Question

A is a set containing n distinct elements. A subset P of A is chosen at random. The set A reconstructed by replacing the elements of P. A subset Q of A is again chosen at random. Find the probability that P & Q have no common elements.

Answer 1

$\because$ Set A contains n elements. So, it has $2^n$ subsets.

$\therefore$ Set A can be chosen in $2^n$ ways, similarly set Q can be chosen $2^n$ ways.

$\therefore P$ and $Q$ can be chosen in $2^n{.2}^n=4^n$ ways.

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

Then, P can be chosen in ${}^n{C}_r$ ways, for Q to be disjoint from P, 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 ${}^n{C}_r{.2}^{n-r}$ ways.

But, R can vary from O to n.

$\therefore$ Total number of disjoint sets P and Q

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

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