TY - JOUR

T1 - Geometric Probability Analysis of Meeting Probability and Intersection Duration for Triple Event Concurrency

AU - Al Bataineh, Mohammad

AU - Al-qudah, Zouhair

AU - Abdrabou, Atef

AU - Sandokah, Ayman N.

N1 - Funding Information:
They also acknowledge the support from their respective institutions. This work was made possible by the UAE University startup grant (Fund Code: 12N105).
Publisher Copyright:
© 2023 by the authors.

PY - 2023/6

Y1 - 2023/6

N2 - 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.

AB - 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.

KW - expected value

KW - geometric probability

KW - joint event

KW - meeting probability

KW - random variables

UR - http://www.scopus.com/inward/record.url?scp=85164193567&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85164193567&partnerID=8YFLogxK

U2 - 10.3390/math11122708

DO - 10.3390/math11122708

M3 - Article

AN - SCOPUS:85164193567

SN - 2227-7390

VL - 11

JO - Mathematics

JF - Mathematics

IS - 12

M1 - 2708

ER -