1.6a Independence of Two Events

Pairwise Independence

Two events $A,B\subseteq \Omega$ in the same probability space are said to be pairwise independent, if their joint probability is equal to the product of their marginal probabilities, such that
$\mathbb{P}[A\cap B]=\mathbb{P}[A]\times \mathbb{P}[B].$
Moreover, if $\mathbb{P}[A]>0$, independence is equivalent to the condition,
$\mathbb{P}[B\mid A]=\mathbb{P}[B],$
and vice-versa.

Intuition

Consider the law of conditional probability, which states that, given events $A,B\in \Omega$, the probability of $B$ given $A$ is $\mathbb{P}[B\mid A]=\frac{\mathbb{P}[A\cap B]}{\mathbb{P}[A]}$. However, if $B$ is not updated given the information that $A$ has occurred, then equivalently, $\mathbb{P}[B\mid A]=\mathbb{P}[B]$. Then, $A$ and $B$ are independent, and in order for the equivalence to hold, $\mathbb{P}[A\cap B]=\mathbb{P}[A]\times \mathbb{P}[B]$.

We adopt the latter definition of independence, since it can be used even when $\mathbb{P}[A]=0$, where $\mathbb{P}[B\mid A]$ is undefined.