Exercise: Consider a theater with $n$ seats that is fully booked for this evening. Each of the $n$ people entering the theater (one by one) has a seat reservation. However, the first person is absent-minded and takes a seat at random. Any subsequent person takes his or her reserved seat if it is free and otherwise picks a free seat at random.
(i) What is the probability that the last person gets his or her reserved seat?
(ii) What is the probability that the $k$th person gets his or her reserved seat?
I get $p = 0.5$ for (i), but I have no idea if it's right.
Can somebody help me with (i) and especially (ii)?
Thank you!
