What is the requirement for the existence of independence in the complement of conditional independence

41 Views Asked by Bumbble Comm At 24 Apr 2026 - 1:46

Given two events A & B which are independent on an event C conditionally, What is the requirement for A&B to to be independent on C's complement.

I mean when,

$P((A\cap B)|C) = P(A|C).P(B|C)$

to be the below,

$P((A\cap B)|C^{c}) = P(A|C^{c}).P(B|C^{c})$

Is below the requirement,

$2.P((A\cap B)|C) = P(A\cap B) - P(A).P(B) + P(B).P(A|C) + P(A).P(B|C)$

Am I correct ?