Question

For two events if $A$ and $B$ are independent then prove that $A^{\prime },B^{\prime }$ are also independent.

Answer 1

Given $A,B$ are independent then $P(A\cap B)=P\left(A\right)P\left(B\right)$.......(1).

Now

$P(A^{\prime }\cap B^{\prime })=P\left\{\right(A\cup B{)}^{\prime }\}$ [ Using De Morgan's Law]

or, $P(A^{\prime }\cap B^{\prime })=1-P(A\cup B)$

or, $P(A^{\prime }\cap B^{\prime })=1-\left\{P\right(A)+P(B)-P(A\cap B\left)\right\}$

or, $P(A^{\prime }\cap B^{\prime })=1-\left\{P\right(A)+P(B)-P(A\left)P\right(B\left)\right\}$ [ Using (1)]

or, $P(A^{\prime }\cap B^{\prime })=1-P\left(A\right)-P\left(B\right)+P\left(A\right)P\left(B\right)$

or, $P(A^{\prime }\cap B^{\prime })=(1-P(A\left)\right)(1-P(B\left)\right)$

or, $P(A^{\prime }\cap B^{\prime })=P\left(A^{\prime }\right)P\left(B^{\prime }\right)$

This given $A^{\prime },B^{\prime }$ are independent.