Assume I have two independent Poisson processes with respective parameters$$ \sim\text{Poisson}(\alpha_1),\sim \text{Poisson}(\alpha_2)$$ that I observe over a time interval $[0,t].$
What is the probability then that the latest jump on this interval was done by the second process?
So what I am looking for is something like an equation that explicitly states this probability. Does anybody know if there is anything like that? My problem is that this event that I want to describe with a probability is too abstract for me to catch.
If anything is unclear, please let me know.