  For $a,b\in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect
product of the Witt algebra and a module of the intermediate series. In this
paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly,
these Lie algebras have symmetric (and skewsymmetric) non-inner biderivations.
As an applications, commutative post-Lie algebra structures on
$\mathcal{W}(a,b)$ are obtained.
