Question

If A and B are two independent events, prove that A' and B are also independent.

Answer 1

$P(A^{\prime }\cap B)=P(B-A)=P\left(B\right)-P(A\cap B)=P\left(B\right)-P\left(A\right)P\left(B\right)$
$[\because P(A\cap B)=P(A\left)P\right(B\left)\right]$, if A and B are independent events.
$\Rightarrow P(A^{\prime }\cap B)=P\left(B\right)[1-P(A\left)\right]\therefore P(A^{\prime }\cap B)=P\left(A^{\prime }\right)P\left(B\right)$
Hence, A' and B are also independent if A and B are independent events.