What are the middle passenger's chances?

Question

100 passengers board a full flight in order. Passenger 1 lost their pass and sits in a uniformly random seat; everyone else takes their own seat if free, otherwise a random free seat. What is the probability that passenger k (say, passenger 99) gets their own seat? Find the general formula.

Accepted Answer

When passenger k boards, their own seat k is either still free (they get it) or already taken (they do not). By the time passenger k arrives, seats 1 through k-1 belong to passengers who have already boarded, so the only relevant question is whether the "loose" seat, the displacement caused by the confused passenger 1, has landed on seat k yet. Consider the first person to sit in a random seat: passenger 1, choosing among all n = 100 seats. Trace the chain: each random chooser either grabs seat 1 (which frees everyone downstream, so passenger k certainly wins), grabs seat k or a seat belongin…

What are the middle passenger's chances?

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