  A relativistic jet has been produced in the single well-localised binary
neutron star (BNS) merger detected to date in gravitational waves (GWs), and
the local rates of BNS mergers and short gamma-ray bursts are of the same order
of magnitude. This suggests that jet formation is not a rare outcome for BNS
mergers, and we show that this intuition can be turned into a quantitative
constraint: at least about $1/3$ of GW-detected BNS mergers, and at least about
$1/5$ of all BNS mergers, should produce a successful jet (90\% credible
level). Whether a jet is launched depends on the properties of the merger
remnant and of the surrounding accretion disc, which in turn are a function of
the progenitor binary masses and equation of state (EoS). The incidence of jets
in the population therefore carries information about the binary component mass
distribution and EoS. Under the assumption that a jet can only be produced by a
black hole remnant surrounded by a non-negligible accretion disc, we show how
the jet incidence can be used to place a joint constraint on the space of BNS
component mass distributions and EoS. The result points to a broad mass
distribution, with particularly strong support for masses in the
$1.3-1.6\,\mathrm{M_\odot}$ range. The constraints on the EoS are shallow, but
we show how they will tighten as the knowledge on the jet incidence improves.
We also discuss how to extend the method to include future BNS mergers, with
possibly uncertain jet associations.
