Problem #WSP-000410

Problem

One hundred people are boarding on a fully-booked plane, and they all have assigned different individual seats. The first person has forgotten his boarding pass, so sits in a seat completely at random (that is, he’s equally like to pick any of the $100$ seats). He might pick his own seat...but the most likely scenario is that he’ll pick someone else’s seat.

Then the second person will sit in their seat if it’s available. But if it’s taken, then they’ll sit in a random seat from those left available. Similarly the third person will sit in their seat if it’s available. But if it’s not, then they’ll sit in a random seat from those available. Each person from the second onwards to the hundredth follows these rules.

What’s the chance that the ${100}^{\text{th}}$ gets to sit in their originally allocated seat?