Question

Assertion :Let $A$ and $B$ are two independent events.
STATEMENT-1 : If $P\left(A\right)=0.3$ and $P(A\cup \overline{B})=0.8$ then $P\left(B\right)=\frac{2}{7}$ Reason: STATEMENT-2 : $P\left(\overline{E}\right)=1-P\left(E\right)$ where $E$ is any event

• A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False
• D. Statement-1 is False, Statement-2 is True

Answer 1

The correct option is A Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1

$P(A\cup B^c)=P\left(A\right)+P\left(B^c\right)-P(A\cap B^c)=P\left(A\right)+\{1-P\left(B\right)\}-P(A\cap B^c)=P\left(A\right)+1-P(A\cup B)=1+P\left(A\right)-P\left(A\right)-P\left(B\right)+P(A\cap B)=1-P\left(B\right)+P\left(A\right)P\left(B\right)$
$\Rightarrow 0.8=1-P\left(B\right)+0.3\times P\left(B\right)\Rightarrow 0.7\times P\left(B\right)=0.2\Rightarrow P\left(B\right)=\frac{2}{7}$
Also $\overline{E}+E=1\Rightarrow P\left(E\right)+P\left(\overline{E}\right)=1\Rightarrow P\left(E\right)=1-P\left(\overline{E}\right)$
hence statement1 and statement 2 both are correct.