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
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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 -