Post History
#2: Post edited
by
DNB
·
2021-07-29T08:11:38Z (over 3 years ago)
- >### 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](https://math.stackexchange.com/a/2922539).
Blitzstein, *Introduction to Probability* (2019 2 ed), p 63.- 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.
- >### 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](https://math.stackexchange.com/a/2922539).
- 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: Initial revision
by
DNB
·
2021-07-29T08:10:47Z (over 3 years ago)
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](https://math.stackexchange.com/a/2922539). Blitzstein, *Introduction to Probability* (2019 2 ed), p 63. 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.