Fair allocation of bandwidth and maximization of channel utilization are two important issues when designing a contention-based wireless medium access control (MAC) protocol. However, fulfilling both design goals at the same time is very difficult. Considering the problem in the IEEE 802.11 wireless local area networks (WLANs), in this work we propose a method using a p-persistent enhanced DCF, called P-IEEE 802.11 DCF, to achieve weighted fairness and efficient channel utilization among multiple priority classes in a WLAN. Its key idea is that when the back-off timer of a node reaches zero, the transmission probability is properly controlled to reflect the relative weights among data traffic flows so as to maximize the aggregate throughput and to minimize the frame delay at the same time. In particular, we obtain the optimal transmission probability based on a theoretical analysis, and also provide an approximation to this probability. The derived optimal and approximation are all evaluated numerically and simulated with different scenarios. The results show that the proposed method can fulfill our design goals under different numbers of priority classes and different numbers of nodes.