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 of P. A subset Q of A is chosen at random. The probability that P and Q have no common element is

• A. $\frac{2^n}{3^n}$
• B. $\frac{2^n}{4^n}$
• C. $\frac{3^n}{4^n}$
• D. $\frac{3^n}{5^n}$

Answer 1

The correct option is C $\frac{3^n}{4^n}$

Let $A=\{a_1,a_2,...a_n\}$
for each $a_i\in A(1\le i\le n)$
we have the following four choices
$i)a_1\in P$ and $a_1\in Q$
$ii)a_1\notin P$ and $a_1\in Q$
$iii)a_1\in P$ and $a_1\notin Q$
$iv)a_1\notin P$ and $a_1\notin Q$
Thus, the total numbers of ways of choosing $P$ and $Q$ is $4^n$.
Out of these four choices, $\left(i\right)$ is not favourable for $P\cap Q=Q$.
thus, $P(P\cap Q)={\left(\frac{3}{4}\right)}^n$