Question

If $A$ and $B$ are two independent events such that $P\left(\overline{A}\right)=\frac{7}{10}$, $P\left(\overline{B}\right)=\alpha$ and $P(A\cup B)=\frac{8}{10}$, then $\alpha$ is

• A. $\frac{2}{7}$
• B. $\frac{5}{7}$
• C. $\frac{6}{7}$
• D. None of these

Answer 1

Answer: (A)

$\frac{2}{7}$

We know $P\left(A^{\prime }\right)=1-P\left(A\right)$

$\Rightarrow P\left(A\right)=1-\frac{7}{10}=\frac{3}{10}$

So, $P\left(A\right)=\frac{3}{10}$

and $P\left(B\right)=1-\alpha$

Since they are both independent events,

So, $P(A\cap B)=P\left(A\right).P\left(B\right)$

$P(A\cap B)=\frac{3}{10}(1-\alpha )$

Now $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow \frac{8}{10}=\frac{3}{10}+1-\alpha -\frac{3}{10}(1-\alpha )$

$\Rightarrow 7\alpha =2$

$\Rightarrow \alpha =\frac{2}{7}$