Generalized Monty Hall: n Doors

You are a contestant on a game show. There are $n \geq 3$ doors. Behind one door is a prize; the other $n - 1$ doors are empty. The prize is placed uniformly at random. You pick a door. The host -- who knows where the prize is -- then opens one of the other $n - 1$ doors, always revealing an empty one. You are now offered a chance to switch to any one of the remaining $n - 2$ unopened doors (not your original pick, not the one the host opened). (a) What is the probability of winning if you switch? (b) For $n = 8$, what is the probability you win by switching? Express as a fraction.