The Lost Ticket
One hundred people are lined up to take their seats in a 100-seat theater. The first in line can’t find her ticket, so she sits on a seat at random. Each remaining theatergoer sits in his or her assigned seat unless it is occupied, in which case he or she sits on a seat at random.
What is the probability that the last person in line takes their assigned seat (to 3 significant figures as a decimal)?
The answer is 0.5.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
WLOG, suppose the seats are numbered 1 to 100, and the n th person's ticket is for seat n . In addition, since we only care about the seat that person 100 ends up in, suppose that when a person (other than person 100) finds their seat occupied, instead of choosing another seat at random, they instead ask the person in their seat to get up. That person then chooses another seat at random. The benefit of this is that each person, apart from person 1 and person 100, is guaranteed to end up in their assigned seat; in addition, the only people who will ever be in a seat other than the one they were assigned are person 1 and person 100.
Now, each time person 1 gets up to move, there are three possibilities.
In both of these cases, person 1 will never be asked to move again, since nobody who would ask them to move will have either of these seats as their assigned seat. Since each time they are forced to move, person 1 chooses their seat at random, each of these cases is equally likely to occur.
In this case, person 1 will be asked to move again, so they will not end up in any of these seats.
What this shows is that in the end, person 1 is equally likely to end up in seat 1 or in seat 100, and cannot end up in any other seat. Therefore, when it's person 100's turn to take their seat, there is a 5 0 % = . 5 chance that they will be able to take their assigned seat at seat 100, and a 5 0 % chance that they will be forced to sit in seat 1.