So imagine there are only two passengers on an aircraft with two passenger seats. Passenger #1 goes onboard, picks a seat at random, and you must pick the other seat. There's no doubt that your chances of getting your assigned seat are one in two; i.e., the probability is exactly 1/2.
Ratcheting up the difficulty only a teeny bit, imagine the number of passengers (and assigned seats) to be three. A little more work quickly leads you to the same probability of 1/2 of finally getting your assigned seat. Ditto when the number of passengers is four.
Now let's take a big step and wonder if the probability is 1/2, no matter now big the number N of passengers is (as long as N is at least two). Here are all the boarding scenarios that allow you to sit in your own seat: First, passenger #1 sits in seat #1 with probability 1/N. In that case, each succeeding passenger sits in her/his assigned seat, and you sit in yours too. On the other hand, passenger #1 sits in seat #K > 1. After that, passengers #2 through K-1 sit in their assigned seats; whereupon passenger #K makes a random choice among seats #1 and those numbered > K. In effect, passenger #K is in the same position as was passenger #1; only the number of available seats is now smaller than N. This allows us to argue by induction.
So, for each K, at least 2 and no more than N-1, there is: first the probability of 1/N that passenger #1 sits in seat #K; second, the probability that you get to sit in your seat after all, once that happens. This latter probability is 1/2, by the inductiion hypothesis; hence the probability of you sitting in your pre-assigned seat is (1/N)+((N-2)/N)(1/2)=(2/2N)+((N-2)/2N)=N/2N=1/2. This establishes, via an induction argument on the number of passengers, that you have exactly a fifty-fifty chance (rather surprising to me) of getting your assigned seat after all the mixup.
The "quick" argument, after you have done all the experimental work, is this: First, either seat #1 is chosen before seat #N or it isn't. Each of these outcomes has probability 1/2. (Not entirely obvious to me at the outset, but hey.) Next, if seat #1 is chosen first, then you're assured of getting your pre-assigned seat. Hence, the probability of you getting your seat is 1/2. (Note: The only alternative to your getting your pre-assinged seat #N is to get seat #1.)