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

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 $E^\pm$.