Question

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

• A. (3/4)${}^n$
• B. (3/8)${}^n$
• C. (3/4)${}^{n-1}$
• D. (3/4)${}^{n+1}$

Answer 1

Answer: (A)

(3/4)${}^n$

Sample space = number of ways in which we can form set A and number of ways in which we can form set B it is $2^n$ in both the cases
$\left({}^n{C}_0+{{}^n{C}_1+}^n{C}_2+..{.}^n{C}_n\right)$
${}^n{C}_0$ when the subset is null set
${}^n{C}_1$ when the subset contain 1 element
it goes on
when the subset contain all the element of the represented
So sample space $=2^n\times 2^n=4^n$
Now number of favorable way when P subset contain no element and Q subset contain n-1 elements
P subsets contain r elements and Q subsets contain n-r elements
....
p subset contain n element and q subset cantain no elements
${\sum }_{r=0}^n{}^n{C}_r\left(2^{n-r}\right)=3^n$
So the probability is ${\left(\frac{3}{4}\right)}^n$