• # question_answer 3) If A and B are two independent events such that $\mathbf{P}\left( A \right)=\frac{3}{10},\mathbf{P}\left( \mathbf{B}' \right)=\mathbf{x}$and $\mathbf{P}\left( \mathbf{A}\cup \mathbf{B} \right)=\frac{8}{10}$, then x is equal to A) $\frac{2}{7}$                          B) $\frac{3}{7}$               C) $\frac{4}{7}$                          D) $\frac{5}{7}$

[a]   $P\left( A\cup B \right)=\frac{8}{10}$ $P\left( B' \right)=x\Rightarrow P\left( B \right)=1-x$ $P(A)=\frac{3}{10}$ Now $A\And B$ be independent. $P\left( A\cap B \right)=P\left( A \right).P\left( B \right)=\left( 1-x \right).\frac{3}{10}$ $\because P\left( A\cup B \right)=P\left( A \right)+P\left( B \right)-P\left( A\cap B \right)$ $\frac{8}{10}=\frac{3}{10}+(1-x)-\frac{-3(1-x)}{10}$ $\Rightarrow \frac{8}{10}=\frac{3+10\,(1-x)-3\,(1-x)}{10}\,\Rightarrow 8=3+7(1-x)$ $\Rightarrow 8=3+7-7x\Rightarrow 7x=10-8=2\Rightarrow x=\frac{2}{7}.$Hence, Option [a]is correct.