Consider a post office that is run by two clerks. Suppose that when you enter the system you discover that Olsen is being served by one of the clerks and Nelson by the other. Suppose also that you are told that your service will begin as soon as either Olsen or Nelsen leaves. If the amount of time that the clerks serving Olsen ans Nelson are exponentially distributed with mean 1/λ1 and 1/λ2, what is the probability that, of the three customers, you are the last to leave the post office?
I conditioned: P(you last) = P(Olsen finished first) $\times$ P(Nelson finishes before you) + P(Nelson finished first)$\times$ P(Olsen finished before you).
This means $\frac{λ1}{λ1 + λ2}$ $\times$ $\frac{λ2}{λ1 + λ2}$ + $\frac{λ2}{λ1 + λ2}$ $\times$ $\frac{λ1}{λ1 + λ2}$ = $\frac{2λ1λ2}{(λ1 + λ2)^2}$.
I think I got that right, but what if you want to know what the probability of Olsen being the last to leave the post office? Or Nelson? Both $\frac{1}{λ1 \timesλ2}$?