The probability that the final passenger to board sits in his
assigned seat is one half.
Note that the final passenger must either sit in his own seat or in the seat
assigned to the first passenger to board. This is true because, once an
intermediate passenger boards, it is guaranteed that the seat assigned to this
passenger will be occupied.
We can then argue that either of these two cases must occur with equal
probability due to symmetry considerations. One can swap the roles of the
seat assigned to the first and last passengers without affecting the boarding
process. Therefore, each possibility occurs with probability one half.