Finding the greatest $k$ numbers with a certainty of $p$
I propose the following algorithm. You take some random element $p$ of your input as pivot element for $k=n/10$. Next, you scan through all your data and split it  into a list of elements $\le p$ and a list with elements $&gt;p$. 

Case 1: The list with the larger elements contains more then $k$ elements. Then take this list and repeat.
Case 2: The list with the larger elements contains less than $k$ elements, say $m$. The repeat with the other list, but set $k=k-m$.
Case 3: Otherwise you are done.

This is very easy to implement and should find the solution quickly. (You said you want to do the math by yourself ;) )

Remark: There is a classic $O(n)$ algorithm for this. You can determine the $k$-th biggest element in $O(n)$ by the algorithm of Blum et al. and then use this element as pivot. However, I think the the constant in the $O(n)$ makes this variant slower then the above solution. Especially if you are fine with an imprecise solution.