Conditional independence problem

[bl00d1]

Help needed.
Let A1, A2 and B be events with P(B)>0. Events A1 and A2 are said to be conditionally independent given B if P(A1nA2|B)=P(A1|B)P(A2|B).

Prove or disprove the following statement:

Suppose 0<P(B)<1. If events A1 and A2 are conditionally independent then A1 and A2 are also condtionally independent of Bcomplement.

 

[Fernando Revilla]

Re: conditional independence problem

Here, you have a counterexample:

Conditional independence - Wikipedia, the free encyclopedia

 

[bl00d1]

Re: conditional independence problem

Here, you have a counterexample:

Conditional independence - Wikipedia, the free encyclopedia

where?

 

[Fernando Revilla]

Re: conditional independence problem

where?

The figure with the squares. If you prefer, consider $\Omega=\{a,b,c\}$ and the probability $p$ on $\mathcal{P}(\Omega)$ defined by $p(a)=p(b)=p(c)=1/3$, and choose $A_1=\{a\}$, $A_2=\{a,b\}$ and $B=\{c\}$. Let's see what do you obtain.

 

[bl00d1]

Re: conditional independence problem

The figure with the squares. If you prefer, consider $\Omega=\{a,b,c\}$ and the probability $p$ on $\mathcal{P}(\Omega)$ defined by $p(a)=p(b)=p(c)=1/3$, and choose $A_1=\{a\}$, $A_2=\{a,b\}$ and $B=\{c\}$. Let's see what do you obtain.

oh gosh.. it's right in front of me and i didnt see it