Randomized Selection
Suppose your array has $n$ elements.  As you have noted, the median is always in the bigger part after the first partition.  The bigger part has size at most $\alpha n$ if the smaller part has size at least $(1-\alpha) n$.  This happens when you pick a pivot that isn't one of the smallest or largest $(1-\alpha) n$ elements.  Because $\alpha &gt; 1/2$, you know these are disjoint sets, so the probability of hitting one of the bad pivots is just $2 - 2\alpha$, and $1- 2 + 2\alpha=2\alpha - 1$.