We model the interaction between a defender of a system and a possible attacker. The defender faces a tradeoff between the cost of missed attacker classifications versus the cost of false alarms, while the attacker faces a tension between attacking more vigorously to exploit their entry into the system versus the increased risk of being correctly classified as an attacker. In this model, both players’ pure strategies are chosen from a continuum (a finite real interval), and moreover both players engage in mixed strategy play. We derive mathematical formula for the mixed Nash equilibria of the game, and also show that the identified equilibria are exhaustive up to variations of measure 0. We further investigate the case that the defender observes beta distributed noise when an attacker is not present and prove that for a wide range of the parameter space describing the game, the Nash equilibrium expected cost to the defender increases as the variance of this distribution increases.