Public transportation systems represent an essential sector of any nation's critical infrastructure. Hence, continuity of their services is deemed important and with a high priority to the nations. Concerns over risks like terrorism, criminal offenses, and business revenue loss impose the need for enhancing situation awareness in these systems. However, practices, such as conducting random patrols or regular spot-checks on passengers to prevent or deter potential violations, are strictly limited by the number of available resources (e.g. security staff or fare inspectors) and by the ability of potential opponents (e.g. criminals, or fare evaders) to predict or observe the inspectors' presence patterns. Casting the interactions between these competitive entities (inspectors/security officials and criminals/fare dodgers) into a game-theoretic model will enable involved system operators to 1) find optimal cost-effective (or multi-goal) human resource allocation or spot-check schedules, 2) capture and treat uncertainty due to imperfectness of information, 3) integrate measurements from heterogeneous natures (e.g. statistics, expert opinions, or simulation results). This work applies a game-theoretical model that uses random probability-distribution-valued payoffs to allow playing spot-checking games with diverging actions' outcomes as well as avoiding information loss due to combining several measurements into one representative (e.g. average).