Solution To Friday Puzzle #2

11 09 2011

Kudos to Al Nikolov, who was the first to correctly answer the Friday Puzzle #2.  The answer is indeed P=1/2, meaning that there is fifty-fifty chance that the Grumpy Old Man will get their seat without fuss.

Here’s the intuitive reasoning: When a businessman finds that their seat is already occupied by someone else and randomly takes another seat, he becomes the virtual Silly Old Lady and the real Silly Old Lady becomes a virtual Energetic Businessman. This adjustment does not affect the outcome but it helps to see that only one person (the real or virtual Silly Old Lady) will ever occupy a wrong seat. For as long as the real or virtual Silly Old Lady takes a seat that belongs to a businessman they will eventually have to move again. But as soon as the real or virtual Silly Old Lady takes either their own seat or the seat that is assigned to the Grumpy Old Man, then all remaining businessmen will be able to board the plane without fuss. Therefore, when Grumpy Old Man enters the plane there are two equiprobable statuses:

  1. All businessmen sit on a seat assigned to some businessman, and the Silly Old Lady sits on the seat assigned to herself
  2. All businessmen sit on a seat assigned to some businessman, and the Silly Old Lady sits on Grumpy Old Man’s seat

Because these two statuses are equiprobable, the answer is 1/2.

 

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