# The Aeroplane Seating Problem

As ever, the publication last week of A level results, and the imminent release of GCSE results later this week, signal the beginning of the end of the wonderful long summer holidays.

I hope you’ve had a fantastic time and, possibly, enjoyed the delights of foreign travel. If you did, I doubt you flew with an airline with such a relaxed seating plan as the one in the problem below …

On this particular flight, MathAir 314, there are 100 hundred passenger seats, and 100 passengers.

The first passenger to board has lost their boarding pass and doesn’t know their seat number. “No worries” declares the helpful steward, sit wherever you like.

The next (and all subsequent passengers) does have her boarding pass – if her seat is free, she sits in that seat. If it is occupied, the helpful steward allows them to chose any unoccupied seat they wish. This continues until all 100 seats are filled by the 100 passengers and the jet departs for its destination, Angle C (Anglesey – geddit?)

The question is:

What is the probability that passenger 100 gets to sit in their own allocated seat?

I’m indebted to Zoe Griffiths, @ZoeLGriffiths for this problem and you can see her video introducing the problem and, more importantly, her solution in the video below. But before you watch it, have a go at solving the problem yourself first. (You can start the video and the pause it after 1min 30 sec to see her intro to the problem.)

How might I use this problem? I might introduce it to a class towards the end of a lesson, and ask them to go away and think about it, reporting back with possible solutions, or even just approaches to a solution, next lesson.

Or I might offer the problem at the end of a weekly department meeting, inviting colleague to think about it before next week’s meeting. PE teachers regularly play their sports for fun, music teachers their instruments, its important for maths teachers to remain engaged with the subject and “do” some maths from time to time.

So, here’s the video with the solution, but give the problem some thought before you hit play.