Consider a race of $n$ runners whose finishing times are i.i.d and drawn from an exponential distribution with parameter $\lambda$. Assign a number id to each racer. What is the probability that the racers finish in ascending order?
My guess would be $\frac{1}{n!}$, since, by symmetry, all finishing orders are supposed to be of equal probability. However, I don't seem to be able to formally prove this... Any help would be appreciated.