Expected Value Problem including permutations

Question

Please help!

Let F denote some permutation that maps from A to A. Where A={1,2,...2n} is a set. For such a permutation, F, let p denote the number of indices i, belonging to A, such that $F(i) > 2i$

Suppose that the permutation F is chosen randomly, from the set of all permutations of the set F, meaning that each permutation is equally liked to be chosen.

How would I go about calculating the expected value of p. I have no idea even how to go about it.

Answer 1

The probability that $F(i)>2i$ is $E(i)=\frac{2n-2i}{2n}$.

Then the expected value of $p$ is the sum of the $E(i)$ for $1\le i \le n-1$.

This is $\frac{1+2+ ... +n-1}{n}=\frac{n-1}{2}.$