"$n$ music lovers have reserved seats in a theater containing a total of $n+k$ seats ($k$ seats are unassigned). The first person who enters the theater, however, lost his seat assignment and chooses a seat at random. Subsequently, people enter the theater one at a time and sit in their assigned seat unless it is already occupied. If it is, they choose a seat at random from the remaining empty seats. What is the probability that person $n$, the last person to enter the theater, finds their seat already occupied?"
I could do this problem for small specific values of $n$ and $k$ but as they grow the expressions seem to get really messy really quick with no discernible pattern. How would one solve this problem?