On the proportion of derangements and on suborbits in finite transitive groups

We find a lower bound on the proportion of derangements in a finite transitive group that depends on the minimal nontrivial subdegree. As a consequence, we prove that, if $\Gamma$ is a $G$-vertex-transitive digraph of valency $d\ge 1$, then the proportion of derangements in $G$ is greater than $1/2d$.