# Intuitively, why does A, B independent $\iff$ A, $B^C$ independent $\iff A^C, B^C$ independent?

## Proposition 2.5.3.

If A and B are independent, then A and $B^C$ are independent, $A^C$ and B are independent, and $A^C$ and $B^C$ are independent.

Blitzstein, *Introduction to Probability* (2019 2 ed) p 64.

I'm seeking solely intuition. I'm NOT asking about how to prove these independences, because I already linked to a proof that I grasped. Proposition 2.5.3 isn't intuitive because

☣ 2.5.8. It is easy to make terrible blunders stemming from confusing independence and conditional independence. Two events can be conditionally independent given E, but not independent given $E^C$. Two events can be conditionally independent given E, but not independent. Two events can be independent, but not conditionally independent given E.

*Op. cit.* p 65.

## 1 answer

If events $A$ and $B$ are independent, then the probability that event $A$ happens is not affected by whether $B$ happens. If it isn't affected by whether $B$ happens, then it isn't affected by whether $B$ doesn't happen. (The rest of your statements can be shown to be true with the same reasoning.)

## 0 comment threads