We use data on gas temperature and velocity dispersion from the Green Bank Ammonia Survey and core masses and sizes from the James Clerk Maxwell Telescope Gould Belt Survey to estimate the virial states of dense cores within the Orion A molecular cloud. Surprisingly, we find that almost none of the dense cores are sufficiently massive to be bound when considering only the balance between self-gravity and the thermal and non-thermal motions present in the dense gas. Including the additional pressure binding imposed by the weight of the ambient molecular cloud material and additional smaller pressure terms, however, suggests that most of the dense cores are pressure-confined.