$What is the probability that the last person to board an airplane gets to sit in their assigned seat?$

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Two hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a random seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats at random otherwise. What is the probability that the last person to board gets to sit in their assigned seat?

Probability Word Problem Putnam Competition

Karl Sch

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The key idea here is that the probability that one of the first 199 people takes the first person's seat (A) or the last person's seat (B) are equal, i.e. $P(A)=P(B)$.

When the last person enters the airplane, the only possible remining seat could be the first person's seat or the last person's sea, and any other seat has been taken.

The last person would get their own seat if their seat is not taken by the first 199 people, and would seat in a different seat if their seat is taken and only the first person's seat is available. Since the probability of these two events are equal, the probability that the last person to board an airplane gets to sit in their assigned seat is
\[P(A)=\frac{1}{2}.\]

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$What is the probability that the last person to board an airplane gets to sit in their assigned seat?$

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