You are getting ready to board a plane. There are exactly 333 seats on the plane

Question

You are getting ready to board a plane. There are exactly 333 seats on the plane, and exactly 333 passengers (counting yourself). You were running rather late to the airport, so you are the last person in line to board. The first individual to board the plane is a terrible person. Rather than look at their boarding pass, they pick a seat at random and sit there — all seats are equally likely. Everyone else in line is a decent human being, so, they look at their boarding passes and sit in their assigned seats, if they can. If a person arrives at their seat and finds it occupied, they randomly sit in an available seat (again, all empty seats are equally likely to be chosen). Since there are exactly the same number of passengers as there are seats, there will be a seat available when you get on the plane. What is the probability that it is your seat (i.e., the one listed on your boarding pass)?

Explanation / Answer

Here, the chances of getting our assigned seat is 50-50

First observe that our fate is determined the moment either the first or the last seat is selected! This is because the last person will either get the first seat or the last seat. Any other seat will necessarily be taken by the time the last guy gets to 'choose'.

Since at each choice step, the first or last is equally probable to be taken, the last person will get either the first or last with equal probability

Hence, the required probability is 1/2

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