How likely are two independent recurrent events to occur simultaneously during a given time?

TL;DR

This paper derives exact and approximate formulas for the probability that two independent recurrent events with specified durations and counts occur simultaneously within a fixed time interval.

Contribution

It introduces new precise and approximate equations for calculating the likelihood of simultaneous occurrences of two independent events over time.

Findings

Derived an exact probability formula for single occurrences.

Proposed a simple approximation for multiple occurrences.

Established a universal probability equation with minimal error.

Abstract

We determine the probability $P$ of two independent events $A$ and $B$, which occur randomly $n_A$ and $n_B$ times during a total time $T$ and last for $t_A$ and $t_B$, to occur simultaneously at some point during $T$. Therefore we first prove the precise equation \begin{equation*} P^* = \dfrac{t_A+t_B}{T} - \dfrac{t_A^2+t_B^2}{2T^2} \end{equation*} for the case $n_A = n_B = 1$ and continue to establish a simple approximation equation \begin{equation*} P \approx 1 - \left( 1 - n_A \dfrac{t_A + t_B}{T} \right)^{n_B} \end{equation*} for any given value of $n_A$ and $n_B$. Finally we prove the more complex universal equation \begin{equation*} P = 1 - \dfrac{ \left( T^+ - t_A n_A - t_B n_B \right)^{n_A + n_B} }{ \left( T^+ - t_A n_A \right)^{n_A} \left( T^+ - t_B n_B \right)^{n_B} } \pm E^\pm, \end{equation*} which yields the probability for $A$ and $B$ to overlap at some point for any given parameter, with $T^+ := T + \frac{t_A + t_B}{2}$ and a small error term $E^\pm$.