Let $X := {1\dots m}$ be the set of indexes corresponding to the elements of the vector of i.i.d. random variables $w:=(w_1, \dots, w_m)$. Let there be the subsets $Y, Z \subseteq X$ which may or may not overlap.
I am trying to obtain a lower bound on the expression
$\mathbb{P}\left(|\prod_{i \in Y} w_i|\geq 1 + |\sum_{i \in Z}w_i)|\right)$.
For example, for random variables $w:=(a, b, c)$, I would be interested e.g. in a lower bound for
$\mathbb{P}(|ab| \geq 1 + |b+c|)$.
Perhaps there is a concentration inequality, transformation, etc. that applies to this case?