Exercise 5.3.5

$\star$ Prove that in the array $P$ in procedure PERMUTE-BY-SORTING, the probability that all elements are unique is at least $1 - 1/n$.

Let $\Pr\{j\}$ be the probability that the element with index $j$ is unique. If there are $n^3$ elements, then the $\Pr\{j\} = 1 - \frac{j-1}{n^3}$.

$$ \begin{align} \Pr\{1 \cap 2 \cap 3 \cap \ldots\} &= \Pr\{1\} \cdot \Pr\{2 | 1\} \cdot \Pr\{3 | 1 \cap 2\} \cdots \\ &= 1 \bigg(1 - \frac{1}{n^3}\bigg) \bigg(1 - \frac{2}{n^3}\bigg) \bigg(1 - \frac{3}{n^3}\bigg) \cdots \\ &\ge 1 \bigg(1 - \frac{n}{n^3}\bigg) \bigg(1 - \frac{n}{n^3}\bigg) \bigg(1 - \frac{n}{n^3}\bigg) \cdots \\ &\ge \bigg(1 - \frac{1}{n^2}\bigg)^n \\ &\ge 1 - \frac{1}{n} \\ \end{align} $$

Why does the last derivation work, you ask? Well, $(1-x)^n \ge 1 - nx$.