I'm working on a problem and came up with two different solutions! I'm not sure which is correct.
Problem: Two clerks with service time $exp(1)$ are helping 2 customers while a third waits. What is the probability that the next customer will be the second to finish? (Leaving one of the two initial customers still being helped)
Solution 1: Exponentials are memoryless, so when the customer in line starts getting served -- the process renews. As such, the probability that one finishes before the other is 1/2.
Solution 2: Let $T_i$ be the time it takes for customer $i$ to be served.
By law of total probability: $P($ Person 3 leaves second $) = P(T_1 + T_3 < T_2 \mid T_1 < T_2) P(T_1<T_2) + P(T_2 + T_3 < T_1 \mid T_2 < T_1) P(T_2<T_1)$
This keeps evaluating to 1/4!
Any thoughts? Thanks
Update: $P(T_1<T_2) = \int_{0}^{\infty}\int_{0}^{T_1}e^{-(t_1+t_2)} dt_1 dt_2 = 1/2$ and $P(T_1 + T_3 <T_2 \mid T_1 < T_2) = \int_{0}^{\infty}\int_{0}^{T_2}e^{-t_2}ze^{-z} dz dt_2 = 1/4 $
Update: Ah -- I see what happened. I was computing the wrong probability: $P(T_1 + T_3 <T_2) = 1/4$ but $P(T_1 + T_3 <T_2 \mid T_1 < T_2) = P(T_1 + T_3 <T_2 \cap T_1 < T_2)/P(T_1 < T_2) = P(T_1 + T_3 <T_2)/P(T_1 < T_2) =(1/4)/(1/2) = 1/2$
The 2nd and 3rd equality is established by noting all $T_i$ are positive so $T_1 + T_3 <T_2$ implies $T_1 < T_2$
The first solution is right. Without knowing how you evaluate the expression in the second solution to $\frac14$, it's hard to say what's going wrong, but it should evaluate as $\frac12\cdot\frac12+\frac12\cdot\frac12=\frac12$.