Assume there is post office that is run by two clerks $1$ and $2$. Customers arrive at times that follow exponential distribution with rate $λ_a$. The amount of time that the clerks 1 and 2 serve are exponentially distributed with mean $1/λ_1$ and $1/λ_2$. The first customer that arrives in the post office is served by clerk $1$ and the second by clerk $2$. The three exponential random variables are independent. Two customers A and B have decided to go to the post office. What is the probability that A arrives before and departs after B?
So, $T_1$ = time taken by clerk $1$ and $T_2$ = time taken by clerk 2. To solve for the probability, if A is the first one there, she goes to clerk $1$ and clerk $1$ has to be faster than clerk $2$ ($T_1 < T_2$). Now what?