without a defined ammount of people getting on the plane, and a answer asking for the probability of the last person to board the plane, I am looking at a trick question?


I'd state that there is a 50% chance that the last person to board the plane gets their assigned seat. I'll justify that by saying this


When I sit down, there are three relevant possibilities:

A) I sit in seat z.  In this case, guy z is guaranteed NOT to get his seat.

B) I sit in seat w.  In this case, guy z is guaranteed to get his seat (since everyone from then on sits in their proper seat).

C) he sits somewhere else.  This essentially just repeats the experiment by forcing some other passenger to sit in a random seat.

So A and B happen with equal probability.  If C happens, then some other passenger (the one who's seat is taken) has to take a random seat, and again, there are the same three relavent possibilities as above.

So this is just a repeating random experiment, that eventually ends in state A or B.  Since A and B are equally likely in each repetition of the experiment, common sense (and I guess some theorem of probability theory) tells us that the whole repeating experiment will end at A or B with equal probability.

More theory than complex equations IMO