Lost boarding pass!

There are n n people in line to board a plane with n n seats. The 1 st 1^{\text{st}} person has lost their boarding pass, so they take a random seat. Everyone that follows takes their assigned seat only if it is available, otherwise they take a random unoccupied seat.

What is the probability the last passenger ends up in their assigned seat?

This problem is not original!

1 n \frac{1}{n} 1 n + 1 \frac{1}{n+1} n + 1 2 \frac{n+1}{2} n 2 \frac{n}{2} 1 2 \frac{1}{2}

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