I am waiting for two friends to arrive at my house. The time until A arrives is exponentially distributed with rate λa, and the time until B arrives is exponentially distributed with rate λb. Once they arrive, both will spend exponentially distributed times, with respective rates μa and μb at my home before departing.

The four exponential random variables are independent.

(a) What is the probability that A arrives before and departs after B?

(b) What is the expected time of the last departure?