Billy is first to board the plane, and he randomly sits in one of the 100 airplane seats. Further, every passenger is very polite, and if Billy (or anyone else) is in their seat they will simply sit randomly in one of the remaining seats.

Suppose you are last to board the plane. What is the probability you get your assigned seat? Hint: first examine smaller number of seats and find pattern/relationship.

I tried this myself and also looked at the solution, but I have a question:

for P(An) where An is the event of you sitting in correct seat with n seats, there is

1/n - Billy sits in his seat -> everyone else will be correct including you -> 1/n * 1

1/n - Billy sits in your seat -> no chance for you to sit in correct -> 1/n * 0

(n-2)/n - Billy sits in another persons seat (not you) -> then the person p that Billy took the seat of can become a new Billy (his seat is Billy's but he still chooses randomly) -> the book says that this is the P(A(n-1)) case so (n-2)/n * P(A(n-1)), but doesn't that assume that person p will board before everyone else? because that was the case for the original problem involving Billy

if person p doesn't board first, everyone before him will find their correct seat, which can't happen in the original Billy case where he goes first always

thanks