Question: 228 people board a short-haul flight from Sydney to Melbourne that is fully booked - there are exactly 228 seats. All passengers are inclined to sit in their allocated seat however the first person to board has spent too long in the bar and ends up taking a completely random seat. All other passengers go to their allocated seat and take it, unless it is already occupied, in which case they too take a random vacant seat. What is the probability that the last passenger to board the flight finds the seat he/she was assigned unoccupied?

Answer: First we need to consider the possible outcomes from this scenario...

If the first passenger, by chance, takes the seat that was indeed allocated to him then every other passenger that boards after him will find their assigned seat. In this case, clearly the last person to board does indeed find his/her allocated seat.

If the first passenger takes a seat that was NOT allocated to him, then at least one of the remaining 227 passengers to board will not find their allocated seat. One of two things can eventuate from this:

  1. One of the following passengers finds his/her seat taken and, by chance, takes the seat that was assigned to the first passenger, in which case all subsequent passengers to board do find their assigned seat; or

  2. None of the next (n-2) passengers to board, by chance, takes the seat that was assigned to the first passenger, in which case the last passenger to board can only take the seat that was assigned to the first passenger since it will be the only one left.

So it's obvious that the last person will either get his/her allocated seat, or the seat allocated to the first person. But what is the probability of each of these outcomes? You might be tempted to get into something complicated but it's actually quite simple. The probability that the last unoccupied seat does indeed belong to the last passenger to board is exactly 1/2. Hence the answer to the puzzle is 50%.

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