Independent Events
Independent events are events where the outcome of one does not affect the probability of the other.
Formula
P(A \text{ and } B) = P(A) \times P(B) \quad [\text{when } A \text{ and } B \text{ are independent}]
Definition
Independent events are events that do not affect each other: knowing that one happened tells you nothing about whether the other happened. Events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A)P(B)$, equivalently $P(A|B) = P(A)$, meaning knowing $B$ occurred does not change the probability of $A$; independence must be verified from the problem context or tested mathematically, not just assumed. Formally, for a collection $\{A_1, \ldots, A_n\}$, mutual independence requires $P(A_{i_1} \cap \ldots \cap A_{i_k}) = \prod P(A_{i_j})$ for all subsets, and pairwise independence does not imply mutual independence.
Example
Flipping a coin and rolling a die are independent: getting heads does not change the chance of rolling a $4$, so $P(\text{heads AND rolling } 4) = 1/2 \times 1/6 = 1/12$. Drawing a card, replacing it, then drawing again keeps the events independent (the deck resets), while drawing without replacement makes them dependent (the second draw is affected by the first). A classic example of pairwise but not mutually independent events: flip two fair coins ($X_1, X_2$) and define $X_3 = X_1 \oplus X_2$; then $X_1$, $X_2$, $X_3$ are pairwise independent but $P(X_1=H, X_2=H, X_3=H) = 0 \neq P(X_1=H)P(X_2=H)P(X_3=H) = 1/8$, so they are not mutually independent.
Key Insight
Independence means you can multiply probabilities: if two things are independent, the "and" probability is just the two individual probabilities multiplied together, but independence is a modeling assumption, not something observable in a single trial, and must be justified by the structure of the experiment (e.g., random sampling with replacement). The i.i.d. (independent and identically distributed) assumption underlies the CLT, law of large numbers, and most estimator properties; testing for independence uses chi-square tests for categorical variables and correlation-based tests for continuous variables.