Suppose $N(\omega,t)$ is a [homogeneous Poisson counting process][1] with a constant parameter $\lambda,\,\forall\omega \in\Omega$ where $\Omega$ is the sample space. Given positive real numbers $T$ and $\tau$, and non-negative integer $n$, what is the probability that $N(\omega,t)$ counts exactly $n$ points within at least one subinterval $[t,t+\tau]$ of $[0,T]$, or Prob$\big(\bigcup_t\big\{\omega\,\big|\, [t,t+\tau]\subseteq [0,T] \wedge N(\omega,t+\tau)-N(\omega,t)=n\big\}\big)$?


  [1]: https://en.wikipedia.org/wiki/Poisson_point_process#Interpreted_as_a_counting_process