We investigate the relationship between low-mass X-ray binaries (LMXBs) and globular clusters (GCs) using United Kingdom Infrared Telescope (UKIRT) observations of M31 and existing Chandra, XMM-Newton and ROSAT catalogues. By fitting King models to these data, we have estimated the structural parameters and stellar collision rates of 239 of its GCs. We show a highly significant trend between the presence of a LMXB and the stellar collision rate of a cluster. The stellar collision rate is found to be a stronger predictor of which clusters will host LMXBs than the host cluster mass. We argue that our results show that the stellar collision rate of the clusters is the fundamental parameter related to the production LMXBs. This is consistent with the formation of LMXBs through dynamical interactions with little direct dependence on the neutron star retention fraction or cluster mass.