$A$ 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

Solution:

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 $2^n{C}_1{\times }^{2^n}{C}_1=2^n\times 2^n=4^r$ ways

Suppose $P$ contains $r$ elements, where $r$ varies from 0 to $n$ . Then, $P$ can be chosen in ${ }^n{C}_r$ ways

For $Q$ to be disjoint from A, it should be chosen from the set of all subsets of set consisting of remaining$n - r$ elements. This can be done in

$2^{n-r}$ ways. Therefore, $P$ and $Q$ can be chosen in

${ }^n{C}_r\times 2^{n-r}$ ways

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

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

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