A system can be in two states, $a$ and $b$. The waiting time for $a$ to become $b$ is modelled by means of an exponential distribution with parameter $\lambda,$ $1-e^{-\lambda t},$ and the waiting time for $b$ to become $a$ is modelled by means of an exponential distribution with parameter $\mu,$ $1-e^{-\mu t}.$
If initially the system is in state $a,$ I'd like to know the probability of the system being in each state after a time $T.$
The exponential distribution gives the probability of a transition happening before time $t,$ but for this particular problem I'd need the probability of the transition taking place exactly at time $t,$ and that's where I'm stuck.