Airplane Seating

The Answer...

The probability that the last passenger will be able to sit in his own seat is 50%.

If the little old lady chooses a seat at random she is just as likely to sit in her own seat as in the last passenger's seat. If she does neither and sits in another passenger's seat, then those options pass to the person whose seat she sits in. Down along the line someone will either sit in the old lady's seat, in which case the problem is solved, or else he will sit in the last passenger's seat, which concludes the matter. He is equally likely to do either.

The Question was...

There are 100 seats on an airplane and they are all assigned. Each person who boards will take his assigned seat unless someone else is sitting in it, in which case he picks a seat at random. The first passenger to board is a little old lady who doesn't bother to look for her seat but sits in the first one she comes to. What is the probability that the last person to board will be able to sit in his assigned seat?