As an innovative technology, the licensed assisted access (LAA) for unlicensed LTE has been recently standardized by the 3rd generation partnership project (3GPP) protocols. The primary issue for LAA deployment is how to friendly and fairly coexist with the existing Wi-Fi networks in the same unlicensed spectrum. In this paper, we propose and analyze an adaptive $p$ -persistent channel access scheme for LAA, namely $p$ -LAA, to balance the tradeoff between the throughput and fairness for the coexistence network. To deal with the asynchronous channel access problem, we first introduce a heterogenous superframe (HS) to facilitate the performance analysis. Based on the HS, we then derive the saturated normalized throughput of the proposed $p$ -LAA protocol. Furthermore, the channel access probability of the $p$ -LAA protocol is optimized to achieve the balance between the throughput and fairness. Finally, the accuracy of our theoretical analysis is verified by sufficient Monte Carlo simulations and the results show that the proposed $p$ -LAA protocol can achieve a better throughput and fairness performance than the conventional LAA protocol.