Independent Events

Definition Two events $A$ and $B$ in the same probability model are independent if

The definition is symmetric in $A$ and $B$ and uses no conditional probability, so it applies even when one of the events has probability $0$. It depends on the probabilities assigned in the model, not only on the events as sets of outcomes.

Definition The product criterion tests independence by computing $P(A)$, $P(B)$ and $P(A \cap B)$ and comparing $P(A \cap B)$ with $P(A)P(B)$. When $P(B) > 0$, it is equivalent to a conditional test,

because dividing the product equation by $P(B)$ gives the conditional equation and multiplying back reverses the step. Likewise, when $P(A) > 0$ it is equivalent to $P(B \mid A) = P(B)$. An event with probability $0$ or probability $1$ satisfies the criterion with every event. If $P(B) = 0$, both sides are $0$. If $P(B) = 1$, then $P(A \cap B) = P(A) = P(A)P(B)$.

Definition Events $A$ and $B$ are dependent when they are not independent, that is, when

An event of probability $0$ or $1$ is independent of every event, so both probabilities of dependent events lie strictly between $0$ and $1$, and then $P(A \mid B) \ne P(A)$ and $P(B \mid A) \ne P(B)$. Disjoint events with $P(A) > 0$ and $P(B) > 0$ are always dependent, since $P(A \cap B) = 0$ while $P(A)P(B) > 0$. Dependence is a statement about probabilities in the model and does not by itself show that one event causes the other.

Definition If $A$ and $B$ are independent, then so are $A$ and $B^{c}$, $A^{c}$ and $B$, and $A^{c}$ and $B^{c}$. For the first pair, $A$ is the disjoint union of $A \cap B$ and $A \cap B^{c}$, so

Swapping the roles of $A$ and $B$ gives the second pair, and applying the result to the pair $A^{c}$ and $B$ gives the third. The statement is about complementing a member of an independent pair. An event $A$ is independent of its own complement exactly when $P(A)$ is $0$ or $1$, since $P(A \cap A^{c}) = 0$ while $P(A)P(A^{c}) = P(A)\bigl(1 - P(A)\bigr)$.

Definition Events $A_1, \dots, A_n$ are pairwise independent if every two of them are independent, that is, if $P(A_i \cap A_j) = P(A_i)P(A_j)$ for all $i \ne j$.

Definition Events $A_1, \dots, A_n$ are mutually independent if every subcollection of at least two of them satisfies the product rule. For all indices $i_1 < i_2 < \cdots < i_k$ with $k \ge 2$,

For three events $A$, $B$ and $C$ this means the three pair identities together with the triple identity

Mutual independence implies pairwise independence, but pairwise independence does not imply mutual independence. The triple identity alone is not enough either. If $C = \varnothing$, it holds for any $A$ and $B$, even dependent ones.