Supersoluble groups and the probability of generating a supersoluble subgroup

Let $G$ be a finite group and let $\mathrm{P}_{\mathcal{U}}(G)$ denote the probability that two randomly chosen elements of $G$ generate a supersoluble subgroup. We prove that if $\mathrm{P}_{\mathcal{U}}(G) > 16/25$ then $G$ is supersoluble, and that the bound $16/25$ is sharp, being attained by th