I am working on a question for class where there are two patients waiting on kidneys. The arrival of transplant kidneys is a Poisson process with rate λ. A will die after an exponential time with rate $μ_A$, and B after an exponential time with rate $μ_B$. A is ahead of B on the waiting list, so the first kidney goes to A as long as they are still alive. We are asked to compute the probability that A and B each survive to receive their kidneys.
For A, I defined $P( A < T_1 )$ and used the total probability law to expand that out into $P( A > T_1 | T_1 = t)P( T_1 = t)$, which simplified pretty easily into $μ_A \exp( - μ_At - λt )$.
For B, I tried a similar approach and got $P( A < T_1)P( B > T_1 ) + P( A > T_1 )P( B > T_1 + T_2)$.
What's the easiest way to approach $P( B > T_1 + T_2)$? Expanding that out with total probability just confused me more. I got this: $P( B > T_1 + T_2 | T_1 + T_2 = t)P( T_1 + T_2 = t | T_1 = s)P( T_1 = s)$ and I just don't know what to do with the variable $s$. Are those supposed to go away? Should I expect the expression for B to depend on two variables? I'm not even sure that my answer for A should be in terms of t.