Let $A,B$ are mutually independent events then show that $\overline{A},B$ are also mutually independent events.

Given the sets A & B are

mutually independent.

Then $P(A\bigcap B)=P\left(A\right).P\left(B\right).$

Now, we are to prove $\overline{A}$ & B

are also mutually indendents

Now,

$P(\overline{A}\bigcap B)$

$=P\left(B\right)-P(A\bigcap B)[\because \overline{A}\bigcap B=B-A\bigcap B]$

$=P\left(B\right)-P\left(A\right).P\left(B\right)$

$=P\left(B\right)[1-P(A\left)\right]$

$=P\left(B\right)P\left(\overline{A}\right)[\because 1-P(A)=P(\overline{A}\left)\right]$

$=P\left(\overline{A}\right).P\left(B\right)$

so, $\overline{A},B$ are mutually independent

sets.