die roll don't affect each other, they are independent: $$ P(A \cap B) = P(\text{heads AND 3}) = 0.5 \times \frac{1}{6} = \frac{1}{12} $$ > [!tip] Quick Check > If knowing one event gives no information about the other, they're independent. ## Multiple Events For $n$ events $E_1, E_2, \ldots, E_n$ to be mutually independent, any subset must satisfy: $$ P\left(\bigcap_{i \in S} E_i\right) = \prod_{i \in S} P(E_i) \quad \text{for all } S \subseteq \{1,2,\ldots,n\} $$ ## Common Misconception > [!warning] Independence vs Mutually Exclusive > **Independent:** $P(A \cap B) = P(A) \cdot P(B)$ > **Mutually Exclusive:** $P(A \cap B) = 0$ > > These are **not** the same! Independent events can occur together (e.g., heads and rolling a 3). $$