  Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the
probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also
a right transversal, thus a two-sided one. Moreover, we define, and denote by
$\mathrm{tp}(G)$, the common transversal probability of $G$ to be the minimum,
taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results
regarding the invariant $\mathrm{tp}(G)$, like lower and upper bounds, and
possible values it can attain. We also show that $\mathrm{tp}(G)$ determines
structural properties of $G$. Finally, several open problems are formulated and
discussed.
