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 and Q are disjoint sets, is

$\left(\frac{3}{4}\right)^n$

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.

Suppose P contains r elements, where r varies from 0 to n.

Then, P can be chosen in ${^nC}_r$, ways.

For Q to be disjoint from A, it should be chosen from the set of all subsets of set consisting remaining n-r elements. This can be done in $2^{n-r}$ ways. Therefore, P and Q can be chosen in ${^nC}_r × 2^{n-r}$ ways.

But, r can vary from 0 to n. Therefore, the total number of ways of selecting P and Q such that they are disjoint is

$\sum\limits^{n}_{r=0}{^nC}_r 2^{n-r}= (1+2)^n = 3^n$

Hence, required probability $=\frac{3^n}{4^n}=\left(\frac{3}{4}\right)^n$