Here is a question I accidentally invented (I wrote an exam to my students and used bad wording so that it could have been interpreted in also unintentional ways. Caught the error on time!) and I was wondering if anyone has thought how to approach it:
Consider a Poisson process with parameter $\lambda$ starting at $t=0$. What is the probability that between $t=0$ and $t=T$ there is a time frame of $\Delta T$ (not specified when it starts or ends) in which exactly $k$ events occurred? For example, if $T=3, k=3, \Delta T=1$ and the events occurred at times $t=0.2,0.3,1.1,2.4$ then such time frame exists (from $0.2$ to $1.2$) but if $k=4$ such time frame does not exist, as there is no segment of length $1$ that includes all the events.
Similar, maybe simpler: What is the probability of the above-mentioned event given that during the entire $[0,T]$ segment there were exactly $K$ events?
Now, if $\Delta T$ was a specific time frame (e.g., from $t=1$ to $t=1+\Delta T$) then both questions are simple (and those are the intendent questions in my exam...). But what if we let it start anytime?