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

$\frac{2}{7}$

We have,

$P(\overline{A}=\frac{7}{10}, P(\overline{B})= \alpha $ and $ P(A ∪ B)=\frac{8}{10}$

Now,

$P(A ∪ B)=\frac{8}{10}$

$⇒ 1- P(\overline{A})P(\overline{B}) =\frac{8}{10}$       [∵ A and B a re independent events]

$⇒ 1-\frac{7\alpha }{10} =\frac{8}{10}⇒ \frac{7 \alpha }{10}=\frac{2}{10} ⇒ \alpha = \frac{2}{7}$