The chances that the last person to board the seat gets to sit in his own seat is 50%.
We can prove this by induction. Let's call this the base case. Consider the case of two passengers. Either passenger one (the crazy guy) sits in his own seat, or the other guys seat. The chances are obviously 50%.
Consider the case of three passengers. If the crazy guy does not sit in the middle passenger's seat, we have the same probability as the case of two passengers, since the middle passenger being permitted to sit in his own seat won't change the situation at all. But even if the crazy guy sits in the middle passenger's seat, we can reduce this case to the base case. Since the middle passenger will now enter the plane and randomly select a seat, he is in essence fulfilling the same role as the crazy guy in the base case with one less seat.
Now consider the case of N passengers. The crazy guy can sit in his own seat, the last seat, or some other seat. When the crazy guy sits down in passenger M's seat (not his own and not the last guy's seat), we can just consider M to be the new crazy guy on the plane with the random choice of seats. Up until M gets on the plane everyone else will be able to sit in their assigned seats.