In multievent wireless sensor networks (WSNs), various events can concurrently require different quality of service (QoS) provision based on different priority levels. For Carrier Sense Multiple Access p-persistent Medium Access Control (MAC) protocol, $\boldsymbol{p}$ is used to reduce the chance of collision and improve efficiency. In this paper, we propose different $\boldsymbol{p}$ values for different priority levels so that higher priority data frames have the higher probability accessing the shared medium and have better QoS in terms of delay and reliability. Mathematic analysis and simulation show that our proposed Priority MAC protocol for Multi-Event wireless sensor network (PMME) protocol with different $\boldsymbol{p}$ values significantly reduce packet delay to all four priority packets and keep high packet success rate compared to other two priority MAC protocols named QoS aware energy-efficient (QAEE) and Multi-priority based QoS (MPQ) in multievent WSNs.