A number $X$ and a sequence of numbers $\{Y_n\vert n\in \mathbb N\}$ are i.i.d draws from the uniform distribution on $[0,1]$. Let $N = \inf\{n \in \mathbb N\vert Y_n > X\}$. The player conducting these draws receives the prize of $(N - 1)$. Calculate the expected value of this prize.
I understand that $(N-1)/X= x \sim Geometric(1 - X)$ but I do not know how to proceed further.
Hint:
Observe that: $$\{N=n\}=\{Y_n>X\geq\max(Y_1,\dots,Y_{n-1})\}$$
concerning the order of $n+1$ iid and continuous random variables.
The orders for $X,Y_1,\dots,Y_n$ are equiprobable and there are $(n+1)!$ orders.
Further there are $(n-1)!$ orders that satisfy the condition $Y_n>X>\max(Y_1,\dots,Y_{n-1})$ leading to: $$P(N=n)=\frac{(n-1)!}{(n+1)!}=\frac1{n(n+1)}$$