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

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Publication date: 22 December 2016

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 �egin{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 �egin{equation*} P approx 1 - left( 1 - n_A dfrac{t_A + t_B}{T} ight)^{n_B} end{equation*} for any given value of

n_{A}

and

n_{B}

. Finally we prove the more complex universal equation �egin{equation*} P = 1 - dfrac{ left( T^+ - t_A n_A - t_B n_B ight)^{n_A + n_B} }{ left( T^+ - t_A n_A ight)^{n_A} left( T^+ - t_B n_B ight)^{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 + f r a c t_{A} + t_{B} 2

and a small error term

E^{p} m

.

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