Question

A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of the subset P. A subset Q of A is again chosen at random. The probability that
where $|X|=$ number of elements in X.

Answer 1

Let $A=\{a_1,a_2,...,a_n\}$. For each $a_1ϵA(1\le i\le n)$ we have
the following four cases:(i) $a_1ϵP$

and $a_1ϵQ$
(ii) $a_1\text{/}ϵP$ and $a_1ϵQ$
(iii) $a_1ϵP$ and $a_1\text{/}ϵQ$
(iv) $a_1\text{/}ϵP$ and $a_1\text{/}ϵQ$ Thus, the total number of ways of choosing P and Q is $4^n$.

A) Out of these four

choices (i) is not favourable for $P\cap Q=\varphi$. Thus, probability that$P\cap Q$ is $(3/4{)}^n$.

B) $P\cap Q$ can
contain exactly one element in ${}^n{C}_1\left(3^{n-1}\right)=n\left(3^{n-1}\right)$ ways.

$\therefore$

Probability that $P\cap Q$ is singleton $=(3^{n-1}/4^n$

C) $P\cup Q=A$ in

$3^n$ ways.

$\therefore$
Probability $P\cup Q=A$ is $(3/4{)}^n$.

D) $|P|=|Q|$ in the following ways.

${(}^n{C}_0{)}^2+{(}^n{C}_1{)}^2+....+{(}^n{C}_n{)}^2{=}^{2n}{C}_n$.