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

Explanation

Let `A={a_1, a_2, …. A_n}`
For an element `a_i in A` and two subset P and Q there four possibilities.

numbers of ways of choosing subset `P=2^n`
As the elements have been replaced after the subset P has been chosen, choosing now a subset Q is as good as a fresh beginning.
Number of ways of choosing subset `Q=2^n`
As the number of ways of choosing P and Q are independent, we have number of ways of choosing two subset P and Q `= (2^n)(2^n)=4^n`
Now a particular element `a_i in A` will not belong to either P or Q in 3 ways. The no. of ways in which no `a_i in A` belong to both P and Q `=3xx3xx...3` to n times
`=3^n`
The required probability = `(3^n)/(4^n)=(3/4)^n`