Question

Set $A$ having $n$ elements. $A$ subset $P$ of $A$ is selected at random and the set $A$ is reconstructed by replacing the elements of P. A. subset $Q$ of $A$ is again selected at random. On the basis of above information answer the following questions. The probability that $Q$ is empty, is $P\cap Q$

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

Answer 1

Answer: (C)

${\left(\frac{3}{4}\right)}^n$

$P$($Q$ is subset of $P$) $=?$
Let the subset $P$ of $A$ contains $r$ elements such that $0\le r\le n$.
Then the number of ways.of chosing $P$ is ${}^{n}C$
The subset $Q$ of $P$ can have at most $r$ elements & the number of ways of selecting $Q$ is $2^r$
Therefore the number of ways of selecting $P$ & $Q$ is ${}^n{C}_r{2}^r$
When $P$ has $r$ elements so $P$ & $Q$ can be chosen in general is eqtial to $\sum _{r=0}^n\cdot$
${}^n{C}_r{2}^r=3^n$ ways $\therefore$
Required probability $=\frac{3^n}{4^n}={\left(\frac{3}{4}\right)}^n$