The classical competitive exclusion principle states that when two populations compete for the same limited resource, only one population can survive; one population will drive the other to extinction. By assuming one of the populations is also subject to Allee effects, we prove in the deterministic model that competitive exclusion and coexistence can occur simultaneously depending on initial populations. We also propose a parallel continuous-time Markov chain model. We simulate both systems by comparing and contrasting their numerical simulations. Although both populations can persist in the deterministic system, it is found numerically that only one population can survive in the stochastic Markov process when the carrying capacities are small. As we increase the carrying capacities of both populations, then this numerical study demonstrates that the probabilities of coexistence as a function of time are positive especially when time is small.