Independent Events
Statistics & ProbabilityIndependent 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.
Example
Flipping a coin and rolling a die are independent. Getting heads on the coin does not change the chance of rolling a $4$. $P(\text{heads AND rolling } 4) = 1/2 \times 1/6 = 1/12$.
Key Insight
Independence means you can multiply probabilities. If two things are independent, the "and" probability is just the two individual probabilities multiplied together.
Definition
Events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A)P(B)$. Equivalently, $P(A|B) = P(A)$: knowing $B$ occurred does not change the probability of $A$. Independence must be verified from the problem context or tested mathematically, not just assumed.
Example
Drawing a card, replacing it, then drawing again: the events are independent (the deck resets). Drawing without replacement: the events are dependent (the second draw is affected by the first).
Key Insight
Independence is a modeling assumption, not something that can be observed in a single trial. It must be justified by the structure of the experiment (e.g., random sampling with replacement).
Definition
Events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$. 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. Pairwise independence does not imply mutual independence.
Example
Pairwise but not mutually independent: 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 (each pair satisfies the product rule) 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 is central to most classical probability and statistical theory. 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.