Taking Seats on a Plane This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless

Imagine there are a 100 people in line to board a plane that seats 100. The first person in line realizes he lost his boarding pass so when he boards he decides to take a random seat instead. Every person that boards the plane after him will either take their "proper" seat, or if that seat is taken, a random seat instead.

Question: What is the probability that the last person that boards will end up in his/her proper seat.

Moreover, and this is the part I'm still pondering about. Can you think of a physical system that would follow this combinatorial statistics? Maybe a spin wave function in a crystal etc...
Here is a rephrasing which simplifies the intuition of this nice puzzle.

Suppose whenever someone finds their seat taken, they politely evict the squatter and take their seat.  In this case, the first passenger keeps getting evicted (and choosing a new random seat) until, by the time everyone else has boarded, he has been forced by a process of elimination into his correct seat.

This process is the same as the original process except for the identities of the people in the seats, so the probability of the last boarder finding their seat occupied is the same.

When the last boarder boards, the first boarder is either in his own seat or in the last boarder's seat, which have both looked exactly the same (i.e. empty) to the first boarder up to now, so there is no way the poor first boarder could be more likely to choose one than the other.
I have come up with my own solution using a revolutionary war analogy, which is quite fun and intuitive I think. 

We classify passengers as two groups: (i) "rebels": the first man and those who do not end up sitting at their own seats; (ii) "rulies": those who end up sitting at their own seat. We will refer to the First Man's seat as "the throne" due to its significance. We will refer to each passenger as "the n-th man" by their order of entrance. We will refer to appointment of the 100-th man (the one true saviour) as a rebel as ultimate victory of the rebellion. Anyone other than the saviour seating on the throne will doom the revolution.

Each rebel (assuming he is the n-th man, including the first man) has 3 choices: 

(i) Doom the rebellion: with probability $\frac{1}{101-n}$, sit at the throne and thus effectively ending the rebellion and ensure no more rebels thereafter; this fails the rebellion 

(ii) Win the rebellion: with probability $\frac{1}{101-n}$ appoint the 100-th man, the saviour, as rebel by sitting at 100-th man's seat, thus winning the rebellion; 

(iii) Leave it to future generations: with remaining probability, appoint a rebel other than the saviour and thus does not doom or win the rebellion on his watch. 

We can see that we have equal chance to win the rebellion or lose it under each rebel's command. So the result is with probability $1/2$ we lose the rebellion, which is the event the last passenger sits at his own seat.