  (Abridged) Chandra and Hubble ACS observations of 11 early-type galaxies
probe the low-mass X-ray binary (LMXB) - globular cluster (GC) connection. We
explore the optical properties of 270 GCs with LMXBs and 6,488 GCs without
LMXBs. More massive, redder, and more compact GCs are more likely to contain
LMXBs. Unlike Galactic GCs, a large number of GCs with LMXBs have half-mass
relaxation times > 2.5 Gyr. We fit the dependence of the expected number of
LMXBs per GC, \lambda_t, on the GC mass M, color (g-z), and half-mass radius
r_{h,cor}, and find that \lambda_t \propto M^{1.24\pm0.08}
10^{0.9^{+0.2}_{-0.1} (g-z)} r_{h,cor}^{-2.2^{+0.3}_{-0.4}}. Our fit rules out
that the number of LMXBs per GC is linearly proportional to GC mass and that
most GCs with high X-ray luminosities contain a single LMXB. The detailed
dependence of \lambda_t on GC properties appears essentially equivalent to a
dependence on the encounter rate \Gamma_h and the metallicity Z, \lambda_t
\propto \Gamma_h^{0.82\pm0.05} Z^{0.39\pm0.07}. Our analysis provides strong
evidence that dynamical formation and metallicity play the primary roles in
determining the presence of an LMXB in extragalactic GCs. The shallower than
linear dependence for our sample requires an explanation by theories of
dynamical binary formation. The abundance dependence is consistent with a
metallicity-dependent variation in the number o neutron stars and black holes
per unit mass GC, effects from irradiation induced winds or suppression of
magnetic braking in metal-poor stars.
