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

$\frac{1}{2^n}$

The set A has n elements. So, it has $2^n$ subsets.

Therefore, set P can be chosen in ${^{2n}C}_1$ ways. Similarly, set Q can also be chosen in ${^{2n}C}_1$ ways.

∴ Sets P and Q can be chosen in ${^{2n}C}_1 × {^{2n}C}_1= 2^n ×2^n = 4^n$ ways.

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

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

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