20 people 20 files in a meeting [duplicate]

Question: Prior to a meeting of 20 people, a file was prepared for each person with their names written. The person who comes first to the meeting picks a random file without looking at the names. Those who came later look for their own file. If their files were not taken by someone else, then they pick their own file. Else, they pick a random file. What is the probability of the last person in the meeting getting his or her own file?

It seems a bit tricky. I tried to solve directly by using conditional probability but failed since the cases are confusing.

This is the airplane seating problem. I believe the answer is $$\frac{1}{2}$$.