$A$ is a set containing n elements.$A$ subset $P$ of A is chosen at random. The set $A$ is reconstructed by replacing the elements of$P. A$ subset $Q$ is again chosen at random. The probability that $P =Q$, is

The set A has $n$ elements. So, it has $2^n$ subsets. Therefore, set $P$ can be chosen in ${{ }^2}^n{C}_1$ ways. Similarly, set $Q$ can also be chosen in

${{ }^2}^n{C}_1$ ways

$\therefore$ Sets $P$ and $Q$ can be chosen in ${ }^{2^n}{C}_1{\times }^{2^n}{C}_1=2^n\times 2^n=4^n$

ways

The total number of ways of selecting $P$ and $Q$ such that $P = Q$ is

$\sum _{r=0}^n{ }^n{C}_r=2^n$

Hence, required probability $=\frac{2^n}{4^n}=\frac{1}{2^n}$