We consider the detection component of a conflict detection and resolution system for air traffic management. A conflict is an event of a close encounter between two aircraft in the near future. The collision alert system has to balance between declaring alerts timely to true threats and reducing the false alarms. We compare two conflict detection schemes, namely, geometric and probabilistic method, for collision alerting based on the predicted trajectory with or without the intent information from the encounter aircraft. We study the threshold selection for each scheme by maximizing the conflict detection probability under an operational cost constraint which is proportional to the false alarm rate and inversely proportional to the response time. Through simulated aircraft encounter scenarios with various types of uncertainties in trajectory prediction, we found that the geometric method performs better than the probabilistic method when the intruder's intent is unknown. However, the probabilistic method is better under relatively large uncertainty of the predicted trajectory. We discuss the implication to the centralized and distributed conflict alert system where each aircraft can optimize its own conflict detection threshold.