Question

Two subsets A and B of a set ‘S’ consisting of ‘n’ elements are constructed randomly. The Probability that $A⊑B$ is equal to:

• A. $(\frac{3}{4}{)}^n$
• B. $\frac{2^n}{n!}$
• C. $\frac{1}{2^n}$
• D. $(\frac{2}{3}{)}^n$

Answer 1

The correct option is A $(\frac{3}{4}{)}^n$

For element x of set S, there are four possibilities:
$\left(i\right)x\in A,x\in B\left(ii\right)x\text{/}\in A,x\in B\left(iii\right)x\text{/}\in A,x\text{/}\in B\left(iv\right)x\in A,x\text{/}\in B$
Out of these four possibilities, only one, the (iv) possibility does not favour the result $A⊑B$ while (i), (ii) and (iii) possibilities favour the result $A⊑B$.
Hence, required probability = $(\frac{3}{4}{)}^n$