I was not able to solve this question on my test.
An aeroplane has $100$ seats (numbered $1$ to $100$) and $100$ passengers waiting to board each having a ticket with a number from $1$ to $100$. No number is on $2$ tickets or on $2$ seats. The rules of boarding are as below:
(i) Passengers board in the order of the number on their respective ticket.
(ii) The first passenger to board can sit on any seat.
(iii) A passenger with ticket number $i(i\ne 1)$, boards the plane and sits on seat number '$i$' if it is empty. However, if that seat is occupied he can sit on any empty seat.If $p$ is the probability that the $100^{th}$ passenger sits in his assigned seat, then $p^2+ (1 - p)^2$ is equal to
So I contacted my teacher and he said to let no. of persons in plane be $n$ and put $n=2,3,4...$ and so on, use induction, and I will see that it does not depend on $n$ and is constant $=0.5$. This is the 1st problem I have encountered of this type. Isn't there any other good method to solve it?