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Is this seat taken?

100 people are waiting to board a plane. The first person’s ticket says Seat 1; the second person in line has a ticket that says Seat 2, and so on until the 100th person, whose ticket says Seat 100. The first person ignores the fact that his ticket says Seat 1, and randomly chooses one of the hundred seats (note: he might randomly choose to sit in Seat 1). From this point on, the next 98 people will always sit in their assigned seats if possible; if their seat is taken, they will randomly choose one of the remaining seats (after the first person, the second person takes a seat; after the second person, the third person takes a seat, and so on). What is the probability the 100th person sits in Seat 100?

This problem can be solved intuitively. The first step to solve this problem is to understand that the last person will either get his seat or the first person's seat. But why?

If the 1st person chooses 1st seat itself, everybody gets their own seat (i.e., the last person will get the last seat too).
Suppose 1st person chooses the 13th seat. Then the 2-12 people will sit in their seats itself. After that, the 13th person must choose at random.

  • Suppose they choose the 1st seat, then from 14 onwards, everybody gets their own seats. 
  • Suppose they choose the 16th seat, then the 16th person must make this decision again. The 16th person can either choose the 1st seat or choose a seat > 16 and pass this decision to them.
  • Suppose they choose the last seat, then from 14-99, everyone will be seating in their own seats. The only option for the 100th person is the 1st seat.
So we now get that the last person can get either their seat or the 1st seat.

Now you have to grasp the fact that a person somewhere between 1st and 100th person, can put an end to this game if they choose 1st or the last seat. If they choose the 1st seat, then the last person gets the last seat. If they choose the last seat, then the last person gets the first seat.

Here comes the conclusion... A person in the middle has an equal probability of choosing the 1st or the 100th seat. This implies that the last person has an equal probability of getting the first or the last seat. Thus, the probability of the last person getting the last seat is $50%$ or $0.5$.
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