Monty Hall I

You made it onto a game show! The host presents you with $n \geq 3$ doors. One door contains a prize, while the other $n-1$ doors are empty. The prize is uniformly random between the doors. You choose an initial door. Afterwards, the host opens a door (not the one you selected) which he knows has no prize behind it. You then are given the option to switch your door. Find the probability that you find the prize if you switch when $n = 8$.