Thanks for your answer! I am a bit confused about the free parameter that remains on the integral. For example, on your first expression, the result depends on t_B in the end. Does that mean "the probability that event A will happen first in the time interval from -infty to t_B"? I'd like to...
For concretness I'll use atoms and photons but this problem is actually just about probabilities.
There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...