  The detection of an electromagnetic transient which may originate from a
binary neutron star merger can increase the probability that a given segment of
data from the LIGO-Virgo ground-based gravitational-wave detector network
contains a signal from a binary coalescence. Additional information contained
in the electromagnetic signal, such as the sky location or distance to the
source, can help rule out false alarms, and thus lower the necessary threshold
for a detection. Here, we develop a framework for determining how much
sensitivity is added to a gravitational-wave search by triggering on an
electromagnetic transient. We apply this framework to a variety of relevant
electromagnetic transients, from short GRBs to signatures of r-process heating
to optical and radio orphan afterglows. We compute the expected rates of
multi-messenger observations in the Advanced detector era, and find that
searches triggered on short GRBs --- with current high-energy instruments, such
as Fermi --- and nucleosynthetic `kilonovae' --- with future optical surveys,
like LSST --- can boost the number of multi-messenger detections by 15% and
40%, respectively, for a binary neutron star progenitor model. Short GRB
triggers offer precise merger timing, but suffer from detection rates decreased
by beaming and the high a priori probability that the source is outside the
LIGO-Virgo sensitive volume. Isotropic kilonovae, on the other hand, could be
commonly observed within the LIGO-Virgo sensitive volume with an instrument
roughly an order of magnitude more sensitive than current optical surveys. We
propose that the most productive strategy for making multi-messenger
gravitational-wave observations is using triggers from future deep, optical
all-sky surveys, with characteristics comparable to LSST, which could make as
many as ten such coincident observations a year.
