  The relative commutativity degree of a subgroup $H$ of a finite group $G$,
denoted by $\Pr(H, G)$, is the probability that an element of $G$ commutes with
an element of $H$. In this article we obtain some lower and upper bounds for
$\Pr(H, G)$ and their consequences. We also study an invariance property of
$\Pr(H, G)$ and its generalizations, under isoclinism of pairs of groups.
