(related to Definition: Pairwise Independent Events)
it does not (!) follow
\[0=p(A_i\cap A_j)=p(A_i)\cdot p(A_j),\quad\quad( * )\]
\[A_i,A_j\subseteq\Omega, i\neq j.\]
In particular, at least one of the probabilities in would have to be \(0\) to fulfill \( ( * ) \), i.e. it would be the impossible event. In other words, e.g. if we assume \(p(A_i) > 0\), then the event \(A_i\) can occur. But then \(p(A_j)\) drops to \(0\). Thus, \(A_j\) cannot occur, if \(A_i\) can, and vice versa. Therefore, both events are not independent (i.e. their occurrence influences each other).