Question

Let $X$ be a universal set such that $n\left(X\right)=k$, The probability of selecting two subsets $A$ and $B$ of the set $X$ such that $B=\overline{A}$ is

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

Answer 1

Answer: (B)

$\frac{1}{2^k-1}$

Numbers of selection of two subsets ${=}^{2^k}{C}_2$
Let $E=$ the event that two selected subsets $A$ and $B$ are such that $A\cap B=\varphi ,A\cup B=X.$
$\therefore n\left(E\right)=\frac{1}{2}({}^k{C}_0{+}^k{C}_1{+}^k{C}_2+...{+}^k{C}_k)=2^{k-1}$
$\therefore p\left(E\right)=\frac{2^{k-1}}{\left(\frac{1}{2}\right).2^k.(2^k-1)}=\frac{1}{2^k-1}$