Geometric Probability Analysis of Meeting Probability and Intersection Duration for Triple Event Concurrency

This study investigates the dynamics of three discrete independent events occurring randomly and repeatedly within the interval (Formula presented.). Each event spans a predetermined fraction γ of the total interval length (Formula presented.) before concluding. Three independent continuous random variables represent the starting times of these events, uniformly distributed over the time interval (Formula presented.). By employing a geometric probability approach, we derive a rigorous closed-form expression for the probability of the joint occurrence of these three events, taking into account various values of the fraction γ. Additionally, we determine the expected value of the intersection duration of the three events within the time interval (Formula presented.). Furthermore, we provide a comprehensive solution for evaluating the expected number of trials required for the simultaneous occurrence of these events. Numerous numerical examples support the theoretical analysis presented in this paper, further validating our findings.