Mixed up airplane seats

[I got this problem from Rajeev Joshi, who I think said he heard it from Jay Misra.]

An airplane has 50 seats, and its 50 passengers havetheir own assigned seats. The first person to enter the plane ignores his seat assignment and instead picks a seat on random. Each subsequent person to enter the plane takes her assigned seat, if available, and otherwise chooses a seat on random. What is the probability that the last passenger gets to sit in her assigned seat?

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