Suppose I have three people A, B, and C that are being checked out by three different cashiers. The amount of time it takes them to finish is exponentially distributed with rates $\lambda$, $1.5\lambda$, and $2\lambda$, respectively. A fourth person D will be served once one of the other three is done being served. What is the probability that D is the last person to be done checking out?
My thought is that, for example, if A is the first one done, then D's waiting time is exponentially distributed with rate $\lambda$. Is this correct?
If so, then I think the probability that D is the last one to be done is as follows:
However, I think my logic is flawed. My other thought is that maybe I need to consider that, if A gets done first then I need to figure out $P(A+D<B, A+D<C)$ where A, B, C, and D represent the amount of time being served rather than the amount of time spent waiting?