There are $n$ seats in a classroom and $n$ students who have been assigned seats on small slips of paper. Unfortunately, the first student to enter the classroom has lost his slip, so he just chooses a seat at random and sits in it. Each of the remaining students enters one at a time and either sits in their assigned seat if it is empty or if someone is sitting in their seat, chooses a seat at random from those that are empty. For $n\geq 2,$ show that the probability that the last student to enter will sit in their assigned seat is $\frac{1}2.$
I tried a few small examples and drew some tree diagrams to list out the possibilities, and got $\frac{1}2$ each time. I did make a few obvious observations though. Firstly, if the first student sits in the right spot, everyone else will too, by the premise. There is a $\frac{1}n$ chance for this. Secondly, if the first student chooses the last person's seat, the last person won't be able to sit in their assigned seat. And thirdly, the only way the last person can sit in their assigned seat is if everyone before him/her did not choose his/her seat. We can label the students $S_1,\cdots, S_n.$
I'm not quite sure how to determine the sample space for this problem (i.e. the total set of points or possibilities to consider).
