The feasibility of practical in-band full-duplex radios has recently been demonstrated experimentally. One way to leverage full-duplex in a network setting is to enable three-node full-duplex, where a full-duplex access point (AP) transmits data to one node yet simultaneously receives data from another node. Such three-node full-duplex communication, however, introduces inter-client interference, directly impacting the full-duplex gain. It hence may not always be beneficial to enable three-node full-duplex transmissions. In this paper, we present a distributed full-duplex medium access control (MAC) protocol that allows an AP to adaptively switch between full-duplex and half-duplex modes. We formulate a model that determines the probabilities of full-duplex and half-duplex access so as to maximize the expected network throughput. A MAC protocol is further proposed to enable the AP and clients to contend for either full-duplex or half-duplex transmissions based on their assigned probabilities in a distributed way. Our evaluation shows that, by combining the advantages of centralized probabilistic scheduling and distributed random access, our design improves the overall throughput by 1.53 times, on average, as compared with the greedy downlink-uplink client pairing.