Question

A set $A$ is containing $n$ elements. A subset $P$ of $A$ is chosen at random. The set is reconstructed by replacing the elements of $P$. A subset $Q$ of $A$ is again chosen at random. The probability that $P$ and $Q$ have no common elements is:

• A. ${\left(\frac{5}{6}\right)}^n$
• B. ${\left(\frac{3}{4}\right)}^n$
• C. ${\left(\frac{3}{5}\right)}^n$
• D. ${\left(\frac{2}{3}\right)}^n$

Answer 1

The correct option is B ${\left(\frac{3}{4}\right)}^n$

$A$ has $n$ element and let $a$ be the subset of $A$
Now, there are $4$ possibilities,
(i) $a$ is a subset of $P$ and $Q$ both
(ii) $a$ is a subset of $P$ but not of $Q$
(iii) $a$ is a subset of $Q$ but not of $P$
(iv) $a$ is not a subset of both $P$ and $Q$
For the condition to satisfy there are $3$ favourable outcomes out of $4$ . Since there are n elements in $A$
Probability = ${\left(\frac{3}{4}\right)}^n$