Probability bounds on size of smaller partition in randomized quicksort
Suppose for simplicity that all elements are distinct, and that after sorting, these elements are $x_1,\ldots,x_n$. A random pivot has the same distribution as $x_i$ for $i$ chosen uniformly at random from $\{1,\ldots,n\}$. The size of the smaller partition is $\min(i-1,n-i)$. If $I$ is the set of indices $i$ such that $\min(i-1,n-i) \geq an$, then the probability you are looking for is $|I|/n$ (by the definition of uniform probability). I'll let you complete the calculation.