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 interval [0,T]. Each event spans a predetermined fraction γ total length T before concluding. Three continuous random variables represent starting times these events, uniformly distributed over time By employing geometric probability approach, we derive rigorous closed-form expression for joint occurrence taking into account various values γ. Additionally, determine expected value intersection duration Furthermore, provide comprehensive solution evaluating number trials required simultaneous events. Numerous numerical examples support theoretical analysis presented in this paper, further validating our findings.