I am starting to study Poisson processes and I came up with this question:
Let there be two Poisson processes with rates $\lambda$ and $\mu$ respectively, monitoring the occurrence of events (e.g. emissions or bus arrivals) $A$ and $B$. $A$ happens with rate $\lambda$ and $B$ with rate $\mu$.
Let $ABA$ be the event that $A$,$B$,$A$ happen in that particular order.
What is the probability that a particular sequence of events happens? For instance, what are $P(ABAAB)=?$ and $P(BBBBBA)=?$