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A is a set having n elements. A subset P...

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.

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Let A en replaced after the subset P has been chosen, choosing now a subset Q ia as good as a fresh beginning.
No. of ways of choosing subset `Q=2^n`
As the no. 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` belongs to both P and Q `=3xx3xx.....3` to n times
`=3^n`
The required probability `=(3^n)/(4^n)=(3/4)^n`
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