# The airplane seat problem

For today’s Family Math project we talked through a classic probability puzzle:

An airplane has 100 seats.  100 passengers are going to board and each one has an assigned seat.  The first person to board ignores the assigned seat requirement, though, and chooses a seat at random (including the 1/100 possibility of actually choosing the correct seat).  After that, everyone else boards taking their assigned seat if it is open, or choosing a seat at random if their seat is taken.  What is the probability that person #100 sits in their correct seat?

Here’s how I introduced the problem to the boys:

At the end of the last video my younger son suggested studying a smaller problem first. He picked though the case with 4 people would be easier – and it would! I suggested starting even lower than that, though, so we started with just one person:

After seeing a pattern in the smaller cases we went back to try to tackle the larger case. We had a little bit of confusion, though – and that confusion may have been only on me misunderstanding what my son was saying! – so we cut this movie a bit short to return to the 4 person case:

Returning to the smaller case with 4 people, my son clarified his argument. That argument was, essentially, an induction argument which was really cool! The boys were able to explain how you extend the same argument to the case with 100 people. Nice solution!

At the end we talked about another fun feature of this problem – what are the possible seats that the last person might sit in?

It is always fun to go over a famous problem. This time was an especially nice discussion surprise since the induction argument was an out of the blue surprise! I think this is a fun problem to talk through with kids.