## HO. Crazies on the Plane

We all know this feeling: someone’s in your seat, and now you’re the nutcase who’s going to take someone else’s seat. After all that what’s the probability the last person on the plane will be able to sit in the correct seat?

The three number trick is just a simple version of this one (but here it is quicker and simpler).

1. ### Shawn said,

March 19, 2012 at 1:05 am

It seems to me that [spoiler] if we sort of think about this in a pseudo-backward-induction type way, we know that seats 2-99 are going to be filled by the time the last passenger arrives regardless of which seat Passenger 1 sits in. So the last passenger is either going to sit in his seat or the first passenger’s seat. Since each displaced passenger moves randomly, either seat is equally likely to have been taken. So the last passenger has 1/2 probability that he will sit in Seat 1, and 1/2 probability that he will sit in Seat 100 (the correct seat). [/spoiler]

2. ### michael thwaites said,

March 22, 2012 at 12:40 pm

I discovered an interesting cool ‘fact’ about this. The last person on the plan always sits in either my seat or their own.
In general, if my assigned seat was no 1 and the nth person was supposed to sit in seat n, then the last person always sit in seat 1 or their own seat. And in general, using this ordering of the seats, everyone either sits in seat 1 or a seat number greater or equal to their assigned seat!