Question

Assertion :A set P contains n elements. Two boys Rakshit and Kunal independently pick up two subsets Q and R of P.

Probability of $Q\cap R=Q$ is ${\left(\frac{3}{4}\right)}^n$.

Reason: Probability of $Q\cup R=P$ is ${\left(\frac{1}{2}\right)}^n$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Assertion :

Given that set $P$ contains $2$ subset $Q$ $\xi$ $R$ choosen by two boys.

Probability of $Q\cap R=Q$ is ${\left(\frac{3}{4}\right)}^n$

$\Rightarrow P(Q\cap R)=P\left(Q\right)+P\left(R\right)-P(Q\cup R)$

$\Rightarrow P\left(Q\right)+P\left(R\right)=1$

$\Rightarrow P(Q\cup R)=\left(\frac{1}{4}\right)$

$\Rightarrow 1-{\left(\frac{1}{4}\right)}^n={\left(\frac{3}{4}\right)}^n$

The reason is given as,

Probability of $Q\cup R=P$ is ${\left(\frac{1}{2}\right)}^n$ is wrong

Hence, the answer is Assertion is correct but reason is incorrect.