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 of A is again chosen at random. Find the probability that P and Q have no common element?

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

Answer 1

Answer: (B)

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

Let $A=\{a_1,a_2,a_3,...........,a_n\}$

For each $a_i,1\le i\le n$, there arise 4 cases

(i) $a_i\in Pand{a}_i\in Q$

(ii) $a_i\notin Pand{a}_i\in Q$

(iii) $a_i\in Pand{a}_i\notin Q$

(iv) $a_i\notin Pand{a}_i\notin Q$

Therefore, total no. of ways of choosing $P$ and $Q$ is $4^n$. Here case (i) is not favourable as $P\cap Q=\varphi$

For each element there are $3$ favourable cases and hence total no. of favourable cases is $3^n$

Hence, Probability$(P\cap Q)=\varphi$ is $\frac{3^n}{4^n}={\left(\frac{3}{4}\right)}^n$