Airplane Seats I

On a sold out flight, $n$ people line up to board the plane. The first passenger in the line has lost their boarding pass, but is allowed in. They take a random seat. If available, every subsequent passenger takes their assigned seat. If their seat is not available, the passenger chooses an open seat uniformly at random to sit in. Find the expected number of passengers that sit in their own seat when $n = 100$ to the nearest thousandth.