If A and B are independent events such that $P(B) =\frac{2}{7}, P( A ∪ \overline{B}) = 0.8 $, then P(A) =

0.3

We have,

$P( A ∪ \overline{B}) = 0.8$ and $P(B) =\frac{2}{7}$

$⇒ P(A) + P(\overline{B}) - P(A ∪ \overline{B})= 0.8 $ and $ P(\overline{B})=\frac{5}{7}$

$⇒ P(A) + P(\overline{B}) - P(A)P(A ∪ \overline{B})= 0.8 $ and $ P(\overline{B})=\frac{5}{7}$

$⇒ P(A)+\frac{5}{7}-\frac{5}{7}P(A)= 0.8⇒\frac{2}{7}P(A)=\frac{3}{35}⇒P(A)= 0.3$