If two subsets A and B are selected at random | vedclass

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Question

(A) Let $S$ be a set with $n$ elements. Each element $x \in S$ can belong to one of the following four disjoint sets: $1$. $x \in A$ and $x \in B$ $2$

Vedclass · Accepted Answer

$\frac{1}{2^n}$

Vedclass · Answer

(A) Let $S$ be a set with $n$ elements. Each element $x \in S$ can belong to one of the following four disjoint sets: $1$. $x \in A$ and $x \in B$ $2$. $x \in A$ and $x 
otin B$ $3$. $x 
otin A$ and $x \in B$ $4$. $x 
otin A$ and $x 
otin B$ Since there are $n$ elements,there are $4^n$ total ways to choose subsets $A$ and $B$. We are given the conditions $A \cap B = \phi$ and $A \cup B = S$. For each element $x \in S$,it must belong to either $A$ or $B$ (since $A \cup B = S$) but not both (since $A \cap B = \phi$). Thus,for each element,there are only $2$ choices: either $x \in A$ (which implies $x 
otin B$) or $x \in B$ (which implies $x 
otin A$). Therefore,the number of favorable outcomes is $2^n$. The probability is $\frac{2^n}{4^n} = \frac{2^n}{(2^2)^n} = \frac{2^n}{2^{2n}} = \frac{1}{2^n}$.

If two subsets $A$ and $B$ are selected at random from a set $S$ containing $n$ elements,then the probability that $A \cap B = \phi$ and $A \cup B = S$ is

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