If the last passenger is displaced, where do they sit?

Question

150 passengers board a full flight in order. Passenger 1 lost their boarding pass and sits in a uniformly random seat. Everyone after sits in their own seat if it is free, otherwise picks a free seat uniformly at random. It is well known that passenger 150 gets their own seat with probability 	frac12. Given that passenger 150 does NOT get their own seat, which seat do they end up in? And what is the probability they end up specifically in seat 1?

Accepted Answer

Here is the sharp fact behind the classic 	frac12: at the very end, the only two seats that can possibly be the "loose" one are seat 1 (the confused passenger's own seat) and seat 150 (the last passenger's own seat). Every other passenger 2 through 149 takes their own seat unless it was already stolen, and the chain of displacements always traces back to seat 1 being the ultimate free seat. Where does passenger 150 land if displaced? When passenger 150 finally boards, exactly one seat is free. If that seat is 150, they win; if it is anything else, it must be seat 1. Why can it only be seat 1…

If the last passenger is displaced, where do they sit?

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