Exercise 7.2.6

$\star$ Argue that for any constant $0 < \alpha \le 1/2$, the probability is approximately $1 - 2\alpha$ that on a random input array, PARTITION produces a split more balanced than $1 - \alpha$ to $\alpha$.

Oh, this is nice!

In order to produce a worse split than $\alpha$ to $1 - \alpha$, PARTITION must pick a pivot that will be either within the smallest $\alpha n$ elements or the largest $\alpha n$ elements. The probability of either is (approximately) $\alpha n / n = \alpha$ and the probability of both is $2\alpha$. Thus, the probability of having a better partition is the complement, $1 - 2\alpha$.