$N$ doors Monty hall type problem

Question

In Peter office there are $N$ doors. Behind each door except one hides a doll. Lucy know what is behind each door while Peter does not. Peter has to select a door and wins if there is no doll behind it. Peter selects a door, instead of opening that door, Lucy opens $k$ other doors behind each of which is a doll , with $1\le k\le N-2$. Now Peter is given a choice to switch the door.

1. What is the probability for Peter to win if Peter does not switch his initial choice?
2. probability if he switch his initial choice
3. prove that whatever are values of $N$ and $k$ , Peter should always switch doors.

Answer 1

Initial selection win probability is $\frac{1}{N}$. What happens after that doesn't effect result of initial selection. Win probability if he switches is then $\frac{N-1}{N}\times\frac{1}{(N-1-k)}$.