Question

Let X be a universal set such that n(X) = k. the probability of selecting two subset A and B such that B = A’, A’ is complement of the set A, is

• A. $\frac{2}{2^k-1}$
• B. $\frac{1}{2^k-1}$
• C. $\frac{1}{2^k}$
• D. $\frac{1}{3^k}$

Answer 1

The correct option is B $\frac{1}{2^k-1}$

Number of selection of two sets $=2^k{C}_2$
Let the event E be the selection of two subsets A and B such that $A=\overline{B}.i.e.A\bigcap B=$
$\varphi andA\bigcup B=X.Andn\left(X\right)=k$
$Thenn\left(E\right)=\frac{1}{2}(k{C}_0+k{C}_1+k{C}_2+...+k{C}_k)=\frac{1}{2}{2}^k.$
Therefore required probabiligy $=\frac{\frac{1}{2}{2}^k.2}{\frac{1}{2}{.2}^k(2^k-1)}=\frac{1}{(2^k-1)}$