In this paper, we study the problem of consensus-based distributed Nash equilibrium (NE) seeking in a network of players represented as a directed graph, where each player aims to minimize their own local cost functions non-cooperatively. To address bandwidth constraints and limited energy, we propose a stochastic event-triggered algorithm that triggers individual players with a probability depending on certain events, thus enhancing communication efficiency through reduced continuous communication. We prove that our developed event-triggered algorithm achieves exponential convergence to the exact NE when the underlying communication graph is strongly connected. Furthermore, we establish that our proposed event-triggered communication scheme does not exhibit Zeno behavior. Finally, through numerical simulations of a spectrum access game and comparisons with existing event-triggered methods, we demonstrate the effectiveness of our proposed algorithm.