Independent Events

Definition of Independence

Two events $A$ and $B$ are called independent if the occurrence of one has no effect on the probability of the other. Formally, $A$ and $B$ are independent if

Equivalence with the Conditional Definition

Whenever $P(B)>0$, this multiplicative condition is exactly equivalent to

and, symmetrically, $P(B\mid A)=P(B)$ whenever $P(A)>0$. This confirms the intuitive meaning of independence directly: knowing that $B$ has occurred leaves the probability of $A$ completely unchanged, and vice versa.

Independence Is Not the Same as Mutual Exclusivity

These two ideas are frequently confused but are in fact almost opposite. Two events $A$ and $B$ (with $P(A)>0,P(B)>0$) are mutually exclusive when $A\cap B=\varnothing$, so $P(A\cap B)=0$. If $A$ and $B$ were also independent, this would force $P(A)P(B)=0$, contradicting $P(A)>0$ and $P(B)>0$. So two events of positive probability can never be both mutually exclusive and independent -- mutual exclusivity means $B$ occurring makes $A$ impossible (the strongest possible dependence), the opposite of independence.

Mutual Independence of Three Events

Three events $A$, $B$, $C$ are said to be (mutually) independent when all four of the following hold simultaneously:

The first three conditions alone (pairwise independence) do not guarantee the fourth -- all four must be checked to conclude mutual independence.

Worked Illustration …