There is an airplane with 100 seats, and there are 100 passengers each of whom has an assigned seat on the plane

There is an airplane with 100 seats, and there are 100 passengers each of whom has an assigned seat on the plane. The passengers line up in a random order to get on the plane. The first woman on line is a confused old lady who doesn’t know how to find her proper seat. She just sits in a random seat. When each person after her gets on the plane, they look to see if their assigned seat is available. If it is, they sit in it. If it is not (i.e., if the old lady or someone else before them has sat in it), they sit in a random seat. What are the odds that the 100th person sits in his/her proper assigned seat?

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There are two very different ways to do this. The brute force long equation way, and the cute, sneaky, but simple way. Hopefully we’ll get both on here.