probability expected winnings per lottery question

Question

I am trying to solve this problem but I'm a little stuck:

You and your friends decide to make a lottery with $n$ tickets, where each ticket has a number between $1$ and $n$, and each ticket is unique. Each ticket is $\$5$, and the lottery works in the following way. Once all $n$ tickets have been purchased, a number $x$ is selected uniformly at random between $1$ and $n$ and all the money is split equally between people with tickets less than $x$. That way, if $x=1$ is selected, you and your friends get to keep the prize pool.

1. What is your expected winnings per lottery as the organizer?

So what I did was make 2 cases. The first case is when $x = 1$ and the organizer keeps all the money, and the second case is where $x > 1$ and the organizer gets no money:

Case 1: $x = 1$

$P(X = 1) = \frac{1}{n}$
organizer gets $5n$ money
$\frac{1}{n}(5n) = 5$

Case 2: $x > 1$

$P(X > 1) = 1 - \frac{1}{n}$
organizer gets no money
$\left(1- \frac{1}{n}\right)(0) = 0$

From there I add the 2 equations and I get that the expected winnings per lottery as the organizer is $5 but the problem is I'm supposed to be getting an equation, the answer shouldn't be a number but I don't know where I'm going wrong. I would really appreciate any advice!

Answer 1

The answer and derivation are correct. As revealed in the comments, the source of the question continues with asking about expected winnings for a player, to which this question is a stepping stone.

Answer 2

The pot is $5n$

You have correctly found expected winnings for the organiser, $\mathbb{E}[X] = 5$

If the organiser doesn't get the pot, the "players" must, and by symmetry, each will earn the same on an average

Thus expected earning for a player $=\Large\frac{5n-5}{n}$